HP (Hewlett Packard) 50g User Manual

HPꢀꢁꢂg graphing calculator

user’s guide

HP part number F2229AA-90006

Edition 1

Preface

You have in your hands a compact symbolic and numerical computer that will

facilitate calculation and mathematical analysis of problems in a variety of

disciplines, from elementary mathematics to advanced engineering and science

subjects. Although referred to as a calculator, because of its compact format

resembling typical hand-held calculating devices, the HP 50g should be

thought of as a graphics/programmable hand-held computer.

The HP 50g can be operated in two different calculating modes, the Reverse

Polish Notation (RPN) mode and the Algebraic (ALG) mode (see page 1-13 for

additional details). The RPN mode was incorporated into calculators to make

calculations more efficient. In this mode, the operands in an operation (e.g., ‘2’

and ‘3’ in the operation ‘2+3’) are entered into the calculator screen, referred

to as the stack, and then the operator (e.g., ‘+’ in the operation ‘2+3’) is

entered to complete the operation. The ALG mode, on the other hand, mimics

the way you type arithmetic expressions in paper. Thus, the operation ‘2+3’, in

ALG mode, will be entered in the calculator by pressing the keys ‘2’, ‘+’, and

‘3’, in that order. To complete the operation we use the ENTER key. Examples

of applications of the different functions and operations in this calculator are

illustrated in this user’s guide in both modes.

This guide contains examples that illustrate the use of the basic calculator

functions and operations. The chapters are organized by subject in order of

difficulty. Starting with the setting of calculator modes and display options, and

continuing with real and complex number calculations, operations with lists,

vectors, and matrices, detailed examples of graph applications, use of strings,

basic programming, graphics programming, string manipulation, advanced

calculus and multivariate calculus applications, advanced differential equations

applications (including Laplace transform, and Fourier series and transforms),

and probability and statistic applications.

For symbolic operations the calculator includes a powerful Computer Algebraic

System (CAS) that lets you select different modes of operation, e.g., complex

numbers vs. real numbers, or exact (symbolic) vs. approximate (numerical)

mode.

The display can be adjusted to provide textbook-type expressions,

which can be useful when working with matrices, vectors, fractions,

summations, derivatives, and integrals. The high-speed graphics of the

calculator produce complex figures in very little time.

Thanks to the infrared port, the RS232 port, and the USB port and cable

provided with your calculator, you can connect your calculator with other

calculators or computers. This allows for fast and efficient exchange of

programs and data with other calculators or computers. The calculator

provides a flash memory card port to facilitate storage and exchange of data

with other users.

The programming capabilities of the calculator allow you or other users to

develop efficient applications for specific purposes. Whether it is advanced

mathematical applications, specific problem solution, or data logging, the

programming languages available in your calculator make it into a very

versatile computing device.

We hope your calculator will become a faithful companion for your school and

professional applications.

Table of contents

Chapter 1 - Getting started ,1-1

Basic Operations ,1-1

Batteries ,1-1

Turning the calculator on and off ,1-2

Adjusting the display contrast ,1-2

Contents of the calculator’s display ,1-2

Menus ,1-3

SOFT menus vs. CHOOSE boxes ,1-4

Selecting SOFT menus or CHOOSE boxes ,1-5

The TOOL menu ,1-7

Setting time and date ,1-7

Introducing the calculator’s keyboard ,1-11

Selecting calculator modes ,1-12

Operating Mode ,1-13

Number Format and decimal dot or comma ,1-17

Angle Measure ,1-23

Coordinate System ,1-24

Beep, Key Click, and Last Stack ,1-25

Selecting CAS settings ,1-26

Selecting Display modes ,1-27

Selecting the display font ,1-27

Selecting properties of the line editor ,1-28

Selecting properties of the Stack ,1-28

Selecting properties of the equation writer (EQW) ,1-29

Selecting the size of the header ,1-30

Selecting the clock display ,1-30

Page TOC-1

Chapter 2 - Introducing the calculator ,2-1

Calculator objects ,2-1

Editing expressions on the screen ,2-3

Creating arithmetic expressions ,2-3

Editing arithmetic expressions ,2-6

Creating algebraic expressions ,2-7

Editing algebraic expressions ,2-8

Using the Equation Writer (EQW) to create expressions ,2-10

Creating arithmetic expressions ,2-11

Editing arithmetic expressions ,2-17

Creating algebraic expressions ,2-19

Editing algebraic expressions ,2-21

Creating and editing summations, derivatives, and integrals ,2-29

Organizing data in the calculator ,2-33

Functions for manipulation of variables ,2-34

The HOME directory ,2-35

The CASDIR sub-directory ,2-35

Typing directory and variable names ,2-37

Creating subdirectories ,2-39

Moving among subdirectories ,2-43

Deleting subdirectories ,2-43

Variables ,2-47

Creating variables ,2-47

Checking variables contents ,2-52

Replacing the contents of variables ,2-55

Copying variables ,2-56

Reordering variables in a directory ,2-59

Moving variables using the FILES menu ,2-60

Deleting variables ,2-61

UNDO and CMD functions ,2-62

Flags ,2-64

Example of flag setting: general solutions vs. principal value ,2-65

Page TOC-2

Other flags of interest ,2-66

CHOOSE boxes vs. Soft MENU ,2-67

Selected CHOOSE boxes ,2-69

Chapter 3 - Calculation with real numbers ,3-1

Checking calculators settings ,3-1

Checking calculator mode ,3-2

Real number calculations ,3-2

Changing sign of a number, variable, or expression ,3-3

The inverse function ,3-3

Addition, subtraction, multiplication, division ,3-3

Using parentheses ,3-4

Absolute value function ,3-4

Squares and square roots ,3-5

Powers and roots ,3-5

Base-10 logarithms and powers of 10 ,3-5

Using powers of 10 in entering data ,3-6

Natural logarithms and exponential function ,3-6

Trigonometric functions ,3-6

Inverse trigonometric functions ,3-6

Differences between functions and operators ,3-7

Real number functions in the MTH menu ,3-7

Hyperbolic functions and their inverses ,3-9

Real number functions ,3-11

Special functions ,3-14

Calculator constants ,3-16

Operations with units ,3-17

The UNITS menu ,3-17

Available units ,3-19

Converting to base units ,3-22

Attaching units to numbers ,3-23

Operations with units ,3-25

Units manipulation tools ,3-27

Page TOC-3

Physical constants in the calculator ,3-29

Special physical functions ,3-32

Function ZFACTOR ,3-32

Function F0λ ,3-33

Function SIDENS ,3-33

Function TDELTA ,3-33

Function TINC ,3-34

Defining and using functions ,3-34

Functions defined by more than one expression ,3-36

The IFTE function ,3-36

Combined IFTE functions ,3-37

Chapter 4 - Calculations with complex numbers ,4-1

Definitions ,4-1

Setting the calculator to COMPLEX mode ,4-1

Entering complex numbers ,4-2

Polar representation of a complex number ,4-3

Simple operations with complex numbers ,4-4

Changing sign of a complex number ,4-5

Entering the unit imaginary number ,4-5

The CMPLX menus ,4-5

CMPLX menu through the MTH menu ,4-6

CMPLX menu in keyboard ,4-7

Functions applied to complex numbers ,4-8

Functions from the MTH menu ,4-9

Function DROITE: equation of a straight line ,4-9

Chapter 5 - Algebraic and arithmetic operations ,5-1

Entering algebraic objects ,5-1

Simple operations with algebraic objects ,5-1

Functions in the ALG menu ,5-3

COLLECT ,5-4

EXPAND ,5-4

Page TOC-4

FACTOR ,5-5

LNCOLLECT ,5-5

LIN ,5-5

PARTFRAC ,5-5

SOLVE ,5-5

SUBST ,5-5

TEXPAND ,5-5

Other forms of substitution in algebraic expressions ,5-6

Operations with transcendental functions ,5-7

Expansion and factoring using log-exp functions ,5-7

Expansion and factoring using trigonometric functions ,5-8

Functions in the ARITHMETIC menu ,5-9

DIVISꢀ,5-9

FACTORS ,5-9

LGCDꢀ,5-10

PROPFRAC ,5-10

SIMP2 ,5-10

INTEGER menu ,5-10

POLYNOMIAL menu ,5-10

MODULO menu ,5-11

Applications of the ARITHMETIC menu ,5-12

Modular arithmetic ,5-12

Finite arithmetic rings in the calculator ,5-14

Polynomials ,5-17

Modular arithmetic with polynomials ,5-17

The CHINREM function ,5-17

The EGCD function ,5-18

The GCD function ,5-18

The HERMITE function ,5-18

The HORNER function ,5-19

The variable VX ,5-19

The LAGRANGE function ,5-19

The LCM function ,5-20

The LEGENDRE function ,5-20

The PCOEF function ,5-21

Page TOC-5

The PROOT function ,5-21

The PTAYL function ,5-21

The QUOT and REMAINDER functions ,5-21

The EPSX0 function and the CAS variable EPS ,5-22

The PEVAL function ,5-22

The TCHEBYCHEFF function ,5-22

Fractions ,5-23

The SIMP2 function ,5-23

The PROPFRAC function ,5-23

The PARTFRAC function ,5-23

The FCOEF function ,5-24

The FROOTS function ,5-24

Step-by-step operations with polynomials and fractions ,5-25

The CONVERT Menu and algebraic operations ,5-26

UNITS convert menu (Option 1) ,5-26

BASE convert menu (Option 2) ,5-27

TRIGONOMETRIC convert menu (Option 3) ,5-27

MATRICES convert menu (Option 5) ,5-27

REWRITE convert menu (Option 4) ,5-27

Chapter 6 - Solution to single equations ,6-1

Symbolic solution of algebraic equations ,6-1

Function ISOL ,6-1

Function SOLVE ,6-2

Function SOLVEVX ,6-3

Function ZEROS ,6-4

Numerical solver menu ,6-5

Polynomial Equations ,6-6

Financial calculations ,6-9

Solving equations with one unknown through NUM.SLV ,6-13

The SOLVE soft menu ,6-26

The ROOT sub-menu ,6-26

Function ROOT ,6-26

Page TOC-6

Variable EQ ,6-26

The SOLVR sub-menu ,6-26

The DIFFE sub-menu ,6-29

The POLY sub-menu ,6-29

The SYS sub-menu ,6-30

The TVM sub-menu ,6-30

Chapter 7 - Solving multiple equations ,7-1

Rational equation systems ,7-1

Example 1 – Projectile motion ,7-1

Example 2 – Stresses in a thick wall cylinder ,7-2

Example 3 - System of polynomial equations ,7-4

Solution to simultaneous equations with MSLV ,7-4

Example 1 - Example from the help facility ,7-5

Example 2 - Entrance from a lake into an open channel ,7-5

Using the Multiple Equation Solver (MES) ,7-9

Application 1 - Solution of triangles ,7-9

Application 2 - Velocity and acceleration in polar coordinates ,7-17

Chapter 8 - Operations with lists ,8-1

Definitions ,8-1

Creating and storing lists ,8-1

Composing and decomposing lists ,8-2

Operations with lists of numbers ,8-2

Changing sign ,8-3

Addition, subtraction, multiplication, division ,8-3

Real number functions from the keyboard ,8-4

Real number functions from the MTH menu ,8-5

Examples of functions that use two arguments ,8-6

Lists of complex numbers ,8-7

Lists of algebraic objects ,8-8

The MTH/LIST menu ,8-8

Manipulating elements of a list ,8-10

Page TOC-7

List size ,8-10

Extracting and inserting elements in a list ,8-10

Element position in the list ,8-11

HEAD and TAIL functions ,8-11

The SEQ function ,8-11

The MAP function ,8-12

Defining functions that use lists ,8-13

Applications of lists ,8-15

Harmonic mean of a list ,8-15

Geometric mean of a list ,8-16

Weighted average ,8-17

Statistics of grouped data ,8-18

Chapter 9 - Vectors ,9-1

Definitions ,9-1

Entering vectors ,9-2

Typing vectors in the stack ,9-2

Storing vectors into variables ,9-3

Using the Matrix Writer (MTRW) to enter vectors ,9-3

Building a vector with ꢀARRY ,9-6

Identifying, extracting, and inserting vector elements ,9-7

Simple operations with vectors ,9-9

Changing sign ,9-9

Addition, subtraction ,9-9

Multiplication by a scalar, and division by a scalar ,9-9

Absolute value function ,9-10

The MTH/VECTOR menu ,9-10

Magnitude ,9-10

Dot product ,9-11

Cross product ,9-11

Decomposing a vector ,9-11

Building a two-dimensional vector ,9-12

Building a three-dimensional vector ,9-12

Page TOC-8

Changing coordinate system ,9-12

Application of vector operations ,9-15

Resultant of forces ,9-15

Angle between vectors ,9-15

Moment of a force ,9-16

Equation of a plane in space ,9-17

Row vectors, column vectors, and lists ,9-18

Function OBJꢀ ,9-19

Function ꢀLIST ,9-20

Function DROP ,9-20

Transforming a row vector into a column vector ,9-20

Transforming a column vector into a row vector ,9-21

Transforming a list into a vector ,9-23

Transforming a vector (or matrix) into a list ,9-24

Chapter 10ꢀ- Creating and manipulating matrices ,10-1

Definitions ,10-1

Entering matrices in the stack ,10-2

Using the Matrix Writer ,10-2

Typing in the matrix directly into the stack ,10-3

Creating matrices with calculator functions ,10-3

Functions GET and PUT ,10-6

Functions GETI and PUTI ,10-6

Function SIZE ,10-7

Function TRN ,10-7

Function CON ,10-8

Function IDN ,10-9

Function RDM ,10-9

Function RANM ,10-11

Function SUB ,10-11

Function REPL ,10-12

Function ꢀDIAG ,10-12

Function DIAGꢀ ,10-13

Page TOC-9

Function VANDERMONDE ,10-13

Function HILBERT ,10-14

A program to build a matrix out of a number of lists ,10-14

Lists represent columns of the matrix ,10-15

Lists represent rows of the matrix ,10-17

Manipulating matrices by columns ,10-17

Function ꢀCOL ,10-18

Function COLꢀ ,10-19

Function COL+ ,10-19

Function COL- ,10-20

Function CSWP ,10-20

Manipulating matrices by rows ,10-21

Function ꢀROW ,10-22

Function ROWꢀ ,10-23

Function ROW+ ,10-23

Function ROW- ,10-24

Function RSWP ,10-24

Function RCI ,10-25

Function RCIJ ,10-25

Chapter 11 - Matrix Operations and Linear Algebra ,11-1

Operations with matrices ,11-1

Addition and subtraction ,11-2

Multiplication ,11-2

Characterizing a matrix (The matrix NORM menu) ,11-7

Function ABS ,11-8

Function SNRM ,11-8

Functions RNRM and CNRM ,11-9

Function SRAD ,11-10

Function COND ,11-10

Function RANK ,11-11

Function DET ,11-12

Function TRACE ,11-14

Page TOC-10

Function TRAN ,11-15

Additional matrix operations (The matrix OPER menu) ,11-15

Function AXL ,11-16

Function AXM ,11-16

Function LCXM ,11-16

Solution of linear systems ,11-17

Using the numerical solver for linear systems ,11-18

Least-square solution (function LSQ) ,11-24

Solution with the inverse matrix ,11-27

Solution by “division” of matrices ,11-27

Solving multiple set of equations with the same coefficient matrix ,11-28

Gaussian and Gauss-Jordan elimination ,11-29

Step-by-step calculator procedure for solving linear systems ,11-38

Solution to linear systems using calculator functions ,11-41

Residual errors in linear system solutions (Function RSD) ,11-44

Eigenvalues and eigenvectors ,11-45

Function PCAR ,11-45

Function EGVL ,11-46

Function EGV ,11-46

Function JORDAN ,11-47

Function MAD ,11-48

Matrix factorization ,11-49

Function LU ,11-50

Orthogonal matrices and singular value decomposition ,11-50

Function SVD ,11-50

Function SVL ,11-51

Function SCHUR ,11-51

Function LQ ,11-51

Function QR ,11-52

Matrix Quadratic Forms ,11-52

The QUADF menu ,11-52

Function AXQ ,11-53

Page TOC-11

Function QXA ,11-53

Function SYLVESTER ,11-54

Function GAUSS ,11-54

Linear Applications ,11-54

Function IMAGE ,11-55

Function ISOM ,11-55

Function KER ,11-56

Function MKISOM ,11-56

Chapter 12 - Graphics ,12-1

Graphs options in the calculator ,12-1

Plotting an expression of the form y = f(x) ,12-2

Some useful PLOT operations for FUNCTION plots ,12-5

Saving a graph for future use ,12-7

Graphics of transcendental functions ,12-8

Graph of ln(X) ,12-8

Graph of the exponential function ,12-10

The PPAR variable ,12-11

Inverse functions and their graphs ,12-11

Summary of FUNCTION plot operation ,12-13

Plots of trigonometric and hyperbolic functions ,12-16

Generating a table of values for a function ,12-17

The TPAR variable ,12-17

Plots in polar coordinates ,12-18

Plotting conic curves ,12-20

Parametric plots ,12-22

Generating a table for parametric equations ,12-25

Plotting the solution to simple differential equations ,12-26

Truth plots ,12-28

Plotting histograms, bar plots, and scatter plots ,12-29

Bar plots ,12-29

Scatter plots ,12-31

Slope fields ,12-33

Page TOC-12

Fast 3D plots ,12-34

Wireframe plots ,12-36

Ps-Contour plots ,12-38

Y-Slice plots ,12-39

Gridmap plots ,12-40

Pr-Surface plots ,12-41

The VPAR variable ,12-42

Interactive drawing ,12-43

DOT+ and DOT- ,12-44

MARK ,12-44

LINE ,12-44

TLINE ,12-45

BOX ,12-45

CIRCL ,12-45

LABEL ,12-45

DEL ,12-46

ERASE ,12-46

MENU ,12-46

SUB ,12-46

REPL ,12-46

PICTꢀ ,12-46

X,Yꢀ ,12-47

Zooming in and out in the graphics display ,12-47

ZFACT, ZIN, ZOUT, and ZLAST ,12-47

BOXZ ,12-48

ZDFLT, ZAUTO ,12-48

HZIN, HZOUT, VZIN and VZOUT ,12-48

CNTR ,12-48

ZDECI ,12-48

ZINTG ,12-48

ZSQR ,12-49

ZTRIG ,12-49

Page TOC-13

The SYMBOLIC menu and graphs ,12-49

The SYMB/GRAPH menu ,12-50

Function DRAW3DMATRIX ,12-52

Chapter 13 - Calculus Applications ,13-1

The CALC (Calculus) menu ,13-1

Limits and derivatives ,13-1

Function lim ,13-2

Derivatives ,13-3

Functions DERIV and DERVX ,13-3

The DERIV&INTEG menu ,13-4

Calculating derivatives with ∂ ,13-4

The chain rule ,13-6

Derivatives of equations ,13-7

Implicit derivatives ,13-7

Application of derivatives ,13-7

Analyzing graphics of functions ,13-8

Function DOMAIN ,13-9

Function TABVAL ,13-9

Function SIGNTAB ,13-10

Function TABVAR ,13-10

Using derivatives to calculate extreme points ,13-12

Higher order derivatives ,13-13

Anti-derivatives and integrals ,13-14

Functions INT, INTVX, RISCH, SIGMA and SIGMAVX ,13-14

Definite integrals ,13-15

Step-by-step evaluation of derivatives and integrals ,13-16

Integrating an equation ,13-17

Techniques of integration ,13-18

Substitution or change of variables ,13-18

Integration by parts and differentials ,13-19

Integration by partial fractions ,13-20

Improper integrals ,13-20

Page TOC-14

Integration with units ,13-21

Infinite series ,13-22

Taylor and Maclaurin’s series ,13-23

Taylor polynomial and reminder ,13-23

Functions TAYLR, TAYLR0, and SERIES ,13-24

Chapter 14 - Multi-variate Calculus Applications ,14-1

Multi-variate functions ,14-1

Partial derivatives ,14-1

Higher-order derivatives ,14-3

The chain rule for partial derivatives ,14-4

Total differential of a function z = z(x,y) ,14-5

Determining extrema in functions of two variables ,14-5

Using function HESS to analyze extrema ,14-6

Multiple integrals ,14-8

Jacobian of coordinate transformation ,14-9

Double integral in polar coordinates ,14-9

Chapter 15 - Vector Analysis Applications ,15-1

Definitions ,15-1

Gradient and directional derivative ,15-1

A program to calculate the gradient ,15-2

Using function HESS to obtain the gradient ,15-2

Potential of a gradient ,15-3

Divergence ,15-4

Laplacian ,15-4

Curl ,15-5

Irrotational fields and potential function ,15-5

Vector potential ,15-6

Chapter 16 - Differential Equations ,16-1

Basic operations with differential equations ,16-1

Entering differential equations ,16-1

Page TOC-15

Checking solutions in the calculator ,16-2

Slope field visualization of solutions ,16-3

The CALC/DIFF menu ,16-3

Solution to linear and non-linear equations ,16-4

Function LDEC ,16-4

Function DESOLVE ,16-7

The variable ODETYPE ,16-8

Laplace Transforms ,16-10

Definitions ,16-10

Laplace transform and inverses in the calculator ,16-11

Laplace transform theorems ,16-12

Dirac’s delta function and Heaviside’s step function ,16-15

Applications of Laplace transform in the solution of linear ODEs ,16-17

Fourier series ,16-26

Function FOURIER ,16-28

Fourier series for a quadratic function ,16-28

Fourier series for a triangular wave ,16-34

Fourier series for a square wave ,16-38

Fourier series applications in differential equations ,16-40

Fourier Transforms ,16-42

Definition of Fourier transforms ,16-45

Properties of the Fourier transform ,16-47

Fast Fourier Transform (FFT) ,16-47

Examples of FFT applications ,16-48

Solution to specific second-order differential equations ,16-51

The Cauchy or Euler equation ,16-51

Legendre’s equation ,16-51

Bessel’s equation ,16-52

Chebyshev or Tchebycheff polynomials ,16-55

Laguerre’s equation ,16-56

Weber’s equation and Hermite polynomials ,16-57

Numerical and graphical solutions to ODEs ,16-57

Page TOC-16

Numerical solution of first-order ODE ,16-57

Graphical solution of first-order ODE ,16-59

Numerical solution of second-order ODE ,16-61

Graphical solution for a second-order ODE ,16-63

Numerical solution for stiff first-order ODE ,16-65

Numerical solution to ODEs with the SOLVE/DIFF menu ,16-67

Function RKF ,16-67

Function RRK ,16-68

Function RKFSTEP ,16-69

Function RRKSTEP ,16-70

Function RKFERR ,16-71

Function RSBERR ,16-71

Chapter 17 - Probability Applications ,17-1

The MTH/PROBABILITY.. sub-menu - part 1 ,17-1

Factorials, combinations, and permutations ,17-1

Random numbers ,17-2

Discrete probability distributions ,17-3

Binomial distribution ,17-4

Poisson distribution ,17-5

Continuous probability distributions ,17-6

The gamma distribution ,17-6

The exponential distribution ,17-6

The beta distribution ,17-7

The Weibull distribution ,17-7

Functions for continuous distributions ,17-7

Continuous distributions for statistical inference ,17-9

Normal distribution pdf ,17-9

Normal distribution cdf ,17-10

The Student-t distribution ,17-10

The Chi-square distribution ,17-11

The F distribution ,17-12

Inverse cumulative distribution functions ,17-13

Page TOC-17

Chapter 18 - Statistical Applications ,18-1

Pre-programmed statistical features ,18-1

Entering data ,18-1

Calculating single-variable statistics ,18-2

Obtaining frequency distributions ,18-5

Fitting data to a function y = f(x) ,18-10

Obtaining additional summary statistics ,18-13

Calculation of percentiles ,18-14

The STAT soft menu ,18-15

The DATA sub-menu ,18-16

The ΣPAR sub-menu ,18-16

The 1VAR sub menu ,18-17

The PLOT sub-menu ,18-17

The FIT sub-menu ,18-18

The SUMS sub-menu ,18-18

Example of STAT menu operations ,18-19

Confidence intervals ,18-22

Estimation of Confidence Intervals ,18-23

Definitions ,18-23

Confidence intervals for the population mean when the population vari-

ance is known ,18-24

Confidence intervals for the population mean when the population vari-

ance is unknown ,18-24

Confidence interval for a proportion ,18-25

Sampling distribution of differences and sums of statistics ,18-25

Confidence intervals for sums and differences of mean values ,18-26

Determining confidence intervals ,18-27

Confidence intervals for the variance ,18-33

Hypothesis testing ,18-35

Procedure for testing hypotheses ,18-35

Errors in hypothesis testing ,18-36

Inferences concerning one mean ,18-37

Inferences concerning two means ,18-39

Page TOC-18

Paired sample tests ,18-41

Inferences concerning one proportion ,18-41

Testing the difference between two proportions ,18-42

Hypothesis testing using pre-programmed features ,18-43

Inferences concerning one variance ,18-47

Inferences concerning two variances ,18-48

Additional notes on linear regression ,18-50

The method of least squares ,18-50

Additional equations for linear regression ,18-51

Prediction error ,18-52

Confidence intervals and hypothesis testing in linear regression ,18-52

Procedure for inference statistics for linear regression using the calcula-

tor ,18-54

Multiple linear fitting ,18-57

Polynomial fitting ,18-59

Selecting the best fitting ,18-62

Chapter 19 - Numbers in Different Bases ,19-1

Definitions ,19-1

The BASE menu ,19-1

Functions HEX, DEC, OCT, and BIN ,19-2

Conversion between number systems ,19-3

Wordsize ,19-4

Operations with binary integers ,19-4

The LOGIC menu ,19-5

The BIT menu ,19-6

The BYTE menu ,19-7

Hexadecimal numbers for pixel references ,19-7

Chapter 20 - Customizing menus and keyboard ,20-1

Customizing menus ,20-1

The PRG/MODES/MENU menu ,20-1

Menu numbers (RCLMENU and MENU functions) ,20-2

Page TOC-19

Custom menus (MENU and TMENU functions) ,20-2

Menu specification and CST variable ,20-4

Customizing the keyboard ,20-5

The PRG/MODES/KEYS sub-menu ,20-5

Recall current user-defined key list ,20-6

Assign an object to a user-defined key ,20-6

Operating user-defined keys ,20-7

Un-assigning a user-defined key ,20-7

Assigning multiple user-defined keys ,20-7

Chapter 21 - Programming in User RPL language ,21-1

An example of programming ,21-1

Global and local variables and subprograms ,21-2

Global Variable Scope ,21-4

Local Variable Scope ,21-5

The PRG menu ,21-5

Navigating through RPN sub-menus ,21-6

Functions listed by sub-menu ,21-7

Shortcuts in the PRG menu ,21-9

Keystroke sequence for commonly used commands ,21-10

Programs for generating lists of numbers ,21-13

Examples of sequential programming ,21-15

Programs generated by defining a function ,21-15

Programs that simulate a sequence of stack operations ,21-17

Interactive input in programs ,21-19

Prompt with an input string ,21-21

A function with an input string ,21-22

Input string for two or three input values ,21-24

Input through input forms ,21-27

Creating a choose box ,21-31

Identifying output in programs ,21-33

Tagging a numerical result ,21-33

Decomposing a tagged numerical result into a number and a tag ,21-33

Page TOC-20

“De-tagging” a tagged quantity ,21-33

Examples of tagged output ,21-34

Using a message box ,21-37

Relational and logical operators ,21-43

Relational operators ,21-43

Logical operators ,21-45

Program branching ,21-46

Branching with IF ,21-47

The IF…THEN…END construct ,21-47

The CASE construct ,21-51

Program loops ,21-53

The START construct ,21-53

The FOR construct ,21-59

The DO construct ,21-61

The WHILE construct ,21-63

Errors and error trapping ,21-64

DOERR ,21-64

ERRN ,21-65

ERRM ,21-65

ERR0 ,21-65

LASTARG ,21-65

Sub-menu IFERR ,21-65

User RPL programming in algebraic mode ,21-67

Chapter 22 - Programs for graphics manipulation ,22-1

The PLOT menu ,22-1

User-defined key for the PLOT menu ,22-1

Description of the PLOT menu ,22-2

Generating plots with programs ,22-14

Two-dimensional graphics ,22-14

Three-dimensional graphics ,22-15

The variable EQ ,22-15

Examples of interactive plots using the PLOT menu ,22-15

Page TOC-21

Examples of program-generated plots ,22-17

Drawing commands for use in programming ,22-19

PICT ,22-20

PDIM ,22-20

LINE ,22-20

TLINE ,22-20

BOX ,22-21

ARC ,22-21

PIX?, PIXON, and PIXOFF ,22-21

PVIEW ,22-22

PXꢀC ,22-22

CꢀPX ,22-22

Programming examples using drawing functions ,22-22

Pixel coordinates ,22-25

Animating graphics ,22-26

Animating a collection of graphics ,22-27

More information on the ANIMATE function ,22-29

Graphic objects (GROBs) ,22-29

The GROB menu ,22-31

A program with plotting and drawing functions ,22-33

Modular programming ,22-35

Running the program ,22-36

A program to calculate principal stresses ,22-38

Ordering the variables in the sub-directory ,22-38

A second example of Mohr’s circle calculations ,22-39

An input form for the Mohr’s circle program ,22-40

Chapter 23 - Charactor strings ,23-1

String-related functions in the TYPE sub-menu ,23-1

String concatenation ,23-2

The CHARS menu ,23-2

The characters list ,23-3

Page TOC-22

Chapter 24 - Calculator objects and flags ,24-1

Description of calculator objects ,24-1

Function TYPE ,24-2

Function VTYPE ,24-2

Calculator flags ,24-3

System flags ,24-3

Functions for setting and changing flags ,24-3

User flags ,24-4

Chapter 25 - Date and Time Functions ,25-1

The TIME menu ,25-1

Setting an alarm ,25-1

Browsing alarms ,25-2

Setting time and date ,25-2

TIME Tools ,25-2

Calculations with dates ,25-3

Calculating with times ,25-4

Alarm functions ,25-4

Chapter 26 - Managing memory ,26-1

Memory Structure ,26-1

The HOME directory ,26-2

Port memory ,26-2

Checking objects in memory ,26-3

Backup objects ,26-4

Backing up objects in port memory ,26-4

Backing up and restoring HOME ,26-5

Storing, deleting, and restoring backup objects ,26-6

Using data in backup objects ,26-7

Using SD cards ,26-7

Inserting and removing an SD card ,26-7

Formatting an SD card ,26-8

Accessing objects on an SD card ,26-9

Page TOC-23

Storing objects on an SD card ,26-9

Recalling an object from an SD card ,26-10

Evaluating an object on an SD card ,26-10

Purging an object from the SD card ,26-11

Purging all objects on the SD card (by reformatting) ,26-11

Specifying a directory on an SD card ,26-11

Using libraries ,26-12

Installing and attaching a library ,26-12

Library numbers ,26-13

Deleting a library ,26-13

Creating libraries ,26-13

Backup battery ,26-13

Chapter 27 - The Equation Library ,27-1

Solving a Problem with the Equation Library ,27-1

Using the Solver ,27-2

Using the menu keys ,27-3

Browsing in the Equation Library ,27-4

Viewing equations ,27-4

Viewing variables and selecting units ,27-5

Viewing the picture ,27-5

Using the Multiple-Equation Solver ,27-6

Defining a set of equations ,27-8

Interpreting results from the Multiple-Equation Solver ,27-10

Checking solutions ,27-11

Appendices

Appendix A - Using input forms ,A-1

Appendix B - The calculator’s keyboard ,B-1

Appendix C - CAS settings ,C-1

Appendix D - Additional character set ,D-1

Appendix E - The Selection Tree in the Equation Writer ,E-1

Page TOC-24

Appendix F - The Applications (APPS) menu ,F-1

Appendix G - Useful shortcuts ,G-1

Appendix H - The CAS help facility ,H-1

Appendix I - Command catalog list ,I-1

Appendix J - MATHS menu ,J-1

Appendix K - MAIN menu ,K-1

Appendix L - Line editor commands ,L-1

Appendix M - Table of Built-In Equations ,M-1

Appendix N - Index ,N-1

Limited Warranty ,LW-1

Serviceꢀ ,LW-2

Regulatory information ,LW-4

Disposal of Waste Equipment by Users in Private Household in the European

Union ,LW-6

Page TOC-25

Chapter 1

Getting started

This chapter provides basic information about the operation of your calculator.

It is designed to familiarize you with the basic operations and settings before

you perform a calculation.

Basic Operations

The following sections are designed to get you acquainted with the hardware of

your calculator.

Batteries

The calculator uses 4 AAA (LR03) batteries as main power and a CR2032

lithium battery for memory backup.

Before using the calculator, please install the batteries according to the

following procedure.

To install the main batteries

a. Make sure the calculator is OFF. Slide up the battery compartment cover as

illustrated.

b. Insert 4 new AAA (LR03) batteries into the main compartment. Make sure

each battery is inserted in the indicated direction.

To install the backup battery

a. Make sure the calculator is OFF. Press down the holder. Push the plate to the

shown direction and lift it.

Page 1-1

b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facing

up.

c. Replace the plate and push it to the original place.

After installing the batteries, press [ON] to turn the power on.

Warning: When the low battery icon is displayed, you need to replace the

batteries as soon as possible. However, avoid removing the backup battery and

main batteries at the same time to avoid data lost.

Turning the calculator on and off

The $ key is located at the lower left corner of the keyboard. Press it once to

turn your calculator on. To turn the calculator off, press the right-shift key @

(first key in the second row from the bottom of the keyboard), followed by the

$ key. Notice that the $ key has a OFF label printed in the upper right

corner as a reminder of the OFF command.

Adjusting the display contrast

You can adjust the display contrast by holding the $ key while pressing the

+ or - keys. The $(hold) + key combination produces a darker

display. The $(hold) - key combination produces a lighter display

Contents of the calculator’s display

Turn your calculator on once more. The display should look as indicated

below.

Page 1-2

At the top of the display you will have two lines of information that describe the

settings of the calculator. The first line shows the characters:

R D XYZ HEX R= 'X'

For details on the meaning of these symbols see Chapter 2.

The second line shows the characters: { HOME } indicating that the HOME

directory is the current file directory in the calculator’s memory. In Chapter 2

you will learn that you can save data in your calculator by storing them in files

or variables. Variables can be organized into directories and sub-directories.

Eventually, you may create a branching tree of file directories, similar to those in

a computer hard drive. You can then navigate through the file directory tree to

select any directory of interest. As you navigate through the file directory the

second line of the display will change to reflect the proper file directory and

sub-directory.

At the bottom of the display you will find a number of labels, namely,

@EDIT @VIEW @@ RCL @@ @@STO@ ! PURGE !CLEAR

associated with the six soft menu keys, F1 through F6:

ABCDEF

The six labels displayed in the lower part of the screen will change depending

on which menu is displayed. But A will always be associated with the first

displayed label, B with the second displayed label, and so on.

Menus

The six labels associated with the keys A through F form part of a menu

of functions. Since the calculator has only six soft menu keys, it only display 6

labels at any point in time. However, a menu can have more than six entries.

Page 1-3

Each group of 6 entries is called a Menu page. The current menu, known as

the TOOL menu (see below), has eight entries arranged in two pages. The next

page, containing the next two entries of the menu is available by pressing the

L (NeXT menu) key. This key is the third key from the left in the third row of

keys in the keyboard. Press L once more to return to the main TOOL menu,

or press the I key (third key in second row of keys from the top of the

keyboard).

The TOOL menu is described in detain in the next section. At this point we will

illustrate some properties of menus that you will find useful while using your

calculator.

SOFT menus vs. CHOOSE boxes

Menus, or SOFT menus, associate labels in the lower part of the screen with the

six soft menu keys (Athrough F). By pressing the appropriate soft menu

key, the function shown in the associated label gets activated. For example,

with the TOOL menu active, pressing the @CLEAR key (F) activates function

CLEAR, which erases (clears up) the contents of the screen. To see this function

in action, type a number, say 123`, and then press the F key.

SOFT menus are typically used to select from among a number of related

functions. However, SOFT menus are not the only way to access collections of

related functions in the calculator. The alternative way will be referred to as

CHOOSE boxes. To see an example of a choose box, activate the TOOL menu

(press I), and then press the keystroke combination ‚ã(associated with

the 3 key). This will provide the following CHOOSE box:

Page 1-4

This CHOOSE box is labeled BASE MENU and provides a list of numbered

functions, from 1. HEX x to 6. BꢀR. This display will constitute the first page of

this CHOOSE box menu showing six menu functions. You can navigate through

the menu by using the up and down arrow keys, —˜, located in the upper

right side of the keyboard, right under the E and Fsoft menu keys. To

activate any given function, first, highlight the function name by using the up

and down arrow keys, —˜, or by pressing the number corresponding to

the function in the CHOOSE box. After the function name is selected, press the

@@@OK@@@ soft menu key (F). Thus, if you wanted to use function RꢀB (Real to

Binary), you could press 6F.

If you want to move to the top of the current menu page in a CHOOSE box, use

„—. To move to the bottom of the current page, use „˜. To move to

the top of the entire menu, use ‚—. To move to the bottom of the entire

menu, use ‚˜.

Selecting SOFT menus or CHOOSE boxes

You can select the format in which your menus will be displayed by changing a

setting in the calculator system flags (A system flag is a calculator variable that

controls a certain calculator operation or mode. For more information about

flags, see Chapter 24). System flag 117 can be set to produce either SOFT

menus or CHOOSE boxes. To access this flag use:

H @FLAGS —„ —˜

Your calculator will show the following screen, highlighting the line starting with

the number 117:

By default, the line will look as shown above. The highlighted line (117

CHOOSE boxes) indicates that CHOOSE boxes are the current menu display

setting. If you prefer to use SOFT menu keys, press the @ꢀ@CHK@@ soft menu key

(C), followed by @@@OK@@@ (F). Press @@@OK@@@ (F) once more to return to

normal calculator display.

Page 1-5

If you now press ‚ã, instead of the CHOOSE box that you saw earlier,

the display will now show six soft menu labels as the first page of the STACK

menu:

To navigate through the functions of this menu, press the L key to move to the

next page, or „«(associated with the L key) to move to the previous

page.

The following figures show the different pages of the BASE menu

accessed by pressing the L key twice:

Pressing the L key once more will takes us back to the first menu page.

Note: With the SOFT menu setting for system flag 117, the keystroke

combination ‚(hold) ˜, will show a list of the functions in the current soft

menu. For example, for the two first pages in the BASE menu, you will get:

To revert to the CHOOSE boxes setting, use:

H @FLAGS —„ —˜@ꢀ@CHK@@ @@@OK@@@ @@@OK@@@.

Notes:

1. The TOOL menu, obtained by pressing I, will always produce a SOFT

menu.

2. Most of the examples in this User’s Manual are shown using both SOFT

menus and CHOOSE boxes. Programming applications (Chapters 21 and

22) use exclusively SOFT menus.

3. Additional information on SOFT menus vs. CHOOSE boxes is presented in

Chapter 2 o f this guide.

Page 1-6

The TOOL menu

The soft menu keys for the menu currently displayed, known as the TOOL menu,

are associated with operations related to manipulation of variables (see pages

for more information on variables):

@EDIT

EDIT the contents of a variable (see Chapter 2 and Appendix

L for more information on editing)

VIEW the contents of a variable

ReCaLl the contents of a variable

STOre the contents of a variable

PURGE a variable

@VIEW

@@ RCL @@ C

@@STO@

! PURGE E

CLEAR F

CLEAR the display or stack

The calculator has only six soft menu keys, and can only display 6 labels at any

point in time. However, a menu can have more than six entries. Each group of

6 entries is called a Menu page. The TOOL menu has eight entries arranged

in two pages. The next page, containing the next two entries of the menu are

available by pressing the L (NeXT menu) key. This key is the third key from

the left in the third row of keys in the keyboard.

In this case, only the first two soft menu keys have commands associated with

them. These commands are:

@CASCM

A CASCMD: CAS CoMmanD, used to launch a command from

the CAS by selecting from a list

@HELP

B HELP facility describing the commands available

Pressing the L key will show the original TOOL menu. Another way to

recover the TOOL menu is to press the I key (third key from the left in the

second row of keys from the top of the keyboard).

Setting time and date

The calculator has an internal real time clock. This clock can be continuously

displayed on the screen and be used for alarms as well as running scheduled

tasks. This section will show not only how to set time and date, but also the

basics of using CHOOSE boxes and entering data in a dialog box. Dialog

boxes on your calculator are similar to a computer dialog box.

To set time and date we use the TIME choose box available as an alternative

function for the 9 key. By combining the right-shift button, ‚, with the

Page 1-7

9 key the TIME choose box is activated. This operation can also be

represented as ‚Ó. The TIME choose box is shown in the figure below:

As indicated above, the TIME menu provides four different options, numbered 1

through 4. Of interest to us as this point is option 3. Set time, date... Using the

down arrow key, ˜, highlight this option and press the !!@@OK#@ soft menu key.

The following input form (see Appendix 1-A) for adjusting time and date is

shown:

Setting the time of the day

Using the number keys, 1234567890, start by

adjusting the hour of the day. Suppose that we change the hour to 11, by

pressing 11 as the hour field in the SET TIME AND DATE input form is

highlighted. This results in the number 11 being entered in the lower line of the

input form:

Press the !!@@OK#@ soft menu key to effect the change. The value of 11 is now

shown in the hour field, and the minute field is automatically highlighted:

Page 1-8

Let’s change the minute field to 25, by pressing: 25 !!@@OK#@ . The seconds

field is now highlighted. Suppose that you want to change the seconds field to

45, use: 45 !!@@OK#@

The time format field is now highlighted. To change this field from its current

setting you can either press the W key (the second key from the left in the fifth

row of keys from the bottom of the keyboard), or press the @CHOOS soft menu key

( B).

ꢀ If using the W key, the setting in the time format field will change to either

of the following options:

AM : indicates that displayed time is AM time

PM : indicates that displayed time is PM time

24-hr : indicates that that the time displayed uses a 24 hour

format where18:00, for example, represents 6pm

The last selected option will become the set option for the time format by

using this procedure.

ꢀ If using the @CHOOS soft menu key, the following options are available.

Use the up and down arrow keys,— ˜, to select among these three

options (AM, PM, 24-hour time). Press the !!@@OK#@ soft menu key to make the

selection.

Page 1-9

Setting the date

After setting the time format option, the SET TIME AND DATE input form will

look as follows:

To set the date, first set the date format. The default format is M/D/Y (month/

day/year). To modify this format, press the down arrow key. This will highlight

the date format as shown below:

Use the @CHOOS soft menu key to see the options for the date format:

Highlight your choice by using the up and down arrow keys,— ˜, and

press the !!@@OK#@ soft menu key to make the selection.

Page 1-10

Introducing the calculator’s keyboard

The figure below shows a diagram of the calculator’s keyboard with the

numbering of its rows and columns.

The figure shows 10 rows of keys combined with 3, 5, or 6 columns. Row 1

has 6 keys, rows 2 and 3 have 3 keys each, and rows 4 through 10 have 5

keys each. There are 4 arrow keys located on the right-hand side of the

keyboard in the space occupied by rows 2 and 3.

Each key has three, four, or five functions. The main key function correspond to

the most prominent label in the key. Also, the left-shift key, key (8,1), the right-

Page 1-11

shift key, key (9,1), and the ALPHA key, key (7,1), can be combined with some

of the other keys to activate the alternative functions shown in the keyboard.

For example, the P key, key(4,4), has the following six functions associated

with it:

Main function, to activate the SYMBolic menu

Left-shift function, to activate the MTH (Math) menu

Right-shift function, to activate the CATalog function

ALPHA function, to enter the upper-case letter P

ALPHA-Left-Shift function, to enter the lower-case letter p

ALPHA-Right-Shift function, to enter the symbol P

„´

… N

~„p

~…p

Of the six functions associated with the key only the first four are shown in the

keyboard itself. This is the way that the key looks in the keyboard:

Notice that the color and the position of the labels in the key, namely, SYMB,

MTH, CAT and P, indicate which is the main function (SYMB), and which of

the other three functions is associated with the left-shift „(MTH), right-shift

… (CAT ) , and ~ (P) keys.

For detailed information on the calculator keyboard operation referee to

Appendix B .

Selecting calculator modes

This section assumes that you are now at least partially familiar with the use of

choose and dialog boxes (if you are not, please refer to Chapter 2).

Press the H button (second key from the left on the second row of keys from

the top) to show the following CALCULATOR MODES input form:

Page 1-12

Press the !!@@OK#@ soft menu key to return to normal display. Examples of selecting

different calculator modes are shown next.

Operating Mode

The calculator offers two operating modes: the Algebraic mode, and the

Reverse Polish Notation (RPN) mode. The default mode is the Algebraic mode

(as indicated in the figure above), however, users of earlier HP calculators may

be more familiar with the RPN mode.

To select an operating mode, first open the CALCULATOR MODES input form

by pressing the H button. The Operating Mode field will be highlighted.

Select the Algebraic or RPN operating mode by either using the \ key

(second from left in the fifth row from the keyboard bottom), or pressing the

@CHOOS soft menu key. If using the latter approach, use up and down arrow

keys, — ˜, to select the mode, and press the !!@@OK#@ soft menu key to

complete the operation.

To illustrate the difference between these two operating modes we will calculate

the following expression in both modes:

⎛

⎞

⎟

⎠

3 ⋅ 5 −

⎜

3 ⋅ 3

⎝

2.5

+ e

Page 1-13

To enter this expression in the calculator we will first use the equation writer,

‚O. Please identify the following keys in the keyboard, besides the

numeric keypad keys:

!@.#*+-/R

Q¸Ü‚Oš™˜—`

The equation writer is a display mode in which you can build mathematical

expressions using explicit mathematical notation including fractions, derivatives,

integrals, roots, etc.

To use the equation writer for writing the expression

shown above, use the following keystrokes:

‚OR3*!Ü5-

1/3*3

———————

/23Q3™™+!¸2.5`

After pressing `the calculator displays the expression:

√ (3*(5-1/(3*3))/23^3+EXP(2.5))

Pressing `again will provide the following value. Accept Approx. mode on,

if asked, by pressing !!@@OK#@. [Note: The integer values used above, e.g., 3, 5,

1, represent exact values. The EXP(2.5), however, cannot be expressed as an

exact value, therefore, a switch to Approx mode is required]:

You could also type the expression directly into the display without using the

equation writer, as follows:

R!Ü3.*!Ü5.-1./

!Ü3.*3.™™

/23.Q3+!¸2.5`

to obtain the same result.

Page 1-14

Change the operating mode to RPN by first pressing the H button. Select the

RPN operating mode by either using the \key, or pressing the @CHOOS soft

menu key. Press the !!@@OK#@ soft menu key to complete the operation. The

display, for the RPN mode looks as follows:

Notice that the display shows several levels of output labeled, from bottom to

top, as 1, 2, 3, etc. This is referred to as the stack of the calculator. The

different levels are referred to as the stack levels, i.e., stack level 1, stack level 2,

etc.

In RPN mode, instead of writing an operation such as 3 + 2 by pressing

3+2`, we write the operands first , in the proper order, and then

the operator, i.e., 3`2+. As you enter the operands, they occupy

different stack levels. Entering 3`puts the number 3 in stack level 1.

Next, entering 2pushes the 3 upwards to occupy stack level 2. Finally, by

pressing +, we are telling the calculator to apply the operator, or program,

+ to the objects occupying levels 1 and 2. The result, 5, is then placed in

level 1.

Let's try some other simple operations before trying the more complicated

expression used earlier for the algebraic operating mode:

123/32

123`32/

4`2Q

³√27

27`3@»

Notice the position of the y and the x in the last two operations. The base in

the exponential operation is y (stack level 2) while the exponent is x (stack level

1) before the key Q is pressed. Similarly, in the cubic root operation, y (stack

level 2) is the quantity under the root sign, and x (stack level 1) is the root.

Try the following exercise involving 3 factors: (5 + 3) × 2

5`3+

Calculates (5 + 3) first.

Completes the calculation.

Let's try now the expression proposed earlier:

Page 1-15

⎛

⎞

⎟

⎠

3 ⋅ 5 −

⎜

3 ⋅ 3

⎝

2.5

+ e

3.` Enter 3 in level 1

5.` Enter 5 in level 1, 3 moves to y

3.` Enter 3 in level 1, 5 moves to level 2, 3 to level 3

3.* Place 3 and multiply, 9 appears in level 1

1/(3×3), last value in lev. 1; 5 in level 2; 3 in level 3

5 - 1/(3×3) , occupies level 1 now; 3 in level 2

3× (5 - 1/(3×3)), occupies level 1 now.

23.`Enter 23 in level 1, 14.66666 moves to level 2.

3.Q Enter 3, calculate 23 into level 1. 14.666 in lev. 2.

(3× (5-1/(3×3)))/23 into level 1

2.5 Enter 2.5 level 1

2.5

!¸

, goes into level 1, level 2 shows previous value.

2.5

(3× (5 - 1/(3×3)))/23

= 12.18369, into lev. 1.

) = 3.4905156, into 1.

2.5

√((3× (5 - 1/(3×3)))/23

Although RPN requires a little bit more thought than the algebraic (ALG) mode,

there are multiple advantages in using RPN. For example, in RPN mode you

can see the equation unfolding step by step. This is extremely useful to detect a

possible input error. Also, as you become more efficient in this mode and learn

more of the tricks, you will be able to calculate expression faster and will much

less keystrokes. Consider, for example the calculation of (4×6 - 5)/(1+4×6 - 5).

In RPN mode you can write:

4 ` 6 * 5 - ` 1 + /

obviously, even In RPN mode, you can enter an expression in the same order as

the algebraic mode by using the Equation writer. For example,

‚OR3.*!Ü5.-

1/3.*3.

———————

/23.Q3™™+!¸2.5`

The resulting expression is shown in stack level 1 as follows:

Page 1-16

Notice how the expression is placed in stack level 1 after pressing `.

Pressing the EVAL key at this point will evaluate the numerical value of that

expression Note: In RPN mode, pressing ENTER when there is no command

line will execute the DUP function which copies the contents of stack level 1 of

the stack onto level 2 (and pushes all the other stack levels one level up). This is

extremely useful as showed in the previous example.

To select between the ALG vs. RPN operating mode, you can also set/clear

system flag 95 through the following keystroke sequence:

H@FLAGS 9 ˜ ˜ ˜ ˜ @ꢀCHK@@ `

Alternatively, you can use one of the following shortcuts:

ꢀ In ALG mode,

CF(-95) selects RPN mode

ꢀ In RPN mode,

95 \` SF selects ALG mode

For more information on calculator’s system flags see Chapter 2.

Number Format and decimal dot or comma

Changing the number format allows you to customize the way real numbers are

displayed by the calculator. You will find this feature extremely useful in

operations with powers of tens or to limit the number of decimals in a result.

To select a number format, first open the CALCULATOR MODES input form by

pressing the H button. Then, use the down arrow key, ˜, to select the

option Number format. The default value is Std, or Standard format. In the

standard format, the calculator will show floating-point numbers with the

maximum precision allowed by the calculator (12 significant digits). To learn

Page 1-17

more about reals, see Chapter 2. To illustrate this and other number formats try

the following exercises:

ꢀ

Standard format:

This mode is the most used mode as it shows numbers in the most familiar

notation.

Press the !!@@OK#@ soft menu key, with the Number format set to Std, to return

to the calculator display. Enter the number 123.4567890123456. Notice

that this number has 16 significant figures. Press the ` key. The number

is rounded to the maximum 12 significant figures, and is displayed as

follows:

In the standard format of decimal display, integer numbers are shown with

no decimal zeros whatsoever. Numbers with different decimal figures will

be adjusted in the display so that only those decimal figures that are

necessary will be shown. More examples of numbers in standard format

are shown next:

ꢀ

Fixed format with no decimals: Press the H button. Next, use the down

arrow key, ˜, to select the option Number format. Press the @CHOOS soft

menu key, and select the option Fixed with the arrow down key ˜.

Page 1-18

Notice that the Number Format mode is set to Fix followed by a zero (0).

This number indicates the number of decimals to be shown after the

decimal point in the calculator’s display. Press the !!@@OK#@ soft menu key to

return to the calculator display. The number now is shown as:

This setting will force all results to be rounded to the closest integer (0 digit

displayed after the comma). However, the number is still stored by the

calculator with its full 12 significant digit precision. As we change the

number of decimals to be displayed, you will see the additional digits

being shown again.

ꢁ

Fixed format with decimals:

This mode is mainly used when working with limited precision. For

example, if you are doing financial calculation, using a FIX 2 mode is

convenient as it can easily represent monetary units to a 1/100 precision.

Press the H button. Next, use the down arrow key, ˜, to select the

option Number format. Press the @CHOOS soft menu key, and select the

option Fixed with the arrow down key ˜.

Press the right arrow key, ™, to highlight the zero in front of the option

Fix. Press the @CHOOS soft menu key and, using the up and down arrow

keys, —˜, select, say, 3 decimals.

Page 1-19

Press the !!@@OK#@ soft menu key to complete the selection:

Press the !!@@OK#@ soft menu key return to the calculator display. The number

now is shown as:

Notice how the number is rounded, not truncated. Thus, the number

123.4567890123456, for this setting, is displayed as 123.457, and not

as 123.456 because the digit after 6 is > 5

ꢁ

Scientific format

The scientific format is mainly used when solving problems in the physical

sciences where numbers are usually represented as a number with limited

precision multiplied by a power of ten.

To set this format, start by pressing the H button. Next, use the down

arrow key, ˜, to select the option Number format. Press the @CHOOS soft

menu key and select the option Scientific with the arrow down key ˜.

Keep the number 3 in front of the Sci. (This number can be changed in the

Page 1-20

same fashion that we changed the Fixed number of decimals in the

example above).

Press the !!@@OK#@ soft menu key return to the calculator display. The number

now is shown as:

This result, 1.23E2, is the calculator’s version of powers-of-ten notation,

i.e., 1.235 x 10 . In this, so-called, scientific notation, the number 3 in

front of the Sci number format (shown earlier) represents the number of

significant figures after the decimal point. Scientific notation always

includes one integer figure as shown above. For this case, therefore, the

number of significant figures is four.

ꢁ

Engineering format

The engineering format is very similar to the scientific format, except that

the powers of ten are multiples of three.

To set this format, start by pressing the H button. Next, use the down

arrow key, ˜, to select the option Number format. Press the @CHOOS soft

menu key and select the option Engineering with the arrow down key ˜.

Keep the number 3 in front of the Eng. (This number can be changed in

the same fashion that we changed the Fixed number of decimals in an

earlier example).

Page 1-21

Press the !!@@OK#@ soft menu key return to the calculator display. The number

now is shown as:

Because this number has three figures in the integer part, it is shown with

four significative figures and a zero power of ten, while using the

Engineering format. For example, the number 0.00256, will be shown as:

ꢁ

Decimal comma vs. decimal point

Decimal points in floating-point numbers can be replaced by commas, if

the user is more familiar with such notation. To replace decimal points for

commas, change the FM option in the CALCULATOR MODES input form to

commas, as follows (Notice that we have changed the Number Format to

Std):

ꢀ Press the H button. Next, use the down arrow key, ˜, once, and the

right arrow key, ™, highlighting the option __FM,. To select commas,

press the @ꢀ@CHK@@ soft menu key. The input form will look as follows:

Page 1-22

ꢀ Press the !!@@OK#@ soft menu key return to the calculator display. The number

123.456789012, entered earlier, now is shown as:

Angle Measure

Trigonometric functions, for example, require arguments representing plane

angles. The calculator provides three different Angle Measure modes for

working with angles, namely:

ꢁ

ꢀ

Degrees: There are 360 degrees (360 ) in a complete circumference, or

90 degrees (90 ) in a right angle. This representation is mainly used when

doing basic geometry, mechanical or structural engineering, and

surveying.

Radians: There are 2π radians (2π ) in a complete circumference, or π/2

radians (π/2 ) in a right angle. This notation is mainly used when solving

mathematics and physics problems. This is the default mode of the

calculator.

Grades: There are 400 grades (400 ) in a complete circumference, or

100 grades (100 ) in a right angle. This notation is similar to the degree

mode, and was introduced in order to “simplify” the degree notation but is

seldom used now.

The angle measure affects the trig functions like SIN, COS, TAN and associated

functions.

To change the angle measure mode, use the following procedure:

ꢀ Press the H button. Next, use the down arrow key, ˜, twice. Select

the Angle Measure mode by either using the \key (second from left in

the fifth row from the keyboard bottom), or pressing the @CHOOS soft menu

Page 1-23

key. If using the latter approach, use up and down arrow keys,— ˜,

to select the preferred mode, and press the !!@@OK#@ soft menu key to

complete the operation. For example, in the following screen, the Radians

mode is selected:

Coordinate System

The coordinate system selection affects the way vectors and complex numbers

are displayed and entered. To learn more about complex numbers and vectors,

see Chapters 4 and 9, respectively.

Two- and three-dimensional vector components and complex numbers can be

represented in any of 3 coordinate systems: The Cartesian (2 dimensional) or

Rectangular (3 dimensional), Cylindrical (3 dimensional) or Polar (2

dimensional), and Spherical (only 3 dimensional). In a Cartesian or

Rectangular coordinate system a point P will have three linear coordinates

(x,y,z) measured from the origin along each of three mutually perpendicular

axes (in 2 d mode, z is assumed to be 0). In a Cylindrical or Polar coordinate

system the coordinates of a point are given by (r,θ,z), where r is a radial

distance measured from the origin on the xy plane, θ is the angle that the radial

distance r forms with the positive x axis -- measured as positive in a

counterclockwise direction --, and z is the same as the z coordinate in a

Cartesian system (in 2 d mode, z is assumed to be 0). The Rectangular and

Polar systems are related by the following quantities:

x = r ⋅cos(θ)

r = x²+ y²

⎛ ⎞

⎜ ⎟

y = r ⋅sin(θ )

θ = tan⁻¹

⎝ ⎠

z = z

In a Spherical coordinate system the coordinates of a point are given by (ρ,θ,φ)

where ρ is a radial distance measured from the origin of a Cartesian system, θ

is an angle representing the angle formed by the projection of the linear

distance ρ onto the xy axis (same as θ in Polar coordinates), and φ is the angle

Page 1-24

from the positive z axis to the radial distance ρ. The Rectangular and Spherical

coordinate systems are related by the following quantities:

x = ρ ⋅sin(φ) ⋅cos(θ)

ρ = x²+ y²+ z²

⎛ ⎞

⎜ ⎟

y = ρ ⋅sin(φ)⋅sin(θ)

θ = tan⁻¹

⎝ ⎠

x²+ y²

⎛

⎞

⎟

φ = tan⁻¹

⎜

z = ρ ⋅cos(φ)

⎜

⎟

⎝

⎠

To change the coordinate system in your calculator, follow these steps:

ꢀ Press the H button. Next, use the down arrow key, ˜, three times.

Select the Angle Measure mode by either using the \ key (second from

left in the fifth row from the keyboard bottom), or pressing the @CHOOS soft

menu key. If using the latter approach, use up and down arrow keys,—

˜, to select the preferred mode, and press the !!@@OK#@ soft menu key to

complete the operation. For example, in the following screen, the Polar

coordinate mode is selected:

Beep, Key Click, and Last Stack

The last line of the CALCULATOR MODES input form include the options:

_Beep _Key Click _Last Stack

By choosing the check mark next to each of these options, the corresponding

option is activated. These options are described next:

_Beep : When selected, , the calculator beeper is active. This operation

mainly applies to error messages, but also some user functions like

BEEP.

_Key Click : When selected, each keystroke produces a “click” sound.

Page 1-25

_Last Stack: Keeps the contents of the last stack entry for use with the functions

UNDO and ANS (see Chapter 2).

The _Beep option can be useful to advise the user about errors. You may want

to deselect this option if using your calculator in a classroom or library.

The _Key Click option can be useful as an audible way to check that each

keystroke was entered as intended.

The _Last Stack option is very useful to recover the last operation in case we

need it for a new calculation.

To select, or deselect, any of these three options, first press the H button.

Next,

ꢀ Use the down arrow key, ˜, four times to select the _Last Stack option.

Use the @ꢀ@CHK@@ soft menu key to change the selection.

ꢀ Press the left arrow key š to select the _Key Click option. Use the @ꢀ@CHK@@

soft menu key to change the selection.

ꢀ Press the left arrow key š to select the _Beep option. Use the @ꢀ@CHK@@ soft

menu key to change the selection.

Press the !!@@OK#@ soft menu key to complete the operation.

Selecting CAS settings

CAS stands for Computer Algebraic System. This is the mathematical core of

the calculator where the symbolic mathematical operations and functions are

programmed and performed. The CAS offers a number of settings can be

adjusted according to the type of operation of interest. These are:

ꢀ The default independent variable

ꢀ Numeric vs. symbolic mode

ꢀ Approximate vs. Exact mode

ꢀ Verbose vs. Non-verbose mode

ꢀ Step-by-step mode for operations

ꢀ Increasing power format for polynomials

ꢀ Rigorous mode

ꢀ Simplification of non-rational expressions

Details on the selection of CAS settings are presented in Appendix C.

Page 1-26

Selecting Display modes

The calculator display can be customized to your preference by selecting

different display modes. To see the optional display settings use the following:

ꢀ First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key

to display the DISPLAY MODES input form.

ꢀ To navigate through the many options in the DISPLAY MODES input form,

use the arrow keys: š™˜—.

ꢀ To select or deselect any of the settings shown above, that require a check

mark, select the underline before the option of interest, and toggle the

@ꢀ@CHK@@ soft menu key until the right setting is achieved. When an option is

selected, a check mark will be shown in the underline (e.g., the Textbook

option in the Stack: line above). Unselected options will show no check

mark in the underline preceding the option of interest (e.g., the _Small,

_Full page, and _Indent options in the Edit: line above).

ꢀ To select the Font for the display, highlight the field in front of the Font:

option in the DISPLAY MODES input form, and use the @CHOOS soft menu key.

ꢀ After having selected and unselected all the options that you want in the

DISPLAY MODES input form, press the @@@OK@@@ soft menu key. This will take

you back to the CALCULATOR MODES input form.

To return to normal

calculator display at this point, press the @@@OK@@@ soft menu key once more.

Selecting the display font

Changing the font display allows you to have the calculator look and feel

changed to your own liking. By using a 6-pixel font, for example, you can

display up to 9 stack levels! Follow these instructions to select your display font:

First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to

display the DISPLAY MODES input form. The Font: field is highlighted, and the

option Ft8_0:system 8 is selected. This is the default value of the display font.

Page 1-27

Pressing the @CHOOS soft menu key will provide a list of available system fonts, as

shown below:

The options available are three standard System Fonts (sizes 8, 7, and 6) and a

Browse.. option.

The latter will let you browse the calculator memory for

additional fonts that you may have created (see Chapter 23) or downloaded

into the calculator.

Practice changing the display fonts to sizes 7 and 6. Press the OK soft menu

key to effect the selection. When done with a font selection, press the @@@OK@@@ soft

menu key to go back to the CALCULATOR MODES input form. To return to

normal calculator display at this point, press the @@@OK@@@ soft menu key once more

and see how the stack display change to accommodate the different font.

Selecting properties of the line editor

First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to

display the DISPLAY MODES input form. Press the down arrow key, ˜, once,

to get to the Edit line. This line shows three properties that can be modified.

When these properties are selected (checked) the following effects are

activated:

_Small

Changes font size to small

_Full page

_Indent

Allows to place the cursor after the end of the line

Auto intend cursor when entering a carriage return

Detailed instructions on the use of the line editor are presented in Chapter 2 in

this guide.

Selecting properties of the Stack

First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to

Page 1-28

display the DISPLAY MODES input form. Press the down arrow key, ˜, twice,

to get to the Stack line. This line shows two properties that can be modified.

When these properties are selected (checked) the following effects are

activated:

_Small

Changes font size to small. This maximized the amount of

information displayed on the screen. Note, this selection

overrides the font selection for the stack display.

Display mathematical expressions in graphical mathematical

notation

_Textbook

To illustrate these settings, either in algebraic or RPN mode, use the equation

writer to type the following definite integral:

‚O…Á0™„è™„¸\x™x`

In Algebraic mode, the following screen shows the result of these keystrokes with

neither _Small nor _Textbook are selected:

With the _Small option selected only, the display looks as shown below:

With the _Textbook option selected (default value), regardless of whether the

_Small option is selected or not, the display shows the following result:

Selecting properties of the equation writer (EQW)

First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to

display the DISPLAY MODES input form. Press the down arrow key, ˜, three

Page 1-29

times, to get to the EQW (Equation Writer) line. This line shows two properties

that can be modified. When these properties are selected (checked) the

following effects are activated:

_Small

Changes font size to small while using the equation

editor

_Small Stack Disp

Shows small font in the stack for textbook style display

Detailed instructions on the use of the equation editor (EQW) are presented

elsewhere in this manual.

∞

−X

For the example of the integral

, presented above, selecting the

e dX

∫

_Small Stack Disp in the EQW line of the DISPLAY MODES input form produces

the following display:

Selecting the size of the header

First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to

display the DISPLAY MODES input form. Press the down arrow key, ˜, four

times, to get to the Header line. The value 2 is assigned to the Header field by

default. This means that the top part of the display will contain two lines, one

showing the current settings of the calculator, and a second one showing the

current sub directory within the calculator’s memory (These lines were described

earlier in the manual). The user can select to change this setting to 1 or 0 to

reduce the number of header lines in the display.

Selecting the clock display

First, press the H button to activate the CALCULATOR MODES input form.

Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to

display the DISPLAY MODES input form. Press the down arrow key, ˜, four

times, to get to the Header line. The Header field will be highlighted. Use the

Page 1-30

right arrow key (™) to select the underline in front of the options _Clock or

_Analog. Toggle the @ꢀ@CHK@@ soft menu key until the desired setting is achieved.

If the _Clock option is selected, the time of the day and date will be shown in

the upper right corner of the display. If the _Analog option is also selected, an

analog clock, rather than a digital clock, will be shown in the upper right corner

of the display. If the _Clock option is not selected, or the header is not present,

or too small, the date and time of day will not be shown in the display.

Page 1-31

Chapter 2

Introducing the calculator

In this chapter we present a number of basic operations of the calculator

including the use of the Equation Writer and the manipulation of data objects in

the calculator. Study the examples in this chapter to get a good grasp of the

capabilities of the calculator for future applications.

Calculator objects

Any number, expression, character, variable, etc., that can be created and

manipulated in the calculator is referred to as an object. Some of the most

useful type of objects are listed below.

Real. These object represents a number, positive or negative, with 12 significant

digits and an exponent ranging from -499 to +499. example of reals are: 1.,

-5., 56.41564 1.5E45, -555.74E-95

When entering a real number, you can use the V key to enter the exponent

and the \ key to change the sign of the exponent or mantissa.

Note that real must be entered with a decimal point, even if the number does

not have a fractional part. Otherwise the number is taken as an integer

number, which is a different calculator objects. Reals behave as you would

expect a number to when used in a mathematical operation.

Integers. These objects represent integer numbers (numbers without fractional

part) and do not have limits (except the memory of the calculator). Example of

integers are: 1, 564654112, -413165467354646765465487. Note how

these numbers do not have a decimal point.

Due to their storage format, integer numbers are always maintain full precision

in their calculation. For example, an operation such as 30/14, with integer

numbers, will return 15/7 and not 2.142…. To force a real (or floating-point)

result, use function ꢀNUM ‚ï.

Integers are used frequently in CAS-based functions as they are designed to

keep full precision in their operation.

If the approximate mode (APPROX) is selected in the CAS (see Appendix C),

integers will be automatically converted to reals. If you are not planning to use

Page 2-1

the CAS, it might be a good idea to switch directly into approximate mode.

Refer to Appendix C for more details.

Mixing integers and reals together or mistaking an integer for a real is a

common occurrence. The calculator will detect such mixing of objects and ask

you if you want to switch to approximate mode.

Complex numbers, are an extension of real numbers that include the unit

imaginary number, i = -1. A complex number, e.g., 3 + 2i, is written as (3, 2)

in the calculator.

Complex numbers can be displayed in either Cartesian or polar mode

depending on the setting selected. Note that complex numbers are always

stored in Cartesian mode and that only the display is affected. This allows the

calculator to keep as much precision as possible during calculations.

Most mathematics functions work on complex numbers. There is no need to use

a special “complex +“ function to add complex numbers, you can use the same

+ function that on reals or integers.

Vector and matrix operations utilize objects of type 3, real arrays, and, if

needed, type 4, complex arrays. Objects type 2, strings, are simply lines of

text (enclosed between quotes) produced with the alphanumeric keyboard.

A list is just a collection of objects enclosed between curly brackets and

separated by spaces in RPN mode (the space key is labeled #), or by

commas in algebraic mode. Lists, objects of type 5, can be very useful when

processing collections of numbers. For example, the columns of a table can be

entered as lists. If preferred, a table can be entered as a matrix or array.

Objects type 8 are programs in User RPL language. These are simply sets of

instructions enclosed between the symbols << >>.

Associated with programs are objects types 6 and 7, Global and Local Names,

respectively. These names, or variables, are used to store any type of objects.

The concept of global or local names is related to the scope or reach of the

variable in a given program.

An algebraic object, or simply, an algebraic (object of type 9), is a valid

algebraic expression enclosed within quotation or tick marks.

Page 2-2

Binary integers, objects of type 10, are used in some computer science

applications.

Graphics objects, objects of type 11, store graphics produced by the calculator.

Tagged objects, objects of type 12, are used in the output of many programs to

identify results. For example, in the tagged object: Mean: 23.2, the word Mean:

is the tag used to identify the number 23.2 as the mean of a sample, for

example.

Unit objects, objects of type 13, are numerical values with a physical unit

attached to them.

Directories, objects of type 15, are memory locations used to organize your

variables in a similar fashion as folders are used in a personal computer.

Libraries, objects of type 16, are programs residing in memory ports that are

accessible within any directory (or sub-directory) in your calculator. They

resemble built-in functions, objects of type 18, and built-in commands, objects

of type 19, in the way they are used.

Editing expressions on the screen

In this section we present examples of expression editing directly into the

calculator display (algebraic history or RPN stack).

Creating arithmetic expressions

For this example, we select the Algebraic operating mode and select a Fix

format with 3 decimals for the display. We are going to enter the arithmetic

expression:

1.0

1.0 +

7.5

3.0 − 2.0³

5.0⋅

To enter this expression use the following keystrokes:

5.*„Ü1.+1./7.5™/

„ÜR3.-2.Q3

Page 2-3

The resulting expression is: 5.*(1.+1./7.5)/(√3.-2.^3).

Press ` to get the expression in the display as follows:

Notice that, if your CAS is set to EXACT (see Appendix C) and you enter your

expression using integer numbers for integer values, the result is a symbolic

quantity, e.g.,

5*„Ü1+1/7.5™/

„ÜR3-2Q3

Before producing a result, you will be asked to change to Approximate mode.

Accept the change to get the following result (shown in Fix decimal mode with

three decimal places – see Chapter 1):

In this case, when the expression is entered directly into the stack. As soon as

you press `, the calculator will attempt to calculate a value for the

expression. If the expression is entered between quotes, however, the calculator

will reproduce the expression as entered. In the following example, we enter

the same expression as above, but using quotes. For this case we set the

operating mode to Algebraic, the CAS mode to Exact (deselect _Approx), and

the display setting to Textbook. The keystrokes to enter the expression are the

following:

³5*„Ü1+1/7.5™/

„ÜR3-2Q3`

The result will be shown as follows:

Page 2-4

To evaluate the expression we can use the EVAL function, as follows:

μ„î`

As in the previous example, you will be asked to approve changing the CAS

setting to Approx. Once this is done, you will get the same result as before.

An alternative way to evaluate the expression entered earlier between quotes is

by using the option …ï. To recover the expression from the existing

stack, use the following keystrokes: ƒƒ…ï

We will now enter the expression used above when the calculator is set to the

RPN operating mode. We also set the CAS to Exact and the display to

Textbook. The keystrokes to enter the expression between quotes are the same

used earlier, i.e.,

³5*„Ü1+1/7.5™/

„ÜR3-2Q3`

Resulting in the output

Press ` once more to keep two copies of the expression available in the

stack for evaluation. We first evaluate the expression using the function EVAL,

and next using the function ꢀNUM.

Here are the steps explained in detail: First, evaluate the expression using

function EVAL. The resulting expression is semi-symbolic in the sense that there

are floating-point components to the result, as well as a √3. Next, we switch

stack locations and evaluate using function ꢀNUM:

™

Exchange stack levels 1 and 2 (the SWAP command)

…ï

Evaluate using function ꢀNUM

Page 2-5

This latter result is purely numerical, so that the two results in the stack, although

representing the same expression, seem different. To verify that they are not, we

subtract the two values and evaluate this difference using function EVAL:

Subtract level 1 from level 2

Evaluate using function EVAL

The result is zero (0.).

Note: Avoid mixing integer and real data to avoid conflicts in the

calculations. For many physical science and engineering applications,

including numerical solution of equation, statistics applications, etc., the

APPROX mode (see Appendix C) works better. For mathematical applications,

e.g., calculus, vector analysis, algebra, etc., the EXACT mode is preferred.

Become acquainted with operations in both modes and learn how to switch

from one to the other for different types of operations (see Appendix C).

Editing arithmetic expressions

Suppose that we entered the following expression, between quotes, with the

calculator in RPN mode and the CAS set to EXACT:

7.5

3 − 2³

rather than the intended expression:

was entered by using:

. The incorrect expression

5⋅

³5*„Ü1+1/1.75™/

„ÜR5-2Q3`

To enter the line editor use „˜. The display now looks as follows:

Page 2-6

The editing cursor is shown as a blinking left arrow over the first character in the

line to be edited. Since the editing in this case consists of removing some

characters and replacing them with others, we will use the right and left arrow

keys, š™, to move the cursor to the appropriate place for editing, and the

delete key, ƒ, to eliminate characters.

The following keystrokes will complete the editing for this case:

ꢀ Press the right arrow key, ™, until the cursor is immediately to the right

of the decimal point in the term 1.75

ꢀ Press the delete key, ƒ, twice to erase the characters 1.

ꢀ Press the right arrow key, ™, once, to move the cursor to the right of

the 7

ꢀ Type a decimal point with .

ꢀ Press the right arrow key, ™, until the cursor is immediately to the right

of the √5

ꢀ Press the delete key, ƒ, once to erase the Character 5

ꢀ Type a 3 with 3

ꢀ Press ` to return to the stack

The edited expression is now available in the stack.

Editing of a line of input when the calculator is in Algebraic operating mode is

exactly the same as in the RPN mode. You can repeat this example in

Algebraic mode to verify this assertion.

Creating algebraic expressions

Algebraic expressions include not only numbers, but also variable names. As

an example, we will enter the following algebraic expression:

2L 1+

+ 2

R + y

Page 2-7

We set the calculator operating mode to Algebraic, the CAS to Exact, and the

display to Textbook. To enter this algebraic expression we use the following

keystrokes:

³2*~l*R„Ü1+~„x/~r™/

„ Ü ~r+~„y™+2*~l/~„b

Press ` to get the following result:

Entering this expression when the calculator is set in the RPN mode is exactly

the same as this Algebraic mode exercise.

Editing algebraic expressions

Editing of an algebraic expression with the line editor is very similar to that of

an arithmetic expression (see exercise above). Suppose that we want to modify

the expression entered above to read

x²

2L 1+

+ 2

R + x

To edit this algebraic expression using the line editor use „˜. This

activates the line editor, showing the expression to be edited as follows:

The editing cursor is shown as a blinking left arrow over the first character in

the line to be edited. As in an earlier exercise on line editing, we will use the

right and left arrow keys, š™, to move the cursor to the appropriate place

for editing, and the delete key, ƒ, to eliminate characters.

The following keystrokes will complete the editing for this case:

Page 2-8

ꢀ Press the right arrow key, ™, until the cursor is to the right of the x

ꢀ Type Q2 to enter the power 2 for the x

ꢀ Press the right arrow key, ™, until the cursor is to the right of the y

ꢀ Press the delete key, ƒ, once to erase the characters y.

ꢀ Type ~„x to enter an x

ꢀ Press the right arrow key, ™, 4 times to move the cursor to the right of

the *

ꢀ Type R to enter a square root symbol

ꢀ Type „Ü to enter a set of parentheses (they come in pairs)

ꢀ Press the right arrow key, ™, once, and the delete key, ƒ, once, to

delete the right parenthesis of the set inserted above

ꢀ Press the right arrow key, ™, 4 times to move the cursor to the right of

the b

ꢀ Type „Ü to enter a second set of parentheses

ꢀ Press the delete key, ƒ, once, to delete the left parenthesis of the set

inserted above.

ꢀ Press ` to return to normal calculator display.

The result is shown next:

Notice that the expression has been expanded to include terms such as

|R|, the absolute value, and SQ(b⋅R), the square of b⋅R. To see if we can

simplify this result, use FACTOR(ANS(1)) in ALG mode:

ꢀ Press „˜ to activate the line editor once more. The result is now:

Page 2-9

ꢀ Pressing ` once more to return to normal display.

To see the entire expression in the screen, we can change the option _Small

Stack Disp in the DISPLAY MODES input form (see Chapter 1). After

effecting this change, the display will look as follows:

Note: To use Greek letters and other characters in algebraic expressions use

the CHARS menu. This menu is activated by the keystroke combination

…

. Details are presented in Appendix D.

Using the Equation Writer (EQW) to create expressions

The equation writer is an extremely powerful tool that not only let you enter or

see an equation, but also allows you to modify and work/apply functions on all

or part of the equation. The equation writer (EQW), therefore, allows you to

perform complex mathematical operations, directly, or in a step-by-step mode,

as you would do on paper when solving, for example, calculus problems.

The Equation Writer is launched by pressing the keystroke combination …

‚O (the third key in the fourth row from the top in the keyboard). The

resulting screen is the following:

Page 2-10

The six soft menu keys for the Equation Writer activate the following functions:

@EDIT: lets the user edit an entry in the line editor (see examples above)

@CURS: highlights expression and adds a graphics cursor to it

@BIG: if selected (selection shown by the character in the label) the font used in

the writer is the system font 8 (the largest font available)

@EVAL: lets you evaluate, symbolically or numerically, an expression highlighted

in the equation writer screen (similar to …μ)

@FACTO: lets you factor an expression highlighted in the equation writer screen (if

factoring is possible)

@SIMP: lets you simplify an expression highlighted in the equation writer screen

(as much as it can be simplified according to the algebraic rules of the CAS)

If you press the L key, two more soft menu options show up as shown below:

The six soft menu keys for the Equation Writer activate the following functions:

@CMDS: allows access to the collection of CAS commands listed in alphabetical

order. This is useful to insert CAS commands in an expression available in the

Equation Writer.

@HELP: activates the calculator’s CAS help facility to provide information and

examples of CAS commands.

Some examples for the use of the Equation Writer are shown below.

Creating arithmetic expressions

Entering arithmetic expressions in the Equation Writer is very similar to entering

an arithmetic expression in the stack enclosed in quotes. The main difference is

that in the Equation Writer the expressions produced are written in “textbook”

style instead of a line-entry style. Thus, when a division sign (i.e., /) is

entered in the Equation Writer, a fraction is generated and the cursor placed in

the numerator. To move to the denominator you must use the down arrow key.

For example, try the following keystrokes in the Equation Writer screen:

5/5+2

Page 2-11

The result is the expression

The cursor is shown as a left-facing key. The cursor indicates the current edition

location. Typing a character, function name, or operation will enter the

corresponding character or characters in the cursor location. For example, for

the cursor in the location indicated above, type now:

*„Ü5+1/3

The edited expression looks as follows:

Suppose that you want to replace the quantity between parentheses in the

denominator (i.e., 5+1/3) with (5+π²/2). First, we use the delete key (ƒ)

delete the current 1/3 expression, and then we replace that fraction with π²/2,

as follows: ƒƒƒ„ìQ2

When we hit this point the screen looks as follows:

In order to insert the denominator 2 in the expression, we need to highlight the

entire π²expression. We do this by pressing the right arrow key (™) once. At

that point, we enter the following keystrokes: /2

The expression now looks as follows:

Page 2-12

Suppose that now you want to add the fraction 1/3 to this entire expression,

i.e., you want to enter the expression:

5 + 2⋅(5 +

)

First, we need to highlight the entire first term by using either the right arrow

(™) or the upper arrow (—) keys, repeatedly, until the entire expression is

highlighted, i.e., seven times, producing:

NOTE: Alternatively, from the original position of the cursor (to the right of the 2

in the denominator of π²/2), we can use the keystroke combination ‚—,

interpreted as (‚ ‘ ).

Once the expression is highlighted as shown above, type +1/3 to

add the fraction 1/3. Resulting in:

Page 2-13

Showing the expression in smaller-size

To show the expression in a smaller-size font (which could be useful if the

expression is long and convoluted), simply press the @BIG soft menu key. For this

case, the screen looks as follows:

To recover the larger-font display, press the @BIG soft menu key once more.

Evaluating the expression

To evaluate the expression (or parts of the expression) within the Equation

Writer, highlight the part that you want to evaluate and press the @EVAL soft

menu key.

For example, to evaluate the entire expression in this exercise, first, highlight the

entire expression, by pressing ‚ ‘. Then, press the @EVAL soft menu key. If

your calculator is set to Exact CAS mode (i.e., the _Approx CAS mode is not

checked), then you will get the following symbolic result:

If you want to recover the unevaluated expression at this time, use the function

UNDO, i.e., …¯(the first key in the third row of keys from the top of the

keyboard). The recovered expression is shown highlighted as before:

Page 2-14

If you want a floating-point (numerical) evaluation, use the ꢀNUM function

(i.e., …ï). The result is as follows:

Use the function UNDO ( …¯) once more to recover the original

expression:

Evaluating a sub-expression

Suppose that you want to evaluate only the expression in parentheses in the

denominator of the first fraction in the expression above. You have to use the

arrow keys to select that particular sub-expression. Here is a way to do it:

™

Highlights only the first fraction

Highlights the numerator of the first fraction

Highlights denominator of the first fraction

Highlights first term in denominator of first fraction

Highlights second term in denominator of first fraction

Highlights first factor in second term in denominator of first fraction

Highlights expression in parentheses in denominator of first fraction

Since this is the sub-expression we want evaluated, we can now press the @EVAL

soft menu key, resulting in:

Page 2-15

A symbolic evaluation once more. Suppose that, at this point, we want to

evaluate the left-hand side fraction only. Press the upper arrow key (—) three

times to select that fraction, resulting in:

Then, press the @EVAL soft menu key to obtain:

Let’s try a numerical evaluation of this term at this point. Use …ï to

obtain:

Let’s highlight the fraction to the right, and obtain a numerical evaluation of that

term too, and show the sum of these two decimal values in small-font format by

using:™ …ï C, we get:

To highlight and evaluate the expression in the Equation Writer we use: —

D, resulting in:

Page 2-16

Editing arithmetic expressions

We will show some of the editing features in the Equation Writer as an exercise.

We start by entering the following expression used in the previous exercises:

And will use the editing features of the Equation Editor to transform it into the

following expression:

In the previous exercises we used the arrow keys to highlight sub-expressions for

evaluation. In this case, we will use them to trigger a special editing cursor.

After you have finished entering the original expression, the typing cursor (a left-

pointing arrow) will be located to the right of the 3 in the denominator of the

second fraction as shown here:

Page 2-17

Press the down arrow key (˜) to trigger the clear editing cursor. The screen

now looks like this:

By using the left arrow key (š) you can move the cursor in the general left

direction, but stopping at each individual component of the expression. For

example, suppose that we will first will transform the expression π /2 into the

expression LN(π /3) . With the clear cursor active, as shown above, press the

left-arrow key (š) twice to highlight the 2 in the denominator of π /2. Next,

press the delete key (ƒ) once to change the cursor into the insertion cursor.

Press ƒ once more to delete the 2, and then 3 to enter a 3. At this point,

the screen looks as follows:

Next, press the down arrow key (˜) to trigger the clear editing cursor

highlighting the 3 in the denominator of π /3. Press the left arrow key (š)

once to highlight the exponent 2 in the expression π /3. Next, press the delete

key (ƒ) once to change the cursor into the insertion cursor. Press ƒ once

more to delete the 2, and then 5 to enter a 5. Press the upper arrow key

(—) three times to highlight the expression π /3. Then, type ‚¹ to

apply the LN function to this expression. The screen now looks like this:

Next, we’ll change the 5 within the parentheses to a ½ by using these

keystrokes:

šƒƒ1/2

Next, we highlight the entire expression in parentheses an insert the square root

symbol by using: ————R

Page 2-18

Next, we’ll convert the 2 in front of the parentheses in the denominator into a

šƒƒ2/3

2/3 by using:

At this point the expression looks as follows:

The final step is to remove the 1/3 in the right-hand side of the expression. This

is accomplished by using: —————™ƒƒƒƒƒ

The final version will be:

In summary, to edit an expression in the Equation Writer you should use the

arrow keys (š™—˜) to highlight expression to which functions will be

applied (e.g., the LN and square root cases in the expression above). Use the

down arrow key (˜) in any location, repeatedly, to trigger the clear editing

cursor. In this mode, use the left or right arrow keys (š™) to move from term

to term in an expression. When you reach a point that you need to edit, use the

delete key (ƒ) to trigger the insertion cursor and proceed with the edition of

the expression.

Creating algebraic expressions

An algebraic expression is very similar to an arithmetic expression, except that

English and Greek letters may be included. The process of creating an

algebraic expression, therefore, follows the same idea as that of creating an

arithmetic expression, except that use of the alphabetic keyboard is included.

To illustrate the use of the Equation Writer to enter an algebraic equation we

will use the following example. Suppose that we want to enter the expression:

x + 2μ ⋅ Δy

⎛

⎜

⎝

⎞

⎟

⎠

λ + e^−μ⋅ LN

1/ 3

Page 2-19

Use the following keystrokes:

2 / R3 ™™ * ~‚n+ „¸\ ~‚m

™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c

~„y ——— / ~‚t Q1/3

This results in the output:

In this example we used several lower-case English letters, e.g., x (~„x),

several Greek letters, e.g., λ (~‚n), and even a combination of Greek

and English letters, namely, Δy (~‚c~„y). Keep in mind that to

enter a lower-case English letter, you need to use the combination: ~„

followed by the letter you want to enter. Also, you can always copy special

characters by using the CHARS menu (…

) if you don’t want to memorize

the keystroke combination that produces it. A listing of commonly used

~‚keystroke combinations was listed in an earlier section.

The expression tree

The expression tree is a diagram showing how the Equation Writer interprets an

expression. See Appendix E for a detailed example.

The CURS function

The CURS function (@CURS) in the Equation Writer menu (the B key) converts

the display into a graphical display and produces a graphical cursor that can

be controlled with the arrow keys (š™—˜) for selecting sub-

expressions. The sub-expression selected with @CURS will be shown framed in the

graphics display. After selecting a sub-expression you can press ` to show

the selected sub-expression highlighted in the Equation writer. The following

figures show different sub-expressions selected with and the corresponding

Equation Writer screen after pressing `.

Page 2-20

Editing algebraic expressions

The editing of algebraic equations follows the same rules as the editing of

algebraic equations. Namely:

ꢀ Use the arrow keys (š™—˜) to highlight expressions

ꢀ Use the down arrow key (˜), repeatedly, to trigger the clear editing

cursor . In this mode, use the left or right arrow keys (š™) to move

from term to term in an expression.

ꢀ At an editing point, use the delete key (ƒ) to trigger the insertion

cursor and proceed with the edition of the expression.

To see the clear editing cursor in action, let’s start with the algebraic expression

that we entered in the exercise above:

Press the down arrow key, ˜, at its current location to trigger the clear editing

cursor. The 3 in the exponent of θ will be highlighted. Use the left arrow key,

š, to move from element to element in the expression. The order of selection

of the clear editing cursor in this example is the following (press the left arrow

key, š, repeatedly):

1. The 1 in the 1/3 exponent

Page 2-21

3. Δy

7. μ in the exponential function

9. 3 in the √3 term

10. the 2 in the 2/√3 fraction

At any point we can change the clear editing cursor into the insertion cursor by

pressing the delete key (ƒ). Let’s use these two cursors (the clear editing

cursor and the insertion cursor) to change the current expression into the

following:

If you followed the exercise immediately above, you should have the clear

editing cursor on the number 2 in the first factor in the expression. Follow these

keystrokes to edit the expression:

™ ~‚2

(entering the factorial changes the cursor to the selection cursor)

˜˜™™

Selects the μ in the exponential function

Enters the factorial for the 3 in the square root

/3*~‚f Modifies exponential function argument

™™™™

Selects Δy

Places a square root symbol on Δy

(this operation also changes the cursor to the selection cursor)

˜˜™—— SSelect θ^1/3and enter the SIN function

The resulting screen is the following:

Page 2-22

Evaluating a sub-expression

1/ 3

Since we already have the sub-expression

the @EVAL soft menu key to evaluate this sub-expression. The result is:

highlighted, let’s press

SIN

(

)

Some algebraic expressions cannot be simplified anymore. Try the following

keystrokes: —D. You will notice that nothing happens, other than the

highlighting of the entire argument of the LN function. This is because this

expression cannot be evaluated (or simplified) any more according to the CAS

rules. Trying the keystrokes: —D again does not produce any changes on

the expression. Another sequence of —D keystrokes, however, modifies

the expression as follows:

One more application of the —D keystrokes produces more changes:

This expression does not fit in the Equation Writer screen. We can see the

entire expression by using a smaller-size font. Press the @BIG soft menu key to

get:

Even with the larger-size font, it is possible to navigate through the entire

expression by using the clear editing cursor. Try the following keystroke

sequence: C˜˜˜˜, to set the clear editing cursor atop the factor

Page 2-23

3 in the first term of the numerator. Then, press the right arrow key, ™, to

navigate through the expression.

Simplifying an expression

Press the @BIG soft menu key to get the screen to look as in the previous figure

(see above). Now, press the @SIMP soft menu key, to see if it is possible to

simplify this expression as it is shown in the Equation Writer. The result is the

following screen:

This screen shows the argument of the SIN function, namely,

, transformed

LN (θ )

into

. This may not seem like a simplification, but it is in the sense that

the cubic root function has been replaced by the inverse functions exp-LN.

Factoring an expression

In this exercise we will try factoring a polynomial expression. To continue the

previous exercise, press the ` key. Then, launch the Equation Writer again

by pressing the ‚O key. Type the equation:

XQ2™+2*X*~y+~y Q2™-

~‚a Q2™™+~‚b Q2

resulting in

Let’s select the first 3 terms in the expression and attempt a factoring of this sub-

expression: ‚—˜‚™‚™ . This produces:

Now, press the @FACTO soft menu key, to get

Page 2-24

Press ‚¯to recover the original expression. Next, enter the following

keystrokes: ˜˜˜™™™™™™™———‚™ to select

the last two terms in the expression, i.e.,

press the @FACTO soft menu key, to get

Press ‚¯to recover the original expression. Now, let’s select the entire

expression by pressing the upper arrow key (—) once. And press the @FACTO

soft menu key, to get

Press ‚¯to recover the original expression.

Note: Pressing the @EVAL or the @SIMP soft menu keys, while the entire original

expression is selected, produces the following simplification of the expression:

Using the CMDS menu key

With the original polynomial expression used in the previous exercise still

selected, press the L key to show the @CMDS and @HELP soft menu keys. These

two commands belong to the second part of the soft menu available with the

Equation Writer. Let’s try this example as an application of the @CMDS soft menu

key: Press the @CMDS soft menu key to get the list of CAS commands:

Page 2-25

Next, select the command DERVX (the derivative with respect to the variable X,

the current CAS independent variable) by using: ~d˜˜˜ .

Command DERVX will now be selected:

Press the @@OK@@ soft menu key to get:

Next, press the L key to recover the original Equation Writer menu, and

press the @EVAL@ soft menu key to evaluate this derivative. The result is:

Using the HELP menu

Press the L key to show the @CMDS and @HELP soft menu keys. Press the @HELP

soft menu key to get the list of CAS commands. Then, press ~ d ˜ ˜

˜ to select the command DERVX. Press the @@OK@@ soft menu key to get

information on the command DERVX:

Page 2-26

Detailed explanation on the use of the help facility for the CAS is presented in

Chapter 1. To return to the Equation Writer, press the @EXIT soft menu key. Press

the ` key to exit the Equation Writer.

Using the editing functions BEGIN, END, COPY, CUT and PASTE

To facilitate editing, whether with the Equation Writer or on the stack, the

calculator provides five editing functions, BEGIN, END, COPY, CUT and PASTE,

activated by combining the right-shift key (‚) with keys (2,1), (2,2), (3,1),

(3,2), and (3,3), respectively. These keys are located in the leftmost part of

rows 2 and 3. The action of these editing functions are as follows:

BEGIN: marks the beginning of a string of characters for editing

END: marks the ending of a string of characters for editing

COPY: copies the string of characters selected by BEGIN and END

CUT: cuts the string of characters selected by BEGIN and END

PASTE: pastes a string of characters, previously copied or cut, into the current

cursor position

To see and example, lets start the Equation Writer and enter the following

expression (used in an earlier exercise):

2 / R3 ™™ * ~‚m + „¸\ ~‚m

™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c

~„y ——— / ~‚t Q1/3

The original expression is the following:

We want to remove the sub-expression x+2⋅λ⋅Δy from the argument of the LN

function, and move it to the right of the λ in the first term. Here is one

possibility: ˜ššš———‚ªšš—*‚¬

The modified expression looks as follows:

Page 2-27

Next, we’ll copy the fraction 2/√3 from the leftmost factor in the expression,

and place it in the numerator of the argument for the LN function. Try the

following keystrokes:

˜˜šš———‚¨˜˜

‚™ššš‚¬

The resulting screen is as follows:

The functions BEGIN and END are not necessary when operating in the

Equation Writer, since we can select strings of characters by using the arrow

keys. Functions BEGIN and END are more useful when editing an expression

with the line editor. For example, let’s select the expression x+2⋅λ⋅Δy from this

expression, but using the line editor within the Equation Writer, as follows:

‚—A

The line editor screen will look like this (quotes shown only if calculator in RPN

mode):

To select the sub-expression of interest, use:

™™™™™™™™‚¢

™™™™™™™™™™‚¤

The screen shows the required sub-expression highlighted:

Page 2-28

We can now copy this expression and place it in the denominator of the LN

argument, as follows:‚¨™™… (27 times) … ™

ƒƒ… (9 times) … ƒ ‚¬

The line editor now looks like this:

Pressing ` shows the expression in the Equation Writer (in small-font format,

press the @BIG soft menu key):

Press ` to exit the Equation Writer.

Creating and editing summations, derivatives, and integrals

Summations, derivatives, and integrals are commonly used for calculus,

probability and statistics applications. In this section we show some examples

of such operations created with the equation writer. Use ALG mode.

Summations

We will use the Equation Writer to enter the following summation:

∞

∑

_k=1k ²

Press ‚O to activate the Equation Writer. Then press ‚½to enter the

summation sign. Notice that the sign, when entered into the Equation Writer

screen, provides input locations for the index of the summation as well as for the

quantity being summed. To fill these input locations, use the following

keystrokes:

~„k™1™„è™1/~„kQ2

The resulting screen is:

Page 2-29

To see the corresponding expression in the line editor, press ‚— and the

A soft menu key, to show:

This expression shows the general form of a summation typed directly in the

stack or line editor:

Σ(index = starting_value, ending_value, summation expression)

Press ` to return to the Equation Writer. The resulting screen shows the value

of the summation,

To recover the unevaluated summation use ‚¯. To evaluate the

summation again, you can use the D soft menu key. This shows again that

∞

∑

_{k =1}k²

You can use the Equation Writer to prove that

∞

= +∞

∑

_k=1k

This summation (representing an infinite series) is said to diverge.

Double summations are also possible, for example:

Derivatives

We will use the Equation Writer to enter the following derivative:

(α ⋅t²+ β ⋅t + δ )

Press ‚O to activate the Equation Writer. Then press ‚¿to enter the

(partial) derivative sign. Notice that the sign, when entered into the Equation

Writer screen, provides input locations for the expression being differentiated

Page 2-30

and the variable of differentiation. To fill these input locations, use the following

keystrokes:

~„t™~‚a*~„tQ2

™™+~‚b*~„t+~‚d

The resulting screen is the following:

To see the corresponding expression in the line editor, press ‚— and the

A soft menu key, to show:

This indicates that the general expression for a derivative in the line editor or in

the stack is:

∂variable(function of variables)

Press ` to return to the Equation Writer. The resulting screen is not the

derivative we entered, however, but its symbolic value, namely,

To recover the derivative expression use ‚¯. To evaluate the derivative

again, you can use the D soft menu key. This shows again that

(α ⋅t²− β ⋅t + δ ) = 2α ⋅t + β .

Second order derivatives are possible, for example:

which evaluates to:

Page 2-31

∂

∂x

Note: The notation

is proper of partial derivatives. The proper

( )

notation for total derivatives (i.e., derivatives of one variable) is

. The

calculator, however, does not distinguish between partial and total derivatives.

Definite integrals

We will use the Equation Writer to enter the following definite

integral:

. Press ‚O to activate the Equation Writer. Then

t ⋅sin(t)⋅ dt

∫

press ‚ Á to enter the integral sign. Notice that the sign, when entered

into the Equation Writer screen, provides input locations for the limits of

integration, the integrand, and the variable of integration. To fill these input

locations, use the following keystrokes:0™~‚u™~ „

t*S~„t™~„t. The resulting screen is the following:

To see the corresponding expression in the line editor, press —— and the

A soft menu key, to show:

This indicates that the general expression for a derivative in the line editor or in

the stack is: ∫(lower_limit, upper_limit,integrand,variable_of_integration)

Press ` to return to the Equation Writer. The resulting screen is not the

definite integral we entered, however, but its symbolic value, namely,

To recover the derivative expression use ‚¯. To evaluate the derivative

again, you can use the D soft menu key. This shows again that

Page 2-32

^τt ⋅sin(t)⋅ dt = sin(τ) −τ ⋅cos(τ)

∫

Double integrals are also possible. For example,

which evaluates to 36. Partial evaluation is possible, for example:

This integral evaluates to 36.

Organizing data in the calculator

You can organize data in your calculator by storing variables in a directory

tree. To understand the calculator’s memory, we first take a look at the file

directory. Press the keystroke combination „¡ (first key in second row of

keys from the top of the keyboard) to get the calculator’s File Manager screen:

This screen gives a snapshot of the calculator’s memory and of the directory

tree. The screen shows that the calculator has three memory ports (or memory

partitions), port 0:IRAM, port 1:ERAM, and port 2:FLASH . Memory ports are

used to store third party application or libraries, as well as for backups. The

size of the three different ports is also indicated. The fourth and subsequent

lines in this screen show the calculator’s directory tree. The top directory

(currently highlighted) is the Home directory, and it has predefined into it a sub-

directory called CASDIR. The File Manager screen has three functions

associated with the soft-menu keys:

Page 2-33

@CHDIR: Change to selected directory

@CANCL: Cancel action

@@OK@@:

Approve a selection

For example, to change directory to the CASDIR, press the down-arrow key,

˜, and press @CHDIR. This action closes the File Manager window and returns

us to normal calculator display. You will notice that the second line from the top

in the display now starts with the characters { HOME CASDIR } indicating that

the current directory is CASDIR within the HOME directory.

Functions for manipulation of variables

This screen includes 20 commands associated with the soft menu keys that can

be used to create, edit, and manipulate variables. The first six functions are the

following:

@EDIT To edit a highlighted variable

@COPY To copy a highlighted variable

@MOVE To move a highlighted variable

@@RCL@ To recall the contents of a highlighted variable

@EVAL To evaluate a highlighted variable

@TREE To see the directory tree where the variable is contained

If you press the L key, the next set of functions is made available:

@PURGE To purge, or delete, a variable

@RENAM To rename a variable

@NEW To create a new variable

@ORDER To order a set of variables in the directory

@SEND To send a variable to another calculator or computer

@RECV To receive a variable from another calculator or computer

If you press the L key, the third set of functions is made available:

@HALT To return to the stack temporarily

@VIEW To see contents of a variable

@EDITB To edit contents of a binary variable (similar to @EDIT)

@HEADE To show the directory containing the variable in the header

@LIST Provides a list of variable names and description

@SORT To sort variables according to a sorting criteria

If you press the L key, the last set of functions is made available:

@XSEND To send variable with X-modem protocol

@CHDIR To change directory

Page 2-34

To move between the different soft menu commands, you can use not only the

NEXT key (L), but also the PREV key („«).

The user is invited to try these functions on his or her own. Their applications

are straightforward.

The HOME directory

The HOME directory, as pointed out earlier, is the base directory for memory

operation for the calculator. To get to the HOME directory, you can press the

UPDIR function („§) -- repeat as needed -- until the {HOME} spec is

shown in the second line of the display header. Alternatively, you can use

„(hold) §, press ` if in the algebraic mode. For this example, the

HOME directory contains nothing but the CASDIR. Pressing J will show the

variables in the soft menu keys:

Subdirectories

To store your data in a well organized directory tree you may want to create

subdirectories under the HOME directory, and more subdirectories within

subdirectories, in a hierarchy of directories similar to folders in modern

computers. The subdirectories will be given names that may reflect the

contents of each subdirectory, or any arbitrary name that you can think of.

The CASDIR sub-directory

The CASDIR sub-directory contains a number of variables needed by the proper

operation of the CAS (Computer Algebraic System, see appendix C). To see

the contents of the directory, we can use the keystroke combination:

„¡which opens the File Manager once more:

Page 2-35

This time the CASDIR is highlighted in the screen. To see the contents of the

directory press the @@OK@@ soft menu key or `, to get the following screen:

The screen shows a table describing the variables contained in the CASDIR

directory. These are variables pre-defined in the calculator memory that

establish certain parameters for CAS operation (see appendix C). The table

above contains 4 columns:

•

The first column indicate the type of variable (e.g., ‘EQ’ means an

equation-type variable, |R indicates a real-value variable, { } means a list,

nam means ‘a global name’, and the symbol

variable.

represents a graphic

•

The second column represents the name of the variables, i.e., PRIMIT,

CASINFO, MODULO, REALASSUME, PERIOD, VX, and EPS.

Column number 3 shows another specification for the variable type, e.g.,

ALG means an algebraic expression, GROB stands for graphics object,

INTG means an integer numeric variable, LIST means a list of data,

GNAME means a global name, and REAL means a real (or floating-point)

numeric variable.

The fourth and last column represents the size, in bytes, of the variable

truncated, without decimals (i.e., nibbles). Thus, for example, variable

PERIOD takes 12.5 bytes, while variable REALASSUME takes 27.5 bytes (1

byte = 8 bits, 1 bit is the smallest unit of memory in computers and

calculators).

•

CASDIR Variables in the stack

Pressing the $ key closes the previous screen and returns us to normal

calculator display. By default, we get back the TOOL menu:

We can see the variables contained in the current directory, CASDIR, by

pressing the J key (first key in the second row from the top of the keyboard).

This produces the following screen:

Page 2-36

Pressing the L key shows one more variable stored in this directory:

•

To see the contents of the variable EPS, for example, use ‚@EPS@. This

shows the value of EPS to be .0000000001

To see the value of a numerical variable, we need to press only the soft

menu key for the variable. For example, pressing cz followed by `,

shows the same value of the variable in the stack, if the calculator is set to

Algebraic. If the calculator is set to RPN mode, you need only press the soft

menu key for `.

To see the full name of a variable, press the tickmark key first, ³, and

then the soft menu key corresponding to the variable. For example, for the

variable listed in the stack as PERIO, we use: ³@PERIO@, which produces

as output the string: 'PERIOD'. This procedure applies to both the

Algebraic and RPN calculator operating modes.

•

Variables in CASDIR

The default variables contained in the CASDIR directory are the following:

PRIMIT

Latest primitive (anti-derivative) calculated, not a default

variable, but one created by a previous exercise

a graph that provides CAS information

CASINFO

MODULO

Modulo for modular arithmetic (default = 13)

REALASSUME List of variable names assumed as real values

PERIOD

Period for trigonometric functions (default = 2π)

Name of default independent variable (default = X)

-10

EPS

Value of small increment (epsilon), (default = 10

)

These variables are used for the operation of the CAS.

Typing directory and variable names

To name subdirectories, and sometimes, variables, you will have to type strings

of letters at once, which may or may not be combined with numbers. Rather

than pressing the ~, ~„, or ~‚ key combinations to type each

letter, you can hold down the ~key and enter the various letter. You can also

Page 2-37

lock the alphabetic keyboard temporarily and enter a full name before

unlocking it again. The following combinations of keystrokes will lock the

alphabetic keyboard:

~~ locks the alphabetic keyboard in upper case. When locked in this

fashion, pressing the „ before a letter key produces a lower case letter, while

pressing the ‚ key before a letter key produces a special character. If the

alphabetic keyboard is already locked in upper case, to lock it in lower case,

type, „~

~~„~ locks the alphabetic keyboard in lower case. When locked

in this fashion, pressing the „ before a letter key produces an upper case

letter. To unlock lower case, press „~

To unlock the upper-case locked keyboard, press ~

Let’s try some exercises typing directory/variable names in the stack. Assuming

that the calculator is in the Algebraic mode of operation (although the

instructions work as well in RPN mode), try the following keystroke sequences.

With these commands we will be typing the words ‘MATH’, ‘Math’, and ‘MatH’

~~math`

~~m„a„t„h`

~~m„~at„h`

The calculator display will show the following (left-hand side is Algebraic mode,

right-hand side is RPN mode):

Note: if system flag 60 is set, you can lock the alphabetical keyboard by just

pressing ~. See Chapter 1 for more information on system flags.

Page 2-38

Creating subdirectories

Subdirectories can be created by using the FILES environment or by using the

command CRDIR. The two approaches for creating sub-directories are

presented next.

Using the FILES menu

Regardless of the mode of operation of the calculator (Algebraic or RPN), we

can create a directory tree, based on the HOME directory, by using the

functions activated in the FILES menu. Press „¡ to activate the FILES

menu. If the HOME directory is not already highlighted in the screen, i.e.,

use the up and down arrow keys (—˜) to highlight it. Then, press the @@OK@@

soft menu key. The screen may look like this:

showing that only one object exists currently in the HOME directory, namely, the

CASDIR sub-directory. Let’s create another sub-directory called MANS (for

MANualS) where we will store variables developed as exercises in this manual.

To create this sub-directory first enter: L @@NEW@@ . This will produce the

following input form:

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The Object input field, the first input field in the form, is highlighted by default.

This input field can hold the contents of a new variable that is being created.

Since we have no contents for the new sub-directory at this point, we simply

skip this input field by pressing the down-arrow key, ˜, once. The Name

input field is now highlighted:

This is where we enter the name of the new sub-directory (or variable, as the

case may be), as follows: ~~mans`

The cursor moves to the _Directory check field. Press the @ꢀ@CHK@@ soft menu key

to specify that you are creating a directory, and press @@OK@@ to exit the input form.

The variable listing for the HOME directory will be shown in the screen as

follows:

The screen indicates that there is a new directory (MANS) within the HOME

directory.

Next, we will create a sub-directory named INTRO (for INTROduction), within

MANS, to hold variables created as exercise in subsequent sections of this

chapter. Press the $ key to return to normal calculator display (the TOOLS

menu will be shown). Then, press J to show the HOME directory contents in

the soft menu key labels. The display may look like this (if you have created

other variables in the HOME directory they will show in the soft menu key labels

too):

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To move into the MANS directory, press the corresponding soft menu key (A

in this case), and ` if in algebraic mode. The directory tree will be shown in

the second line of the display as {HOME M NS}. However, there will be no

labels associated with the soft menu keys, as shown below, because there are

no variables defined within this directory.

Let’s create the sub-directory INTRO by using:

„¡@@OK@@ L @@NEW@@ ˜ ~~intro` @ꢀ@CHK@@ @@OK@@

Press the $ key, followed by the J key, to see the contents of the MANS

directory as follows:

Press the !INTRO soft menu key to move into the INTRO sub-directory. This will

show an empty sub-directory. Later on, we will do some exercises in creating

variables.

Using the command CRDIR

The command CRDIR can be used to create directories. This command is

available through the command catalog key (the ‚N key, second key in

fourth row of keys from the top of the keyboard), through the programming

menus (the „°key, same key as the ‚N key), or by simply typing it.

•

Through the catalog key

Press ‚N~c. Use the up and down arrow keys (—˜) to

locate the CRDIR command. Press the @@OK@@ soft menu key to activate the

command.

•

Through the programming menus

Press „°. This will produce the following pull-down menu for

programming:

Page 2-41

Use the down arrow key (˜) to select the option 2. MEMORY… , or just

press 2. Then, press @@OK@@. This will produce the following pull-down

menu:

Use the down arrow key (˜) to select the 5. DIRECTORY option, or just

press 5. Then, press @@OK@@. This will produce the following pull-down

menu:

Use the down arrow key (˜) to select the 5. CRDIR option, and press

@@OK@@.

Command CRDIR in Algebraic mode

Once you have selected the CRDIR through one of the means shown above, the

command will be available in your stack as follows:

At this point, you need to type a directory name, say chap1 :

~~„~chap1~`

The name of the new directory will be shown in the soft menu keys, e.g.,

Command CRDIR in RPN mode

To use the CRDIR in RPN mode you need to have the name of the directory

already available in the stack before accessing the command. For example:

~~„~chap2~`

Then access the CRDIR command by either of the means shown above, e.g.,

through the ‚N key:

Page 2-42

Press the @@OK@ soft menu key to activate the command, to create the sub-

directory:

Moving among subdirectories

To move down the directory tree, you need to press the soft menu key

corresponding to the sub-directory you want to move to. The list of variables in

a sub-directory can be produced by pressing the J (VARiables) key. To move

up in the directory tree, use the function UPDIR, i.e., enter „§.

Alternatively, you can use the FILES menu, i.e., press „¡. Use the up and

down arrow keys (—˜) to select the sub-directory you want to move to, and

then press the !CHDIR (CHange DIRectory) or A soft menu key. This will show

the contents of the sub-directory you moved to in the soft menu key labels.

Deleting subdirectories

To delete a sub-directory, use one of the following procedures:

Using the FILES menu

Press the „¡ key to trigger the FILES menu. Select the directory containing

the sub-directory you want to delete, and press the !CHDIR if needed. This will

close the FILES menu and display the contents of the directory you selected. In

this case you will need to press `. Press the @@OK@@ soft menu key to list the

contents of the directory in the screen. Select the sub-directory (or variable) that

you want to delete. Press L@PURGE. A screen similar to the following will be

shown:

Page 2-43

The ‘S2’ string in this form is the name of the sub-directory that is being deleted.

The soft menu keys provide the following options:

@YES@

@ALL@

!ABORT

@@NO@@

Proceed with deleting the sub-directory (or variable)

Proceed with deleting all sub-directories (or variables)

Do not delete sub-directory (or variable) from a list

Do not delete sub-directory (or variable)

After selecting one of these four commands, you will be returned to the screen

listing the contents of the sub-directory. The !ABORT command, however, will

show an error message:

and you will have to press @@OK@@, before returning to the variable listing.

Using the command PGDIR

The command PGDIR can be used to purge directories. Like the command

CRDIR, the PGDIR command is available through the ‚N or through the

„°key, or it can simply be typed in.

•

Through the catalog key

Press ‚N~~pg. This should highlight the PGDIR command.

Press the @@OK@@ soft menu key to activate the command.

•

Through the programming menus

Press „°. This will produce the following pull-down menu for

programming:

Page 2-44

Use the down arrow key (˜) to select the option 2. MEMORY… Then,

press @@OK@@. This will produce the following pull-down menu:

Use the down arrow key (˜) to select the 5. DIRECTORY option. Then,

press @@OK@@. This will produce the following pull-down menu:

Use the down arrow key (˜) to select the 6. PGDIR option, and press

@@OK@@.

Command PGDIR in Algebraic mode

Once you have selected the PGDIR through one of the means shown above, the

command will be available in your stack as follows:

At this point, you need to type the name of an existing directory, say S4 :

~s4`

As a result, sub-directory @@S4@@ is deleted:

Instead of typing the name of the directory, you can simply press the

corresponding soft menu key at the listing of the PGDIR( ) command, e.g.,

Page 2-45

Press @@OK@@, to get:

Then, press @@S3@@ to enter ‘S3’ as the argument to PGDIR.

Press ` to delete the sub-directory:

Command PGDIR in RPN mode

To use the PGDIR in RPN mode you need to have the name of the directory,

between quotes, already available in the stack before accessing the command.

For example: ³~s2`

Then access the PGDIR command by either of the means shown above, e.g.,

through the ‚N key:

Press the @@OK@ soft menu key to activate the command and delete the sub-

directory:

Page 2-46

Using the PURGE command from the TOOL menu

The TOOL menu is available by pressing the I key (Algebraic and RPN

modes shown):

The PURGE command is available by pressing the @PURGE soft menu key. In the

following examples we want to delete sub-directory S1:

•

Algebraic mode: Enter @PURGE J@@S1@@`

RPN mode: Enter J³@S1@@ `I@PURGE J

Variables

Variables are like files on a computer hard drive. One variable can store one

object (numerical values, algebraic expressions, lists, vectors, matrices,

programs, etc). Even sub-directories can be through of as variables (in fact, in

the calculator, a subdirectory is also a type of calculator object).

Variables are referred to by their names, which can be any combination of

alphabetic and numerical characters, starting with a letter (either English or

Greek). Some non-alphabetic characters, such as the arrow (→) can be used in

a variable name, if combined with an alphabetical character. Thus, ‘→A’ is a

valid variable name, but ‘→’ is not. Valid examples of variable names are: ‘A’,

‘B’, ‘a’, ‘b’, ‘α’, ‘β’, ‘A1’, ‘AB12’, ‘ꢀA12’,’Vel’,’Z0’,’z1’, etc.

A variable can not have the same name than a function of the calculator. You

can not have a SIN variable for example as there is a SIN command in the

calculator. The reserved calculator variable names are the following:

ALRMDAT, CST, EQ, EXPR, IERR, IOPAR, MAXR, MINR, PICT, PPAR, PRTPAR,

VPAR, ZPAR, der_, e, i, n1,n2, …, s1, s2, …, ΣDAT, ΣPAR, π, ∞

Variables can be organized into sub-directories.

Creating variables

To create a variable, we can use the FILES menu, along the lines of the

examples shown above for creating a sub-directory. For example, within the

sub-directory {HOME M NS INTRO}, created in an earlier example, we want

to store the following variables with the values shown:

Page 2-47

Name

A12

Contents

12.5

Type

real

-0.25

3×10

real

algebraic

vector

‘r/(m+r)'

[3,2,1]

3+5i

<< → r 'π*r^2' >>

complex

program

Using the FILES menu

We will use the FILES menu to enter the variable A. We assume that we are in

the sub-directory {HOME M NS INTRO}. To get to this sub-directory, use

the following: „¡ and select the INTRO sub-directory as shown in this

screen:

Press @@OK@@ to enter the directory. You will get a files listing with no entries (the

INTRO sub-directory is empty at this point)

Press the L key to move to the next set of soft menu keys, and press the @@NEW@@

soft menu key. This will produce the NEW VARIABLE input form:

Page 2-48

To enter variable A (see table above), we first enter its contents, namely, the

number 12.5, and then its name, A, as follows: 12.5

@@OK@@ ~a@@OK@@. Resulting in the following screen:

Press @@OK@@ once more to create the variable. The new variable is shown in the

following variable listing:

The listing indicates a real variable (|R), whose name is A, and that occupies

10.5 bytes of memory. To see the contents of the variable in this screen, press

L@VIEW@.

•

Press the @GRAPH soft menu key to see the contents in a graphical format.

•

Press the @TEXT soft menu key to see the contents in text format.

Press @@OK@@ to return to the variable list

Press $ once more to return to normal display. Variable A should now

be featured in the soft menu key labels:

Page 2-49

Using the STOꢀcommand

A simpler way to create a variable is by using the STO command (i.e., the K

key). We provide examples in both the Algebraic and RPN modes, by creating

the remaining of the variables suggested above, namely:

Name

A12

Contents

Type

real

-0.25

3×10

real

algebraic

vector

‘r/(m+r)'

[3,2,1]

3+5i

<< → r 'π*r^2' >>

complex

program

•

Algebraic mode

Use the following keystrokes to store the value of –0.25 into variable α:

0.25\ K ~‚a. AT this point, the screen will

look as follows:

This expression means that the value –0.25 is being stored into α (the

symbol ꢁsuggests the operation). Press ` to create the variable.

The variable is now shown in the soft menu key labels when you press

The following are the keystrokes required to enter the remaining

variables:

A12: 3V5K~a12`

Q: ~„r/„Ü

~„m+~„r™™ K~q`

R: „Ô3‚í2‚í1™ K~r`

Page 2-50

z1: 3+5*„¥ K~„z1` (if needed,

accept change to Complex mode)

p1: ‚å‚é~„r³„ì*

~„rQ2™™™ K~„p1`.

The screen, at this point, will look as follows:

You will see six of the seven variables listed at the bottom of the screen:

p1, z1, R, Q, A12, α.

•

RPN mode

Use the following keystrokes to store the value of –0.25 into variable α:

.25\`³~‚a`. At this point, the screen

will look as follows:

With –0.25 on the level 2 of the stack and 'α' on the level 1 of the

stack, you can use the K key to create the variable. The variable is

now shown in the soft menu key labels when you press J:

To enter the value 3×10 into A12, we can use a shorter version of the

procedure: 3V5³~a12` K

Here is a way to enter the contents of Q:

Q: ~„r/„Ü

~„m+~„r™™ ³~q` K

To enter the value of R, we can use an even shorter version of the

procedure:

R: „Ô3#2#1™ ³~r K

Notice that to separate the elements of a vector in RPN mode we can

use the space key (#), rather than the comma (‚í ) used

above in Algebraic mode.

Page 2-51

z1: ³3+5*„¥ ³~„z1 K(if

needed, accept change to Complex mode)

p1: ‚å‚é~„r³„ì*

~„rQ2™™™ ³ ~„p1™` K.

The screen, at this point, will look as follows:

You will see six of the seven variables listed at the bottom of the screen:

p1, z1, R, Q, A12, α.

Checking variables contents

As an exercise on peeking into the contents of variables we will use the seven

variables entered in the exercise above. We showed how to use the FILES menu

to view the contents of a variable in an earlier exercise when we created the

variable A. In this section we will show a simple way to look into the contents

of a variable.

Pressing the soft menu key label for the variable

This procedure will show the contents of a variable as long as the variable

contains a numerical value or an algebraic value, or an array. For example, for

the variables listed above, press the following keys to see the contents of the

variables:

Algebraic mode

Type these keystrokes: J@@z1@@ ` @@@R@@ `@@@Q@@@ `. At this point, the

screen looks as follows:

Next, type these keystrokes: @@A12@ `

screen looks as follows:

` L@@@A@@@ `. At this point, the

Page 2-52

Pressing the soft menu key corresponding to p1 will provide an error message

(try L@@@p1@@ `):

Note: By pressing @@@p1@@ ` we are trying to activate (run) the p1 program.

However, this program expects a numerical input. Try the following exercise:

$@@@p1@ „Ü5`. The result is:

The program has the following structure: << → r 'π*r^2' >>

The « » symbols indicate a program in User RPL language. The characters → r

indicate that an input, to be referred to as r, is to be provided to the program.

The action from the program is to take that value of r and evaluate the

algebraic 'π*r^2'. In the example shown above, r took the value of 5, and thus

the value of πr = π⋅25 is returned. This program, therefore, calculates the area

of a circle given its radius r.

RPN mode

In RPN mode, you only need to press the corresponding soft menu key label to

get the contents of a numerical or algebraic variable. For the case under

consideration, we can try peeking into the variables z1, R, Q, A12, α, and A,

created above, as follows: J@@z1@@ @@@R@@ @@@Q@@ @@A12@@

Page 2-53

At this point, the screen looks like this:

To see the contents of A, use: L@@@A@@@.

To run program p1 with r = 5, use: L5@@@p1@@@.

Notice that to run the program in RPN mode, you only need to enter the input

(5) and press the corresponding soft menu key. (In algebraic mode, you need

to place parentheses to enter the argument).

Using the right-shift key ‚ followed by soft menu key labels

In Algebraic mode, you can display the content of a variable by pressing J

@ and then the corresponding soft menu key. Try the following examples:

J‚@@p1@@ ‚ @@z1@@ ‚ @@@R@@ ‚@@@Q@@ ‚@@A12@@

Note: In RPN mode, you don’t need to press @ (just J and then the

corresponding soft menu key).

This produces the following screen (Algebraic mode in the left, RPN in the right)

Page 2-54

Notice that this time the contents of program p1 are listed in the screen. To see

the remaining variables in this directory, press L:

Listing the contents of all variables in the screen

Use the keystroke combination ‚˜ to list the contents of all variables in the

screen. For example:

Press $ to return to normal calculator display.

Replacing the contents of variables

Replacing the contents of a variable can be thought of as storing a different

value in the same variable name. Thus, the examples for creating variables

shown above can be used to illustrate the replacement of a variable’s content.

Using the STOꢁcommand

Using as illustration the six variables, p1, z1, R, Q, A12, a, and A, created

earlier, we will proceed to change the contents of variable A12 (currently a

numerical variable) with the algebraic expression ‘β/2’, using the STOꢁ

command. First, using the Algebraic operating mode:

³~‚b/2™ K @@A12@@ `

Check the new contents of variable A12 by using ‚@@A12@@ .

Using the RPN operating mode:

³~‚b/2` ³@@A12@@ ` K

or, in a simplified way,

³~‚b/2™ ³@@A12@@ K

Using the left-shift „ key followed by the variable’s soft menu key (RPN)

This is a very simple way to change the contents of a variable, but it only works

in the RPN mode. The procedure consists in typing the new contents of the

variable and entering them into the stack, and then pressing the left-shift key

Page 2-55

followed by the variable’s soft menu key. For example, in RPN, if we want to

change the contents of variable z1 to ‘a+b⋅i ’, use:

³~„a+~„b*„¥`

This will place the algebraic expression ‘a+b⋅i ’ in level 1: in the stack. To enter

this result into variable z1, use: J„@@@z1@@

To check the new contents of z1, use: ‚@@@z1@@

An equivalent way to do this in Algebraic mode is the following:

~„a+~„b*„¥` K @@@z1@@ `

To check the new contents of z1, use: ‚@@@z1@@

Using the ANS(1) variable (Algebraic mode)

In Algebraic mode one can use the ANS(1) variable to replace the contents of

a variable. For example, the procedure for changing the contents of z1 to

‘a+bi’ is the following:

of z1, use: ‚@@@z1@@

„î K @@@z1@@ `. To check the new contents

Copying variables

The following exercises show different ways of copying variables from one sub-

directory to another.

Using the FILES menu

To copy a variable from one directory to another you can use the FILES menu.

For example, within the sub-directory {HOME MANS INTRO}, we have

variables p1, z1, R, Q, A12, α, and A. Suppose that we want to copy variable

A and place a copy in sub-directory {HOME MANS}. Also, we will copy

variable R and place a copy in the HOME directory. Here is how to do it:

Press „¡@@OK@@ to produce the following list of variables:

Use the down-arrow key ˜ to select variable A (the last in the list), then press

@@COPY@. The calculator will respond with a screen labeled PICK DESTINATION:

Page 2-56

Use the up arrow key — to select the sub-directory MANS and press @@OK@@. If

you now press „§, the screen will show the contents of sub-directory

MANS (notice that variable A is shown in this list, as expected):

Press $ @INTRO@ `(Algebraic mode), or $ @INTRO@ (RPN mode) to return to

the INTRO directory. Press „¡@@OK@@ to produce the list of variables in

{HOME MANS INTRO}. Use the down arrow key (˜) to select variable R,

then press @@COPY@. Use the up arrow key (—) to select the HOME directory,

and press @@OK@@. If you now press „§, twice, the screen will show the

contents of the HOME directory, including a copy of variable R:

Using the history in Algebraic mode

Here is a way to use the history (stack) to copy a variable from one directory to

another with the calculator set to the Algebraic mode. Suppose that we are

within the sub-directory {HOME MANS INTRO}, and want to copy the contents

of variable z1 to sub-directory {HOME MANS}. Use the following procedure:

‚@@z1@ K@@z1@ ` This simply stores the contents of z1 into itself (no change

effected on z1). Next, use „§` to move to the {HOME MANS} sub-

directory. The calculator screen will look like this:

Page 2-57

Next, use the delete key three times, to remove the last three lines in the display:

ƒ ƒ ƒ. At this point, the stack is ready to execute the command

ANS(1)ꢁz1. Press ` to execute this command. Then, use ‚@@z1@, to

verify the contents of the variable.

Using the stack in RPN mode

To demonstrate the use of the stack in RPN mode to copy a variable from one

sub-directory to another, we assume you are within sub-directory {HOME

MANS INTRO}, and that we will copy the contents of variable z1 into the

HOME directory. Use the following procedure:‚@@z1@ `³@@z1@ `

This procedure lists the contents and the name of the variable in the stack. The

calculator screen will look like this:

Now, use „§„§ to move to the HOME directory, and press K to

complete the operation. Use ‚@@z1@, to verify the contents of the variable.

Copying two or more variables using the stack in Algebraic mode

The following is an exercise to demonstrate how to copy two or more variables

using the stack when the calculator is in Algebraic mode. Suppose, once more,

that we are within sub-directory {HOME MANS INTRO} and that we want to

copy the variables R and Q into sub-directory {HOME MANS}. The keystrokes

necessary to complete this operation are shown following:

‚@@ @R@@ K@@@R@@ `

‚@@ @Q@@ K@@@Q@@ `

„§`

ƒ ƒ ƒ`

ƒ ƒ ƒ ƒ `

To verify the contents of the variables, use ‚@@ @R@ and ‚@@ @Q.

This procedure can be generalized to the copying of three or more variables.

Page 2-58

Copying two or more variables using the stack in RPN mode

The following is an exercise to demonstrate how to copy two or more variables

using the stack when the calculator is in RPN mode. We assume, again, that

we are within sub-directory {HOME MANS INTRO} and that we want to copy

the variables R and Q into sub-directory {HOME MANS}. The keystrokes

necessary to complete this operation are shown following:

‚@@ @R@@ ³@@@R@@ `

‚@@ @Q@@ ³@@@Q@@ `

„§K K

To verify the contents of the variables, use ‚@@ @R@ and ‚@@ @Q.

This procedure can be generalized to the copying of three or more variables.

Reordering variables in a directory

In this section we illustrate the use of the ORDER command to reorder the

variables in a directory. We assume we start within the sub-directory {HOME

MANS} containing the variables, A12, R, Q, z1, A, and the sub-directory

INTRO, as shown below. (Copy A12 from INTRO into MANS).

Algebraic mode

In this case, we have the calculator set to Algebraic mode. Suppose that we

want to change the order of the variables to INTRO, A, z1, Q, R, A12.

Proceed as follows to activate the ORDER function:

„°˜@@OK@@

˜˜˜˜ @@OK@@

—— @@OK@@

Select MEMORY from the programming menu

Select DIRECTORY from the MEMORY menu

Select ORDER from the DIRECTORY menu

The screen will show the following input line:

Next, we’ll list the new order of the variables by using their names typed

between quotes:

„ä ³@INTRO ™‚í³@@@@A@@@

™‚í³@@@z1@@™‚í³@@@Q@@@™

‚í³@@@@R@@@ ™‚í³@@A12@@ `

Page 2-59

The screen now shows the new ordering of the variables:

RPN mode

In RPN mode, the list of re-ordered variables is listed in the stack before

applying the command ORDER. Suppose that we start from the same situation

as above, but in RPN mode, i.e.,

The reordered list is created by using:

„ä @INTRO @@@@A@@@ @@@z1@@ @@@Q@@@ @@@@R@@@ @@A12@@ `

Then, enter the command ORDER, as done before, i.e.,

„°˜@@OK@@

˜˜˜˜ @@OK@

—— @@OK@@

Select MEMORY from the programming menu

Select DIRECTORY from the MEMORY menu

Select ORDER from the DIRECTORY menu

The result is the following screen:

Moving variables using the FILES menu

To move a variable from one directory to another you can use the FILES menu.

For example, within the sub-directory {HOME MANS INTRO}, we have

variables p1, z1, R, Q, A12, α, and A. Suppose that we want to move

variable A12 to sub-directory {HOME MANS}. Here is how to do it: Press

„¡@@OK@@ to show a variable list. Use the down-arrow key ˜ to select

variable A12, then press @@MOVE@. The calculator will respond with a PICK

DESTINATION screen. Use the up arrow key — to select the sub-directory

MANS and press @@OK@@. The screen will now show the contents of sub-directory

{HOME MANS INTRO}:

Page 2-60

Notice that variable A12 is no longer there. If you now press „§, the

screen will show the contents of sub-directory MANS, including variable A12:

Note: You can use the stack to move a variable by combining copying with

deleting a variable. Procedures for deleting variables are demonstrated in

the next section.

Deleting variables

Variables can be deleted using function PURGE. This function can be accessed

directly by using the TOOLS menu (I), or by using the FILES menu

„¡@@OK@@ .

Using the FILES command

The FILES command can be used to purge one variable at a time. To delete a

variable from a given directory you can use the FILES menu. For example, within

the sub-directory {HOME MANS INTRO}, we have variables p1, z1, R, Q, α,

and A left. Suppose that we delete variable A. Here is how to do it: Press

„¡@@OK@@ to produce the variable list. Use the down-arrow key ˜ to

select variable A (the last in the list), then press L@PURGE@ @@@YES@@@. The screen

will now show the contents of sub-directory INTRO without variable A.

Using function PURGE in the stack in Algebraic mode

We start again at subdirectory {HOME MANS INTRO} containing now only

variables p1, z1, Q, R, and α. We will use command PURGE to delete

Page 2-61

variable p1. Press I @PURGE@ J@@p1@@ `. The screen will now show

variable p1 removed:

You can use the PURGE command to erase more than one variable by placing

their names in a list in the argument of PURGE. For example, if now we wanted

to purge variables R and Q, simultaneously, we can try the following exercise.

Press :

I @PURGE@ „ä³ J@@@R!@@ ™ ‚í ³ J@@@Q!@@

At this point, the screen will show the following command ready to be executed:

To finish deleting the variables, press `. The screen will now show the

remaining variables:

Using function PURGE in the stack in RPN mode

We start again at subdirectory {HOME MANS INTRO} containing variables

p1, z1, Q, R, and α. We will use command PURGE to delete variable p1.

Press ³@@p1@@ ` I @PURGE@. The screen will now show variable p1

removed:

To delete two variables simultaneously, say variables R and Q, first create a list

(in RPN mode, the elements of the list need not be separated by commas as in

Algebraic mode): J „ä³ @@@R!@@ ™ ³ @@@Q!@@ `. Then, press

I@PURGE@ use to purge the variables.

UNDO and CMD functions

Functions UNDO and CMD are useful for recovering recent commands, or to

revert an operation if a mistake was made. These functions are associated with

Page 2-62

the HIST key: UNDO results from the keystroke sequence ‚¯, while CMD

results from the keystroke sequence „®.

To illustrate the use of UNDO, try the following exercise in algebraic (ALG)

mode: 5*4/3`. The UNDO command (‚¯) will

simply erase the result. The same exercise in RPN mode, will follow these

keystrokes: 5`4`*3`/. Using ‚¯ at this point

will undo the most recent operation (20/3), leaving the original terms back in

the stack:

To illustrate the use of CMD, let’s enter the following entries in ALG mode. Press

` after each entry.

Next, use the CMD function („®) to show the four most recent commands

entered by the user, i.e.,

You can use the up and down arrow keys (—˜) to navigate through these

commands and highlight any of them that you want to entry anew. Once you

have selected the command to enter, press @@@OK@@@.

The CMD function operates in the same fashion when the calculator is in RPN

mode, except that the list of commands only shows numbers or algebraics. It

does not show functions entered. For example, try the following exercise in RPN

mode:

5`2`3/*S

³S5*2`.

Pressing „® produces the following selection box:

Page 2-63

As you can see, the numbers 3, 2, and 5, used in the first calculation above,

are listed in the selection box, as well as the algebraic ‘SIN(5x2)’, but not the

SIN function entered previous to the algebraic.

Flags

A flag is a Boolean value, that can be set or cleared (true or false), that

specifies a given setting of the calculator or an option in a program. Flags in

the calculator are identified by numbers. There are 256 flags, numbered from -

128 to 128. Positive flags are called user flags and are available for

programming purposes by the user. Flags represented by negative numbers are

called system flags and affect the way the calculator operates.

To see the current system flag setting press the H button, and then the @FLAGS!

soft menu key (i.e., F1). You will get a screen labeled SYSTEM FLAGS listing

flag numbers and the corresponding setting.

(Note: In this screen, as only system flags are present, only the absolute value

of the flag number Is displayed). A flag is said to be set if you see a check

mark (ꢁ) in front of the flag number. Otherwise, the flag is not set or cleared.

To change the status of a system flag press the @ꢀ@CHK@@! soft menu key while the

flag you want to change is highlighted, or use the \ key. You can use the up

and down arrow keys (—˜) to move about the list of system flags.

Although there are 128 system flags, not all of them are used, and some of

them are used for internal system control. System flags that are not accessible to

the user are not visible in this screen. A complete list of flags is presented in

Chapter 24.

Page 2-64

Example of flag setting: general solutions vs. principal value

For example, the default value for system flag 01 is General solutions. What

this means is that, if an equation has multiple solutions, all the solutions will be

returned by the calculator, most likely in a list. By pressing the @ꢀ@CHK@@ soft menu

key you can change system flag 01 to Principal value. This setting will force the

calculator to provide a single value known as the principal value of the solution.

To see this at work, first set system flag 01 (i.e., select Principal Value). Press

@@OK@@ twice to return to normal calculator display. We will try solving a

quadratic equation solution, say, t +5t+6 = 0, with command QUAD.

Algebraic mode

Use the following keystroke sequence: ‚N~q (use the up and down

arrow keys, —˜ , to select command QUAD) , press @@OK@@ .

To enter the equation as the first argument of function QUAD, use the following

keystrokes:

‚O~ „t Q2™+5*~ „t+6——

‚Å0`

‚í ~ „t`

The result is:

Now, change the setting of flag 1 to General solutions: H@FLAGS@ @ꢀ@CHK@@ @@OK@@

@@OK@@ . And try the solution again: ——``. The solution now includes

two values:

RPN mode

First set system flag 01 (i.e., Principal Value). Press @@OK@@ twice to return to

normal calculator display. Then, type the quadratic equation as follows:

‚O~ „t Q2™+5*~ „t+6——

‚Å0`

Page 2-65

` (keeping a second copy in the RPN stack)

³~ „t`

Use the following keystroke sequence to enter the QUAD command:

‚N~q (use the up and down arrow keys, —˜ , to select

command QUAD) , press @@OK@@ . The screen shows the principal solution:

Now, change the setting of flag 01 to General solutions: H@FLAGS@ @ꢀ@CHK@@

@@OK@@ @@OK@@ . And try the solution again:

ƒ³ ~ „t`

‚N~q (use the up and down arrow keys, —˜ , to select

command QUAD) , press @@OK@@ . The screen now shows the two solutions:

Other flags of interest

Bring up once more the current flag setting by pressing the H button, and

then the @FLAGS! soft menu key. Make sure to clear system flag 01, which was

left set from the previous exercise. Use the up and down arrow keys (—˜)

to move about the system flag list.

Some flags of interest and their preferred value for the purpose of the exercises

that follow in this manual are:

02 Constant → symb: Constant values (e.g., π) are kept as symbols

03 Function → symb:

Functions are not automatically evaluated, instead

they are loaded as symbolic expressions.

Complex numbers are represented as ordered pairs

The sequence ~~locks the alphabetic keyboard

27 ‘X+Y*i’ → (X,Y):

60 [α][α] locks:

Press @@OK@@ twice to return to normal calculator display.

Page 2-66

CHOOSE boxes vs. Soft MENU

In some of the exercises presented in this chapter we have seen menu lists of

commands displayed in the screen. These menu lists are referred to as

CHOOSE boxes. For example, to use the ORDER command to reorder

variables in a directory, we use, in algebraic mode:

„°˜

Show PROG menu list and select MEMORY

@@OK@@ ˜˜˜˜ Show the MEMORY menu list and select DIRECTORY

@@OK@@ ——

Show the DIRECTORY menu list and select ORDER

@@OK@@

activate the ORDER command

There is an alternative way to access these menus as soft MENU keys, by setting

flag 117. To set this flag try the following:

H @FLAGS! ———————

Page 2-67

The screen shows flag 117 not set (CHOOSE boxes), as shown here:

Press the @ꢀ@CHK@@ soft menu key to set flag 117 to soft MENU. The screen will

reflect that change:

Press @@OK@@ twice to return to normal calculator display.

Now, we’ll try to find the ORDER command using similar keystrokes to those

used above, i.e., we start with „°.

Notice that instead of a menu list, we get soft menu labels with the different

options in the PROG menu, i.e.,

Press B to select the MEMORY soft menu (@@MEM@@). The display now shows:

Press E to select the DIRECTORY soft menu (@@DIR@@)

The ORDER command is not shown in this screen. To find it we use the L key

to find it:

To activate the ORDER command we press the C(@ORDER) soft menu key.

Although not applied to a specific example, this exercise shows the two options

for menus in the calculator (CHOOSE boxes and soft MENUs).

Page 2-68

Note: most of the examples in this user guide assume that the current setting of

flag 117 is its default setting (that is, not set). If you have set the flag but want

to strictly follow the examples in this guide, you should clear the flag before

continuing.

Selected CHOOSE boxes

Some menus will only produce CHOOSE boxes, e.g.,

•

The APPS (APPlicationS menu), activated with the G key, first key in

the second row of keys from the top of the keyboard:

•

The CAT (CATalog menu), activated with the ‚N key, second key

in the fourth row of keys from the top of the keyboard:

•

The HELP menu, activated with I L @HELP

Page 2-69

•

The CMDS (CoMmanDS) menu, activated within the Equation Writer,

i.e., ‚O L @CMDS

Page 2-70

Chapter 3

Calculation with real numbers

This chapter demonstrates the use of the calculator for operations and functions

related to real numbers. Operations along these lines are useful for most

common calculations in the physical sciences and engineering. The user should

be acquainted with the keyboard to identify certain functions available in the

keyboard (e.g., SIN, COS, TAN, etc.). Also, it is assumed that the reader

knows how to adjust the calculator's operation, i.e., select operating mode (see

Chapter 1), use menus and choose boxes (see Chapter 1), and operate with

variables (see Chapter 2).

Checking calculators settings

To check the current calculator and CAS settings you need to just look at the top

line in the calculator display in normal operation. For example, you may see

the following setting:

RAD XYZ DEC R = ‘X’

This stands for RADians for angular measurements, XYZ for Rectangular

(Cartesian) coordinates, DECimal number base, Real numbers preferred, =

means “exact” results, and ‘X’ is the value of the default independent variable.

Another possible listing of options could be

DEG R∠Z HEX C ~ ‘t’

This stands for DEGrees as angular measurements, R∠Z for Polar coordinates,

HEXagesimal number base, Complex numbers allowed, ~ stands for

“approximate” results, and ‘t’ as the default independent variable.

In general, this part of the display contains seven elements. Each element is

identified next under numbers 1 through 7. The possible values for each

element are shown between parentheses following the element description. The

explanation of each of those values is also shown:

1. Angle measure specification (DEG, RAD, GRD)

DEG: degrees, 360 degrees in a complete circle

RAD: radians, 2π radians in a complete circle

GRD: grades, 400 grades in a complete circle

Page 3-1

2. Coordinate system specification (XYZ, R∠Z, R∠∠). The symbol ∠

stands for an angular coordinate.

XYZ: Cartesian or rectangular (x,y,z)

R∠Z: cylindrical Polar coordinates (r,θ,z)

R∠∠: Spherical coordinates (ρ,θ,φ)

3. Number base specification (HEX, DEC, OCT, BIN)

HEX: hexadecimal numbers (base 16)

DEC: decimal numbers (base 10)

OCT: octal numbers (base 8)

BIN: binary numbers (base 2)

4. Real or complex mode specification (R, C)

real numbers

complex numbers

5. Exact or approximate mode specification (=, ~)

exact (symbolic) mode

approximate (numerical) mode

6. Default CAS independent variable (e.g., ‘X’, ‘t’, etc.)

Checking calculator mode

When in RPN mode the different levels of the stack are listed in the left-hand

side of the screen. When the ALGEBRAIC mode is selected there are no

numbered stack levels, and the word ALG is listed in the top line of the display

towards the right-hand side. The difference between these operating modes

was described in detail in Chapter 1.

Real number calculations

To perform real number calculations it is preferred to have the CAS set to Real

(as opposite to Complex) mode. In some cases, a complex result may show up,

and a request to change the mode to Complex will be made by the calculator.

Exact mode is the default mode for most operations. Therefore, you may want

to start your calculations in this mode. Any change to Approx mode required to

complete an operation will be requested by the calculator. There is no

preferred selection for the angle measure or for the number base specification.

Page 3-2

Real number calculations will be demonstrated in both the Algebraic (ALG) and

Reverse Polish Notation (RPN) modes.

Changing sign of a number, variable, or expression

Use the \ key. In ALG mode, you can press \ before entering the

number, e.g., \2.5`. Result = -2.5. In RPN mode, you need to

enter at least part of the number first, and then use the \ key, e.g.,

2.5\. Result = -2.5. If you use the \ function while there is no

command line, the calculator will apply the NEG function (inverse of sign) to the

object on the first level of the stack.

The inverse function

Use the Ykey. In ALG mode, press Y first, followed by a number or

algebraic expression, e.g., Y2. Result = ½ or 0.5. In RPN mode, enter

the number first, then use the key, e.g., 4`Y. Result = ¼ or 0.25.

Addition, subtraction, multiplication, division

Use the proper operation key, namely, + - * /. In ALG mode,

press an operand, then an operator, then an operand, followed by an ` to

obtain a result. Examples:

3.7 + 5.2 `

6.3 - 8.5 `

4.2 * 2.5 `

2.3 / 4.5 `

The first three operations above are shown in the following screen shot:

In RPN mode, enter the operands one after the other, separated by an `,

then press the operator key. Examples:

3.7` 5.2 +

6.3` 8.5 -

4.2` 2.5 *

2.3` 4.5 /

Page 3-3

Alternatively, in RPN mode, you can separate the operands with a space (#)

before pressing the operator key. Examples:

3.7#5.2 +

6.3#8.5 -

4.2#2.5 *

2.3#4.5 /

Using parentheses

Parentheses can be used to group operations, as well as to enclose arguments

of functions. The parentheses are available through the keystroke combination

„Ü. Parentheses are always entered in pairs. For example, to calculate

(5+3.2)/(7-2.2):

In ALG mode:

„Ü5+3.2™/„Ü7-2.2`

In RPN mode, you do not need the parenthesis, calculation is done directly on

the stack:

5`3.2+7`2.2-/

In RPN mode, typing the expression between quotes will allow you to enter the

expression like in algebraic mode:

³„Ü5+3.2™/

„Ü7-2.2`μ

For both, ALG and RPN modes, using the Equation Writer:

‚O5+3.2™/7-2.2

The expression can be evaluated within the Equation writer, by using:

————@EVAL@ or, ‚—@EVAL@

Absolute value function

The absolute value function, ABS, is available through the keystroke

combination: „Ê. When calculating in the stack in ALG mode, enter the

function before the argument, e.g., „Ê \2.32`

In RPN mode, enter the number first, then the function, e.g.,

2.32\„Ê

Page 3-4

Squares and square roots

The square function, SQ, is available through the keystroke combination:

„º. When calculating in the stack in ALG mode, enter the function

before the argument, e.g., „º\2.3`

In RPN mode, enter the number first, then the function, e.g.,

2.3\„º

The square root function, √, is available through the R key. When calculating in

the stack in ALG mode, enter the function before the argument, e.g.,

R123.4`

In RPN mode, enter the number first, then the function, e.g.,

123.4R

Powers and roots

The power function, ^, is available through the Q key. When calculating in

the stack in ALG mode, enter the base (y) followed by the Q key, and then the

exponent (x), e.g., 5.2Q1.25

In RPN mode, enter the number first, then the function, e.g.,

5.2`1.25`Q

The root function, XROOT(y,x), is available through the keystroke combination

‚». When calculating in the stack in ALG mode, enter the function

XROOT followed by the arguments (y,x), separated by commas, e.g.,

‚»3‚í 27`

In RPN mode, enter the argument y, first, then, x, and finally the function call,

e.g.,

27`3`‚»

Base-10 logarithms and powers of 10

Logarithms of base 10 are calculated by the keystroke combination ‚Ã

(function LOG) while its inverse function (ALOG, or antilogarithm) is calculated

by using „Â. In ALG mode, the function is entered before the argument:

‚Ã2.45`

„Â\2.3`

In RPN mode, the argument is entered before the function

2.45` ‚Ã

2.3\` „Â

Page 3-5

Using powers of 10 in entering data

Powers of ten, i.e., numbers of the form -4.5´10 , etc., are entered by using the

V key. For example, in ALG mode:

-2

\4.5V\2`

Or, in RPN mode:

4.5\V2\`

Natural logarithms and exponential function

Natural logarithms (i.e., logarithms of base e = 2.7182818282) are calculated

by the keystroke combination ‚¹ (function LN) while its inverse function,

the exponential function (function EXP) is calculated by using „¸. In ALG

mode, the function is entered before the argument:

‚¹2.45`

„¸\2.3`

In RPN mode, the argument is entered before the function

2.45` ‚¹

2.3\` „¸

Trigonometric functions

Three trigonometric functions are readily available in the keyboard: sine (S),

cosine (T), and tangent (U). The arguments of these functions are angles,

therefore, they can be entered in any system of angular measure (degrees,

radians, grades). For example, with the DEG option selected, we can calculate

the following trigonometric functions:

In ALG mode:

S30`

T45`

U135`

In RPN mode:

30`S

45`T

135`U

Inverse trigonometric functions

The inverse trigonometric functions available in the keyboard are the arcsine

(ASIN), arccosine (ACOS), and arctangent (ATAN), available through the

keystroke combinations „¼, „¾, and „À, respectively. Since

Page 3-6

the inverse trigonometric functions represent angles, the answer from these

functions will be given in the selected angular measure (DEG, RAD, GRD).

Some examples are shown next:

In ALG mode:

„¼0.25`

„¾0.85`

„À1.35`

In RPN mode:

0.25`„¼

0.85`„¾

1.35`„À

All the functions described above, namely, ABS, SQ, √, ^, XROOT, LOG,

ALOG, LN, EXP, SIN, COS, TAN, ASIN, ACOS, ATAN, can be combined with

the fundamental operations (+-*/) to form more complex

expressions. The Equation Writer, whose operations is described in Chapter 2,

is ideal for building such expressions, regardless of the calculator operation

mode.

Differences between functions and operators

Functions like ABS, SQ, √, LOG, ALOG, LN, EXP, SIN, COS, TAN, ASIN,

ACOS, ATAN require a single argument. Thus, their application is ALG mode

is straightforward, e.g., ABS(x). Some functions like XROOT require two

arguments, e.g., XROOT(x,y). This function has the equivalent keystroke

sequence ‚».

Operators, on the other hand, are placed after a single argument or between

two arguments. The factorial operator (!), for example, is placed after a

number, e.g., 5~‚2`. Since this operator requires a single

argument it is referred to as a unary operator. Operators that require two

arguments, such as + - * / Q, are binary operators, e.g.,

3*5, or 4Q2.

Real number functions in the MTH menu

The MTH (MaTHematics) menu include a number of mathematical functions

mostly applicable to real numbers. To access the MTH menu, use the keystroke

Page 3-7

combination „´. With the default setting of CHOOSE boxes for system

flag 117 (see Chapter 2), the MTH menu is shown as the following menu list:

As they are a great number of mathematic functions available in the calculator,

the MTH menu is sorted by the type of object the functions apply on. For

example, options 1. VECTOR.., 2. MATRIX., and 3. LIST.. apply to those data

types (i.e., vectors, matrices, and lists) and will discussed in more detail in

subsequent chapters. Options 4. HYPERBOLIC.. and 5. REAL.. apply to real

numbers and will be discussed in detailed herein. Option 6. BASE.. is used for

conversion of numbers in different bases, and is also to be discussed in a

separate chapter. Option 7. PROBABILITY.. is used for probability applications

and will be discussed in an upcoming chapter. Option 8. FFT.. (Fast Fourier

Transform) is an application of signal processing and will be discussed in a

different chapter. Option 9. COMPLEX.. contains functions appropriate for

complex numbers, which will be discussed in the next chapter. Option 10.

CONSTANTS provides access to the constants in the calculator. This option

will be presented later in this section. Finally, option 11. SPECIAL

FUNCTIONS.. includes functions of advanced mathematics that will be

discussed in this section also.

In general, to apply any of these functions you need to be aware of the number

and order of the arguments required, and keep in mind that, in ALG mode you

should select first the function and then enter the argument, while in RPN mode,

you should enter the argument in the stack first, and then select the function.

Using calculator menus:

1. Since the operation of MTH functions (and of many other calculator menus)

is very similar, we will describe in detail the use of the 4. HYPERBOLIC..

menu in this section, with the intention of describing the general operation

of calculator menus. Pay close attention to the process for selecting

different options.

2. To quickly select one of the numbered options in a menu list (or CHOOSE

box), simply press the number for the option in the keyboard. For

example, to select option 4. HYPERBOLIC.. in the MTH menu, simply press

Page 3-8

Hyperbolic functions and their inverses

Selecting Option 4. HYPERBOLIC.. , in the MTH menu, and pressing @@OK@@,

produces the hyperbolic function menu:

The hyperbolic functions are:

-1

Hyperbolic sine, SINH, and its inverse, ASINH or sinh

-1

Hyperbolic cosine, COSH, and its inverse, ACOSH or cosh

-1

Hyperbolic tangent, TANH, and its inverse, ATANH or tanh

This menu contains also the functions:

EXPM(x) = exp(x) – 1,

LNP1(x) = ln(x+1).

Finally, option 9. MATH, returns the user to the MTH menu.

For example, in ALG mode, the keystroke sequence to calculate, say, tanh(2.5),

is the following:

„´

Select MTH menu

4 @@OK@@

Select the 4. HYPERBOLIC.. menu

Select the 5. TANH function

Evaluate tanh(2.5)

5 @@OK@@

2.5`

The screen shows the following output:

In the RPN mode, the keystrokes to perform this calculation are the following:

2.5`

Enter the argument in the stack

Select MTH menu

„´

4 @@OK@@

Select the 4. HYPERBOLIC.. menu

Select the 5. TANH function

5 @@OK@@

Page 3-9

The result is:

The operations shown above assume that you are using the default setting for

system flag 117 (CHOOSE boxes). If you have changed the setting of this flag

(see Chapter 2) to SOFT menu, the MTH menu will show as labels of the soft

menu keys, as follows (left-hand side in ALG mode, right –hand side in RPN

mode):

Pressing L shows the remaining options:

Note: Pressing „«will return to the first set of MTH options. Also, using

the combination ‚˜ will list all menu functions in the screen, e.g.

Thus, to select, for example, the hyperbolic functions menu, with this menu

format press @@HYP@ , to produce:

Finally, in order to select, for example, the hyperbolic tangent (tanh) function,

simply press @@TANH@.

Page 3-10

Note: To see additional options in these soft menus, press the L key or the

„«keystroke sequence.

For example, to calculate tanh(2.5), in the ALG mode, when using SOFT menus

over CHOOSE boxes, follow this procedure:

„´

@@HYP@

Select MTH menu

Select the HYPERBOLIC.. menu

Select the TANH function

Evaluate tanh(2.5)

@@TANH@

2.5`

In RPN mode, the same value is calculated using:

2.5`

„´

@@HYP@

Enter argument in the stack

Select MTH menu

Select the HYPERBOLIC.. menu

Select the TANH function

@@TANH@

As an exercise of applications of hyperbolic functions, verify the following

values:

SINH (2.5) = 6.05020..

COSH (2.5) = 6.13228..

TANH(2.5) = 0.98661..

EXPM(2.0) = 6.38905….

ASINH(2.0) = 1.4436…

ACOSH (2.0) = 1.3169…

ATANH(0.2) = 0.2027…

LNP1(1.0) = 0.69314….

Once again, the general procedure shown in this section can be applied for

selecting options in any calculator menu.

Real number functions

Selecting option 5. REAL.. in the MTH menu, with system flag 117 set to

CHOOSE boxes, generates the following menu list:

Page 3-11

Option 19. MATH.. returns the user to the MTH menu. The remaining functions

are grouped into six different groups described below.

If system flag 117 is set to SOFT menus, the REAL functions menu will look like

this (ALG mode used, the same soft menu keys will be available in RPN mode):

The very last option, @@MTH@, returns the user to the MTH menu.

Percentage functions

These functions are used to calculate percentages and related values as follows:

% (y,x)

: calculates the x percentage of y

%CH(y,x) : calculates 100(y-x)/x, i.e., the percentage change, the

difference between two numbers.

: calculates 100 x/y, i.e., the percentage total, the portion

that one number (x) is of another (y).

%T(y,x)

These functions require two arguments, we illustrate the calculation of

%T(15,45), i.e., calculation 15% of 45, next. We assume that the calculator is

set to ALG mode, and that system flag 117 is set to CHOOSE boxes. The

procedure is as follows:

„´

5 @@OK@@

3 @@OK@@

‚í

Select MTH menu

Select the 5. REAL.. menu

Select the 5. %T function

Enter first argument

Enter a comma to separate arguments

Enter second argument

Calculate function

Page 3-12

The result is shown next:

In RPN mode, recall that argument y is located in the second level of the stack,

while argument x is located in the first level of the stack. This means, you should

enter x first, and then, y, just as in ALG mode. Thus, the calculation of

%T(15,45), in RPN mode, and with system flag 117 set to CHOOSE boxes, we

proceed as follows:

15`

45`

„´

Enter first argument

Enter second argument

Select MTH menu

5 @@OK@@

Select the 5. REAL.. menu

Select the 5. %T function

3 @@OK@@

Note: The exercises in this section illustrate the general use of calculator

functions having 2 arguments. The operation of functions having 3 or more

arguments can be generalized from these examples.

As an exercise for percentage-related functions, verify the following values:

%(5,20) = 1, %CH(22,25) = 13.6363.., %T(500,20) = 4

Minimum and maximum

Use these functions to determine the minimum or maximum value of two

arguments.

MIN(x,y) : minimum value of x and y

MAX(x,y) : maximum value of x and y

As an exercise, verify that MIN(-2,2) = -2, MAX(-2,2) = 2

Modulo

MOD: y mod x = residual of y/x, i.e., if x and y are integer numbers, y/x = d

+ r/x, where d = quotient, r = residual. In this case, r = y mod x.

Page 3-13

Please notice that MOD is not a function, but rather an operator, i.e., in ALG

mode, MOD should be used as y MOD x, and not as MOD(y,x). Thus, the

operation of MOD is similar to that of +, -, *, /.

As an exercise, verify that 15 MOD 4 = 15 mod 4 = residual of 15/4 = 3

Absolute value, sign, mantissa, exponent, integer and fractional parts

ABS(x)

: calculates the absolute value, |x|

SIGN(x) : determines the sign of x, i.e., -1, 0, or 1.

MANT(x) : determines the mantissa of a number based on log .

XPON(x) : determines the power of 10 in the number

IP(x)

FP(x)

: determines the integer part of a real number

: determines the fractional part of a real number

As an exercise, verify that ABS(-3) = |-3| = 3, SIGN(-5) = -1, MANT(2540) =

2.540, XPON(2540) = 3, IP(2.35) = 2, FP(2.35) = 0.35.

Rounding, truncating, floor, and ceiling functions

RND(x,y) : rounds up y to x decimal places

TRNC(x,y) :ꢁtruncate y to x decimal places

FLOOR(x) : closest integer that is less than or equal to x

CEIL(x)

: closest integer that is greater than or equal to x

As an exercise, verify that RND(1.4567,2) = 1.46, TRNC(1.4567,2) = 1.45,

FLOOR(2.3) = 2, CEIL(2,3) = 3

Radians-to-degrees and degrees-to-radians functions

DꢀR (x)

RꢀD (x)

: converts degrees to radians

: converts radians to degrees.

rad

As an exercise, verify that DꢀR(45) = 0.78539 (i.e., 45 = 0.78539 ),

RꢀD(1.5) = 85.943669.. (i.e., 1.5 = 85.943669.. ).

rad

Special functions

Option 11. Special functions… in the MTH menu includes the following

functions:

Page 3-14

GAMMA:

PSI:

Psi:

The Gamma function Γ(α)

N-th derivative of the digamma function

Digamma function, derivative of the ln(Gamma)

∞

Γ(α) = x^{α −1}e^−xdx

The Gamma function is defined by

. This function has

∫

applications in applied mathematics for science and engineering, as well as in

probability and statistics.

Factorial of a number

The factorial of a positive integer number n is defined as n!=n⋅(n-1)×(n-2)

…3×2×1, with 0! = 1. The factorial function is available in the calculator by

using ~‚2. In both ALG and RPN modes, enter the number first,

followed by the sequence ~‚2. Example: 5~‚2`.

The Gamma function, defined above, has the property that

Γ(α) = (α−1) Γ(α−1), for α > 1.

Therefore, it can be related to the factorial of a number, i.e., Γ(α) = (α−1)!,

when α is a positive integer. We can also use the factorial function to calculate

the Gamma function, and vice versa. For example, Γ(5) = 4! or,

4~‚2`. The factorial function is available in the MTH menu,

through the 7. PROBABILITY.. menu.

The PSI function, Ψ(x,y), represents the y-th derivative of the digamma function,

d ⁿ

dxⁿ

i.e., Ψ(n, x) =

ψ (x), where Ψ(x) is known as the digamma function, or

Psi function. For this function, y must be a positive integer.

ψ (x) = ln[Γ(x)]

The Psi function, Ψ(x), or digamma function, is defined as

Page 3-15

Examples of these special functions are shown here using both the ALG and

RPN modes. As an exercise, verify that GAMMA(2.3) = 1.166711…,

PSI(1.5,3) = 1.40909.., and Psi(1.5) = 3.64899739..E-2.

These calculations are shown in the following screen shot:

Calculator constants

The following are the mathematical constants used by your calculator:

ꢀ e:

ꢀ i:

the base of natural logarithms.

the imaginary unit, i = -1.

i 2

ꢀ

π:

the ratio of the length of the circle to its diameter.

ꢀ MINR: the minimum real number available to the calculator.

ꢀ MAXR: the maximum real number available to the calculator.

To have access to these constants, select option 11. CONSTANTS.. in the MTH

menu,

The constants are listed as follows:

Page 3-16

Selecting any of these entries will place the value selected, whether a symbol

(e.g., e, i, π, MINR, or MAXR) or a value (2.71.., (0,1), 3.14.., 1E-499,

9.99..E499) in the stack.

Please notice that e is available from the keyboard as exp(1), i.e.,

„¸1`, in ALG mode, or 1` „¸, in RPN mode. Also, π

is available directly from the keyboard as „ì. Finally, i is available by

using „¥.

Operations with units

Numbers in the calculator can have units associated with them. Thus, it is

possible to calculate results involving a consistent system of units and produce

a result with the appropriate combination of units.

The UNITS menu

The units menu is launched by the keystroke combination ‚Û(associated

with the 6 key). With system flag 117 set to CHOOSE boxes, the result is

the following menu:

Option 1. Tools.. contains functions used to operate on units (discussed later).

Options 3. Length.. through 17.Viscosity.. contain menus with a number of units

for each of the quantities described. For example, selecting option 8. Force..

shows the following units menu:

Page 3-17

The user will recognize most of these units (some, e.g., dyne, are not used very

often nowadays) from his or her physics classes: N = newtons, dyn = dynes, gf

= grams – force (to distinguish from gram-mass, or plainly gram, a unit of

mass), kip = kilo-poundal (1000 pounds), lbf = pound-force (to distinguish from

pound-mass), pdl = poundal.

To attach a unit object to a number, the number must be followed by an

underscore. Thus, a force of 5 N will be entered as 5_N.

For extensive operations with units SOFT menus provide a more convenient way

of attaching units. Change system flag 117 to SOFT menus (see Chapter 1),

and use the keystroke combination ‚Û to get the following menus. Press

L to move to the next menu page.

Pressing on the appropriate soft menu key will open the sub-menu of units for

that particular selection. For example, for the @SPEED sub-menu, the following

units are available:

Pressing the soft menu key @UNITS will take you back to the UNITS menu.

Recall that you can always list the full menu labels in the screen by using

‚˜, e.g., for the @ENRG set of units the following labels will be listed:

Page 3-18

Note: Use the L key or the „«keystroke sequence to navigate through

the menus.

Available units

The following is a list of the units available in the UNITS menu. The unit symbol

is shown first followed by the unit name in parentheses:

LENGTH

m (meter), cm (centimeter), mm (millimeter), yd (yard), ft (feet), in (inch), Mpc

(Mega parsec), pc (parsec), lyr (light-year), au (astronomical unit), km

(kilometer), mi (international mile), nmi (nautical mile), miUS (US statute mile),

chain (chain), rd (rod), fath (fathom), ftUS (survey foot), Mil (Mil), μ (micron), Å

(Angstrom), fermi (fermi)

AREA

m^2 (square meter), cm^2 (square centimeter), b (barn), yd^2 (square yard),

ft^2 (square feet), in^2 (square inch), km^2 (square kilometer), ha (hectare), a

(are), mi^2 (square mile), miUS^2 (square statute mile), acre (acre)

VOLUME

m^3 (cubic meter), st (stere), cm^3 (cubic centimeter), yd^3 (cubic yard), ft^3

(cubic feet), in^3 (cubic inch), l (liter), galUK (UK gallon), galC (Canadian

gallon), gal (US gallon), qt (quart), pt (pint), ml (mililiter), cu (US cup), ozfl (US

fluid ounce), ozUK (UK fluid ounce), tbsp (tablespoon), tsp (teaspoon), bbl

(barrel), bu (bushel), pk (peck), fbm (board foot)

TIME

yr (year), d (day), h (hour), min (minute), s (second), Hz (hertz)

Page 3-19

SPEED

m/s (meter per second), cm/s (centimeter per second), ft/s (feet per second),

kph (kilometer per hour), mph (mile per hour), knot (nautical miles per hour), c

(speed of light), ga (acceleration of gravity )

MASS

kg (kilogram), g (gram), Lb (avoirdupois pound), oz (ounce), slug (slug), lbt

(Troy pound), ton (short ton), tonUK (long ton), t (metric ton), ozt (Troy ounce), ct

(carat), grain (grain), u (unified atomic mass), mol (mole)

FORCE

N (newton), dyn (dyne), gf (gram-force), kip (kilopound-force), lbf (pound-

force), pdl (poundal)

ENERGY

J (joule), erg (erg), Kcal (kilocalorie), Cal (calorie), Btu (International table btu),

ft×lbf (foot-pound), therm (EEC therm), MeV (mega electron-volt), eV (electron-

volt)

POWER

W (watt), hp (horse power),

PRESSURE

Pa (pascal), atm (atmosphere), bar (bar), psi (pounds per square inch), torr

(torr), mmHg (millimeters of mercury), inHg (inches of mercury), inH20 (inches

of water),

TEMPERATURE

C (degree Celsius), F (degree Fahrenheit), K (Kelvin), R (degree Rankine),

ELECTRIC CURRENT (Electric measurements)

V (volt), A (ampere), C (coulomb), Ω (ohm), F (farad), W (watt), Fdy (faraday),

H (henry), mho (mho), S (siemens), T (tesla), Wb (weber )

Page 3-20

ANGLE (planar and solid angle measurements)

(sexagesimal degree), r (radian), grad (grade), arcmin (minute of arc), arcs

(second of arc), sr (steradian)

LIGHT (Illumination measurements)

fc (footcandle), flam (footlambert), lx (lux), ph (phot), sb (stilb), lm (lumem), cd

(candela), lam (lambert)

RADIATION

Gy (gray), rad (rad), rem (rem), Sv (sievert), Bq (becquerel), Ci (curie), R

(roentgen)

VISCOSITY

P (poise), St (stokes)

Units not listed

Units not listed in the Units menu, but available in the calculator include: gmol

(gram-mole), lbmol (pound-mole), rpm (revolutions per minute), dB (decibels).

These units are accessible through menu 117.02, triggered by using

MENU(117.02) in ALG mode, or 117.02 ` MENU in RPN mode. The menu

will show in the screen as follows (use ‚˜to show labels in display):

These units are also accessible through the catalog, for example:

gmol:

lbmol:

rpm:

dB:

‚N~„g

‚N~„l

‚N~„r

‚N~„d

Page 3-21

Converting to base units

To convert any of these units to the default units in the SI system, use the function

UBASE. For example, to find out what is the value of 1 poise (unit of viscosity)

in the SI units, use the following:

In ALG mode, system flag 117 set to CHOOSE boxes:

‚Û

Select the UNITS menu

Select the TOOLS menu

Select the UBASE function

@@OK@@

˜ @@OK@@

1 ‚Ý Enter 1 and underline

‚Û

Select the UNITS menu

Select the VISCOSITY option

Select the UNITS menu

Convert the units

—

@@OK@@

This results in the following screen (i.e., 1 poise = 0.1 kg/(m⋅s)):

In RPN mode, system flag 117 set to CHOOSE boxes:

‚Û

Enter 1 (no underline)

Select the UNITS menu

Select the VISCOSITY option

Select the unit P (poise)

Select the UNITS menu

Select the TOOLS menu

Select the UBASE function

—

@@OK@@

‚Û

@@OK@@

˜ @@OK@@

In ALG mode, system flag 117 set to SOFT menus:

‚Û

@TOOLS

@UBASE

Select the UNITS menu

Select the TOOLS menu

Select the UBASE function

1 ‚Ý Enter 1 and underline

‚Û

„«@VISC Select the VISCOSITY option

@@@P@@ Select the unit P (poise)

Select the UNITS menu

Page 3-22

Convert the units

In RPN mode, system flag 117 set to SOFT menus:

Enter 1 (no underline)

Select the UNITS menu

‚Û

„«@VISC Select the VISCOSITY option

@@@P@@

Select the unit P (poise)

Select the UNITS menu

Select the TOOLS menu

Select the UBASE function

‚Û

@TOOLS

@UBASE

Attaching units to numbers

To attach a unit object to a number, the number must be followed by an

underscore (‚Ý, key(8,5)). Thus, a force of 5 N will be entered as 5_N.

Here is the sequence of steps to enter this number in ALG mode, system flag

117 set to CHOOSE boxes:

5‚Ý

‚Û

8@@OK@@

@@OK@@

Enter number and underscore

Access the UNITS menu

Select units of force (8. Force..)

Select Newtons (N)

Enter quantity with units in the stack

The screen will look like the following:

Note: If you forget the underscore, the result is the expression 5*N, where N

here represents a possible variable name and not Newtons.

To enter this same quantity, with the calculator in RPN mode, use the following

keystrokes:

Enter number (do not enter underscore)

‚Û

8@@OK@@

@@OK@@

Access the UNITS menu

Select units of force (8. Force..)

Select Newtons (N)

Page 3-23

Notice that the underscore is entered automatically when the RPN mode is

active. The result is the following screen:

As indicated earlier, if system flag 117 is set to SOFT menus, then the UNITS

menu will show up as labels for the soft menu keys. This set up is very

convenient for extensive operations with units.

The keystroke sequences to enter units when the SOFT menu option is selected,

in both ALG and RPN modes, are illustrated next. For example, in ALG mode,

to enter the quantity 5_N use:

5‚Ý

Enter number and underscore

Access the UNITS menu

‚Û

L@@FORCE

@ @@N@@

Select units of force

Select Newtons (N)

Enter quantity with units in the stack

The same quantity, entered in RPN mode uses the following keystrokes:

Enter number (no underscore)

Access the UNITS menu

Select units of force

‚Û

L@@FORCE

@ @@N@@

Select Newtons (N)

Note: You can enter a quantity with units by typing the underline and units

with the ~keyboard, e.g., 5‚Ý~n will produce the entry:

5_N

Unit prefixes

You can enter prefixes for units according to the following table of prefixes from

the SI system.

The prefix abbreviation is shown first, followed by its name, and by the

exponent x in the factor 10 corresponding to each prefix:

___________________________________________________

Prefix Name x

___________________________________________________

Page 3-24

yotta

zetta

exa

peta

tera

giga

+24

+21

+18

+15

+12

deci

centi

milli

-1

-2

-3

micro -6

nano -9

pico

-12

k,K

h,H

D(*)

mega +6

femto -15

kilo

hecto +2

deka +1

atto

-18

zepto -21

yocto -24

___________________________________________________

(*) In the SI system, this prefix is da rather than D. Use D for deka in the

calculator, however.

To enter these prefixes, simply type the prefix using the ~ keyboard. For

example, to enter 123 pm (1 picometer), use:

123‚Ý~„p~„m

Using UBASE to convert to the default unit (1 m) results in:

Operations with units

Once a quantity accompanied with units is entered into the stack, it can be

used in operations similar to plain numbers, except that quantities with units

cannot be used as arguments of functions (say, SQ or SIN). Thus, attempting to

calculate LN(10_m) will produce an error message: Error: Bad Argument Type.

Here are some calculation examples using the ALG operating mode. Be

warned that, when multiplying or dividing quantities with units, you must

enclosed each quantity with its units between parentheses. Thus, to enter, for

example, the product 12.5m × 5.2_yd, type it to read (12.5_m)*(5.2_yd) `:

Page 3-25

which shows as 65_(m⋅yd). To convert to units of the SI system, use function

UBASE:

Note: Recall that the ANS(1) variable is available through the keystroke

combination „î(associated with the ` key).

To calculate a division, say, 3250 mi / 50 h, enter it as (3250_mi)/(50_h)

which transformed to SI units, with function UBASE, produces:

Addition and subtraction can be performed, in ALG mode, without using

parentheses, e.g., 5 m + 3200 mm, can be entered simply as 5_m + 3200_mm

More complicated expression require the use of parentheses, e.g.,

(12_mm)*(1_cm^2)/(2_s) `:

Page 3-26

Stack calculations in the RPN mode, do not require you to enclose the different

terms in parentheses, e.g.,

12_m ` 1.5_yd ` *

3250_mi ` 50_h ` /

These operations produce the following output:

Also, try the following operations:

5_m ` 3200_mm ` +

12_mm ` 1_cm^2 `* 2_s ` /

These last two operations produce the following output:

Note: Units are not allowed in expressions entered in the equation writer.

Units manipulation tools

The UNITS menu contains a TOOLS sub-menu, which provides the following

functions:

CONVERT(x,y): convert unit object x to units of object y

UBASE(x):

UVAL(x):

convert unit object x to SI units

extract the value from unit object x

Page 3-27

UFACT(x,y):

ꢀUNIT(x,y):

factors a unit y from unit object x

combines value of x with units of y

The UBASE function was discussed in detail in an earlier section in this chapter.

To access any of these functions follow the examples provided earlier for

UBASE. Notice that, while function UVAL requires only one argument, functions

CONVERT, UFACT, and ꢀUNIT require two arguments.

Try the following exercises. The output shown below was developed in ALG

mode with system flat 117 set to SOFT menu:

Examples ofꢁCONVERT

These examples produce the same result, i.e., to convert 33 watts to btus

CONVERT(33_W,1_hp) `

CONVERT(33_W,11_hp) `

These operations are shown in the screen as:

Examples of UVAL:

UVAL(25_ft/s) `

UVAL(0.021_cm^3) `

Examples of UFACT

UFACT(1_ha,18_km^2) `

UFACT(1_mm,15.1_cm) `

Page 3-28

Examples of ꢀUNIT

ꢀUNIT(25,1_m) `

ꢀUNIT(11.3,1_mph) `

Physical constants in the calculator

Following along the treatment of units, we discuss the use of physical constants

that are available in the calculator’s memory. These physical constants are

contained in a constants library activated with the command CONLIB. To

launch this command you could simply type it in the stack:

~~conlib~`

or, you can select the command CONLIB from the command catalog, as

follows: First, launch the catalog by using: ‚N~c. Next, use the up

and down arrow keys —˜ to select CONLIB. Finally, press the F(@@OK@@)

soft menu key. Press `, if needed.

The constants library screen will look like the following (use the down arrow key

to navigate through the library):

Page 3-29

The soft menu keys corresponding to this CONSTANTS LIBRARY screen include

the following functions:

when selected, constants values are shown in SI units

ENGL when selected, constants values are shown in English units (*)

UNIT

when selected, constants are shown with units attached (*)

VALUE when selected, constants are shown without units

ꢀSTK copies value (with or without units) to the stack

QUIT

exit constants library

(*) Active only if the function VALUE is active.

This is the way the top of the CONSTANTS LIBRARY screen looks when the

option VALUE is selected (units in the SI system):

Page 3-30

To see the values of the constants in the English (or Imperial) system, press the

@ENGL option:

If we de-select the UNITS option (press @UNITS ) only the values are shown

(English units selected in this case):

To copy the value of Vm to the stack, select the variable name, and press !

then, press @QUIT@. For the calculator set to the ALG, the screen will look like this:

The display shows what is called a tagged value, Vm:359.0394. In here,

Vm, is the tag of this result. Any arithmetic operation with this number will

ignore the tag. Try, for example: ‚¹2*„î`, which

produces:

The same operation in RPN mode will require the following keystrokes (after the

value of Vm was extracted from the constants library): 2`*‚ ¹

Page 3-31

Special physical functions

Menu 117, triggered by using MENU(117) in ALG mode, or 117 ` MENU

in RPN mode, produces the following menu (labels listed in the display by using

‚˜):

The functions include:

ZFACTOR: gas compressibility Z factor function

FANNING: Fanning friction factor for fluid flow

DARCY: Darcy-Weisbach friction factor for fluid flow

F0λ: Black body emissive power function

SIDENS: Silicon intrinsic density

TDELTA: Temperature delta function

In the second page of this menu (press L) we find the following items:

In this menu page, there is one function (TINC) and a number of units described

in an earlier section on units (see above). The function of interest is:

TINC: temperature increment command

Out of all the functions available in this MENU (UTILITY menu), namely,

ZFACTOR, FANNING, DARCY, F0λ, SIDENS, TDELTA, and TINC, functions

FANNING and DARCY are described in Chapter 6 in the context of solving

equations for pipeline flow. The remaining functions are described following.

Function ZFACTOR

Function ZFACTOR calculates the gas compressibility correction factor for

nonideal behavior of hydrocarbon gas. The function is called by using

Page 3-32

ZFACTOR(x , y ), where x is the reduced temperature, i.e., the ratio of actual

temperature to pseudo-critical temperature, and y is the reduced pressure, i.e.,

the ratio of the actual pressure to the pseudo-critical pressure. The value of x

must be between 1.05 and 3.0, while the value of y must be between 0 and

30. Example, in ALG mode:

Function F0λ

Function F0λ (T, λ) calculates the fraction (dimensionless) of total black-body

emissive power at temperature T between wavelengths 0 and λ. If no units are

attached to T and λ, it is implied that T is in K and λ in m. Example, in ALG

mode:

Function SIDENS

Function SIDENS(T) calculates the intrinsic density of silicon (in units of 1/cm )

as a function of temperature T (T in K), for T between 0 and 1685 K. For

example,

Function TDELTA

Function TDELTA(T ,T ) yields the temperature increment T – T . The result is

returned with the same units as T , if any. Otherwise, it returns simply the

difference in numbers. For example,

The purpose of this function is to facilitate the calculation of temperature

differences given temperatures in different units. Otherwise, it’s simply

calculates a subtraction, e.g.,

Page 3-33

Function TINC

Function TINC(T ,ΔT) calculates T +DT. The operation of this function is similar

to that of function TDELTA in the sense that it returns a result in the units of T .

Otherwise, it returns a simple addition of values, e.g.,

Defining and using functions

Users can define their own functions by using the DEF command available

thought the keystroke sequence „à (associated with the 2 key). The

function must be entered in the following format:

Function_name(arguments) = expression_containing_arguments

For example, we could define a simple functionH(x) = ln(x+1) + exp(-x).

Suppose that you have a need to evaluate this function for a number of discrete

values and, therefore, you want to be able to press a single button and get the

result you want without having to type the expression in the right-hand side for

each separate value. In the following example, we assume you have set your

calculator to ALG mode. Enter the following sequence of keystrokes:

„à³~h„Ü~„x™‚Å

‚¹~„x+1™+„¸~„x`

The screen will look like this:

Page 3-34

Press the J key, and you will notice that there is a new variable in your soft

menu key (@@@H@@). To see the contents of this variable press ‚@@@H@@. The screen

will show now:

Thus, the variable H contains a program defined by:

<< ꢀ x ‘LN(x+1) + EXP(x)’ >>

This is a simple program in the default programming language of the calculator.

This programming language is called UserRPL. The program shown above is

relatively simple and consists of two parts, contained between the program

containers << >>:

ꢀ Input:

ꢀ x

ꢀ Process:

‘LN(x+1) + EXP(x) ‘

This is to be interpreted as saying: enter a value that is temporarily assigned to

the name x (referred to as a local variable), evaluate the expression between

quotes that contain that local variable, and show the evaluated expression.

To activate the function in ALG mode, type the name of the function followed by

the argument between parentheses, e.g., @@@H@@@ „Ü2`. Some

examples are shown below:

In the RPN mode, to activate the function enter the argument first, then press the

soft menu key corresponding to the variable name @@@H@@@ . For example, you

could try: 2@@@H@@@ . The other examples shown above can be entered by

using: 1.2@@@H@@@ , 2`3/@@@H@@@ .

Functions can have more than 2 arguments. For example, the screen below

shows the definition of the function K(α,β) = α+β, and its evaluation with the

arguments K(√2,π), and K(1.2,2.3):

Page 3-35

The contents of the variable K are: << ꢀ α β ‘α+β’ >>.

Functions defined by more than one expression

In this section we discuss the treatment of functions that are defined by two or

more expressions. An example of such functions would be

2 ⋅ x − 1,

x < 0

x > 0

⎧

⎨

⎩

⎫

⎬

⎭

f ( x ) =

− 1,

The function IFTE (IF-Then-Else) describes such functions.

The IFTE function

The IFTE function is written as IFTE(condition, operation_if_true, operation_if_false)

If condition is true then operation_if_true is performed, else operation_if_false is

performed. For example, we can write ‘f(x) = IFTE(x>0, x^2-1, 2*x-1)’, to

describe the function listed above. Function IFTE is accessible from the function

catalog (‚N). The symbol ‘>’ (greater than) is available as (associated

with the Y key). To define this function in ALG mode use the command:

DEF(f(x) = IFTE(x>0, x^2-1, 2*x-1)) then, press `. In RPN mode, type the

function definition between apostrophes:

‘f(x) = IFTE(x>0, x^2-1, 2*x-1)’

then press „à.

Press J to recover your variable menu. The function @@@f@@@ should be available

in your soft key menu. Press ‚@@@f@@@ to see the resulting program:

<< ꢀ x ‘IFTE(x>0, x^2-1, 2*x-1)’ >>

To evaluate the function in ALG mode, type the function name, f, followed by the

number at which you want to evaluate the function, e.g., f(2), then press `.

In RPN mode, enter a number and press @@@f@@@. Check, for example, that f(2) =

3, while f(-2) = -5.

Page 3-36

Combined IFTE functions

To program a more complicated function such as

− x, x < −2

x +1, − 2 ≤ x < 0

x −1, 0 ≤ x < 2

x², x ≥ 2

⎧

⎪

⎨

g(x) =

⎪

⎩

you can combine several levels of the IFTE function, i.e.,

‘g(x) = IFTE(x<-2, -x, IFTE(x<0, x+1, IFTE(x<2, x-1, x^2)))’,

Define this function by any of the means presented above, and check that

g(-3) = 3, g(-1) = 0, g(1) = 0, g(3) = 9.

Page 3-37

Chapter 4

Calculations with complex numbers

This chapter shows examples of calculations and application of functions to

complex numbers.

Definitions

A complex number z is a number written as z = x + iy, where x and y are real

numbers, and i is the imaginary unit defined by i = -1. The complex number

x+iy has a real part, x = Re(z), and an imaginary part, y = Im(z). We can

think of a complex number as a point P(x,y) in the x-y plane, with the x-axis

referred to as the real axis, and the y-axis referred to as the imaginary axis.

Thus, a complex number represented in the form x+iy is said to be in its

Cartesian representation. An alternative Cartesian representation is the ordered

pair z = (x,y). A complex number can also be represented in polar coordinates

iθ

(polar representation) as z = re = r⋅cosθ + i r⋅sinθ, where r = |z|

x²+ y²

is the magnitude of the complex number z, and θ = Arg(z) =

arctan(y/x) is the argument of the complex number z. The relationship between

the Cartesian and polar representation of complex numbers is given by the

iθ

Euler formula: e = cos θ + i sin θ. The complex conjugate of a complex

iθ

-iθ

number z = x + iy = re , is⎯z = x – iy = re . The complex conjugate of i can

be thought of as the reflection of z about the real (x) axis. Similarly, the

iθ

negative of z, –z = -x-iy = - re , can be thought of as the reflection of z about

the origin.

Setting the calculator to COMPLEX mode

When working with complex numbers it is a good idea to set the calculator to

complex mode, using the following keystrokes:

H@@CAS@ ˜˜™@ꢀ@CHK@

The COMPLEX mode will be selected if the CAS MODES screen shows the

option _Complex checked, i.e.,

Page 4-1

Press @@OK@@ , twice, to return to the stack.

Entering complex numbers

Complex numbers in the calculator can be entered in either of the two

Cartesian representations, namely, x+iy, or (x,y). The results in the calculator

will be shown in the ordered-pair format, i.e., (x,y). For example, with the

calculator in ALG mode, the complex number (3.5,-1.2), is entered as:

„Ü3.5‚í\1.2`

A complex number can also be entered in the form x+iy. For example, in ALG

mode, 3.5-1.2i is entered as:

3.5 -1.2*„¥`

The following screen results after entering these complex numbers:

In RPN mode, these numbers will be entered using the following keystrokes:

„Ü3.5‚í1.2\`

(Notice that the change-sign keystroke is entered after the number 1.2 has been

entered, in the opposite order as the ALG mode exercise).

The resulting RPN screen will be:

Page 4-2

Notice that the last entry shows a complex number in the form x+iy. This is so

because the number was entered between single quotes, which represents an

algebraic expression. To evaluate this number use the EVAL key( μ).

Once the algebraic expression is evaluated, you recover the complex number

(3.5,1.2).

Polar representation of a complex number

The result shown above represents a Cartesian (rectangular) representation of

the complex number 3.5-1.2i. A polar representation is possible if we change

the coordinate system to cylindrical or polar, by using function CYLIN. You can

find this function in the catalog (‚N). Changing to polar shows the resultꢁ

in RPN mode:

For this result, it is in standard notation and the angular measure is set to

radians (you can always change to radians by using function RAD). The result

shown above represents a magnitude, 3.7, and an angle 0.33029…. The

angle symbol (∠) is shown in front of the angle measure.

Return to Cartesian or rectangular coordinates by using function RECT

(available in the catalog, ‚N). A complex number in polar

iθ

representation is written as z = r⋅e . You can enter this complex number into

the calculator by using an ordered pair of the form (r, ∠θ). The angle symbol

(∠) can be entered as ~‚6. For example, the complex number z =

1.5i

5.2e , can be entered as follows (the figures show the stack, before and after

entering the number):

Because the coordinate system is set to rectangular (or Cartesian), the

calculator automatically converts the number entered to Cartesian coordinates,

i.e., x = r cos θ, y = r sin θ, resulting, for this case, in (0.3678…, 5.18…).

Page 4-3

On the other hand, if the coordinate system is set to cylindrical coordinates (use

CYLIN), entering a complex number (x,y), where x and y are real numbers, will

produce a polar representation. For example, in cylindrical coordinates, enter

the number (3.,2.). The figure below shows the RPN stack, before and after

entering this number:

Simple operations with complex numbers

Complex numbers can be combined using the four fundamental operations

(+-*/). The results follow the rules of algebra with the caveat that

i = -1. Operations with complex numbers are similar to those with real

numbers. For example, with the calculator in ALG mode and the CAS set to

Complex, we’ll attempt the following sum: (3+5i) + (6-3i):

Notice that the real parts (3+6) and imaginary parts (5-3) are combined

together and the result given as an ordered pair with real part 9 and imaginary

part 2. Try the following operations on your own:

(5-2i) - (3+4i) = (2,-6)

(3-i)·(2-4i) = (2,-14)

(5-2i)/(3+4i) = (0.28,-1.04)

1/(3+4i) = (0.12, -0.16)

Notes:

The product of two numbers is represented by: (x +iy )(x +iy ) = (x x - y y )

1 2

+ i (x y + x y ).

1 2

2 1

The division of two complex numbers is accomplished by multiplying both

numerator and denominator by the complex conjugate of the denominator,

i.e.,

Page 4-4

x₁+ iy₁

x₁+ iy₁x₂− iy₂x₁x₂+ y₁y₂

x₂y₁− x₁y₂

⋅

+ i ⋅

x₂²+ y₂²

x₂+ iy₂x₂+ iy₂x₂− iy₂

Thus, the inverse function INV (activated with the Y key) is defined as

x − iy

⋅

+ i ⋅

x²+ y²

x + iy x + iy x − iy

Changing sign of a complex number

Changing the sign of a complex number can be accomplished by using the

\ key, e.g., -(5-3i) = -5 + 3i

Entering the unit imaginary number

To enter the unit imaginary number type : „¥

Notice that the number i is entered as the ordered pair (0,1) if the CAS is set to

APPROX mode. In EXACT mode, the unit imaginary number is entered as i.

Other operations

Operations such as magnitude, argument, real and imaginary parts, and

complex conjugate are available through the CMPLX menus detailed later.

The CMPLX menus

There are two CMPLX (CoMPLeX numbers) menus available in the calculator.

One is available through the MTH menu (introduced in Chapter 3) and one

directly into the keyboard (‚ß). The two CMPLX menus are presented

next.

Page 4-5

CMPLX menu through the MTH menu

Assuming that system flag 117 is set to CHOOSE boxes (see Chapter 2), the

CMPLX sub-menu within the MTH menu is accessed by using: „´9

@@OK@@ . The following sequence of screen shots illustrates these steps:

The first menu (options 1 through 6) shows the following functions:

RE(z) : Real part of a complex number

IM(z) : Imaginary part of a complex number

C→R(z) : Takes a complex number (x,y) and separates it into its real and

imaginary parts

R→C(x,y): Forms the complex number (x,y) out of real numbers x and y

ABS(z) : Calculates the magnitude of a complex number or the absolute value

of a real number.

ARG(z): Calculates the argument of a complex number.

The remaining options (options 7 through 10) are the following:

SIGN(z) : Calculates a complex number of unit magnitude as z/|z|.

NEG : Changes the sign of z

CONJ(z): Produces the complex conjugate of z

Examples of applications of these functions are shown next. Recall that, for ALG

mode, the function must precede the argument, while in RPN mode, you enter

the argument first, and then select the function. Also, recall that you can get

these functions as soft menus by changing the setting of system flag 117 (See

Chapter 3).

Page 4-6

This first screen shows functions RE, IM, and CꢀR. Notice that the last function

returns a list {3. 5.} representing the real and imaginary components of the

complex number:

The following screen shows functions RꢀC, ABS, and ARG. Notice that the

ABS function gets translated to |3.+5.·i|, the notation of the absolute value.

Also, the result of function ARG, which represents an angle, will be given in the

units of angle measure currently selected. In this example, ARG(3.+5.·i) =

1.0303… is given in radians.

In the next screen we present examples of functions SIGN, NEG (which shows

up as the negative sign - ), and CONJ.

CMPLX menu in keyboard

A second CMPLX menu is accessible by using the right-shift option associated

with the 1 key, i.e., ‚ß. With system flag 117 set to CHOOSE boxes,

the keyboard CMPLX menu shows up as the following screens:

Page 4-7

The resulting menu include some of the functions already introduced in the

previous section, namely, ARG, ABS, CONJ, IM, NEG, RE, and SIGN. It also

includes function i which serves the same purpose as the keystroke combination

„¥, i.e., to enter the unit imaginary number i in an expression.

The keyboard-base CMPLX menu is an alternative to the MTH-based CMPLX

menu containing the basic complex number functions. Try the examples shown

earlier using the keyboard-based CMPLX menu for practice.

Functions applied to complex numbers

Many of the keyboard-based functions defined in Chapter 3 for real numbers,

e.g., SQ, ,LN, e , LOG, 10 , SIN, COS, TAN, ASIN, ACOS, ATAN, can be

applied to complex numbers. The result is another complex number, as

illustrated in the following examples. To apply this functions use the same

procedure as presented for real numbers (see Chapter 3).

Page 4-8

Note: When using trigonometric functions and their inverses with complex

numbers the arguments are no longer angles. Therefore, the angular measure

selected for the calculator has no bearing in the calculation of these functions

with complex arguments. To understand the way that trigonometric functions,

and other functions, are defined for complex numbers consult a book on

complex variables.

Functions from the MTH menu

The hyperbolic functions and their inverses, as well as the Gamma, PSI, and Psi

functions (special functions) were introduced and applied to real numbers in

Chapter 3. These functions can also be applied to complex numbers by

following the procedures presented in Chapter 3. Some examples are shown

below:

The following screen shows that functions EXPM and LNP1 do not apply to

complex numbers. However, functions GAMMA, PSI, and Psi accept complex

numbers:

Function DROITE: equation of a straight line

Function DROITE takes as argument two complex numbers, say, x +iy and

x +iy , and returns the equation of the straight line, say, y = a+bx, that

contains the points (x ,y ) and (x ,y ). For example, the line between points

A(5,-3) and B(6,2) can be found as follows (example in Algebraic mode):

Page 4-9

Function DROITE is found in the command catalog (‚N).

Using EVAL(ANS(1)) simplifies the result to:

Page 4-10

Chapter 5

Algebraic and arithmetic operations

An algebraic object, or simply, algebraic, is any number, variable name or

algebraic expression that can be operated upon, manipulated, and combined

according to the rules of algebra. Examples of algebraic objects are the

following:

•

A number: 12.3, 15.2_m, ‘π’, ‘e’, ‘i’

A variable name: ‘a’, ‘ux’, ‘width’, etc.

An expression: ‘p*D^2/4’,’f*(L/D)*(V^2/(2*g))’

An equation: ‘Q=(Cu/n)*A(y)*R(y)^(2/3)*So^0.5’

Entering algebraic objects

Algebraic objects can be created by typing the object between single quotes

directly into stack level 1 or by using the equation writer ‚O. For

example, to enter the algebraic object ‘π*D^2/4’ directly into stack level 1

use:³„ì*~dQ2/4`. The resulting screen is

shown next for both the ALG mode (left-hand side) and the RPN mode (right-

hand side):

An algebraic object can also be built in the Equation Writer and then sent to

the stack. The operation of the Equation Writer was described in Chapter 2.

As an exercise, build the following algebraic object in the Equation Writer:

After building the object, press to show it in the stack (ALG and RPN modes

shown below):

Simple operations with algebraic objects

Algebraic objects can be added, subtracted, multiplied, divided (except by

zero), raised to a power, used as arguments for a variety of standard functions

Page 5-1

(exponential, logarithmic, trigonometry, hyperbolic, etc.), as you would any real

or complex number. To demonstrate basic operations with algebraic objects,

let’s create a couple of objects, say ‘π*R^2’ and ‘g*t^2/4’, and store them in

variables A1 and A2 (See Chapter 2 to learn how to create variables and store

values in them). Here are the keystrokes for storing variables A1 in ALG

mode:³„ì*~rQ2™ K ~a1 `, resulting in:

The keystrokes corresponding to RPN mode are: ³„ì*~r

Q2`~a1 K

After storing the variable A2 and pressing the key, the screen will show the

variables as follows:

In ALG mode, the following keystrokes will show a number of operations with

the algebraics contained in variables @@A1@@ and @@A2@@ (press J to recover

variable menu):

@@A1@@ + @@A2@@ `

@@A1@@ - @@A2@@ `

@@A1@@ * @@A2@@ `

@@A1@@ / @@A2@@ `

Page 5-2

‚¹@@A1@@

„¸@@A2@@

The same results are obtained in RPN mode if using the following keystrokes:

@@A1@@ @@A2@@ +μ

@@A1@@ @@A2@@ *μ

@@A1@@ [ ‚¹ μ

@@A1@@ @@A2@@ -μ

@@A1@@ @@A2@@ /μ

@@A2@@ [„¸ μ

Functions in the ALG menu

The ALG (Algebraic) menu is available by using the keystroke sequence

‚× (associated with the 4 key). With system flag 117 set to

CHOOSE boxes, the ALG menu shows the following functions:

Rather than listing the description of each function in this manual, the user is

invited to look up the description using the calculator’s help facility: I L

@HELP@ `. To locate a particular function, type the first letter of the function.

For example, for function COLLECT, we type ~c, then use the up and down

arrow keys, —˜, to locate COLLECT within the help window.

To complete the operation press @@OK@@. Here is the help screen for function

COLLECT:

Page 5-3

We notice that, at the bottom of the screen, the line See: EXPAND FACTOR

suggests links to other help facility entries, the functions EXPAND and FACTOR.

To move directly to those entries, press the soft menu key @SEE1! for EXPAND, and

@SEE2! for FACTOR. Pressing @SEE1!, for example, shows the following

information for EXPAND:

Help facility

A help facility accessible via TOOL NEXT CASCMD allows you to browse

through all the CAS commands. It provides not only information on each

command, but also provides an example of its application. This example can

be copied onto your stack by pressing the @ECHO! soft menu key. For example, for

the EXPAND entry shown above, press the @ECHO! soft menu key to get the

following example copied to the stack (press ` to execute the command):

We leave for the user to explore the list of CAS functions available. Here are a

couple of examples:

The help facility will show the following information on the commands:

COLLECT:

EXPAND:

Page 5-4

FACTOR:

LNCOLLECT:

PARTFRAC:

SUBST:

LIN:

SOLVE:

TEXPAND:

Note: Recall that, to use these, or any other functions in the RPN mode,

you must enter the argument first, and then the function. For example, the

example for TEXPAND, in RPN mode will be set up as:

³„¸+~x+~y`

At this point, select function TEXPAND from menu ALG (or directly from the

catalog ‚N), to complete the operation.

Page 5-5

Other forms of substitution in algebraic expressions

Functions SUBST, shown above, is used to substitute a variable in an expression.

A second form of substitution can be accomplished by using the ‚¦

(associated with the I key). For example, in ALG mode, the following entry will

substitute the value x = 2 in the expression x+x . The figure to the left shows the

way to enter the expression (the substituted value, x=2, must be enclosed in

parentheses) before pressing `. After the ` key is pressed, the result is

shown in the right-hand figure:

In RPN mode, this can be accomplished by entering first the expression where

the substitution will be performed (x+x ), followed by a list (see Chapter 8)

containing the substitution variable, an space, and the value to be substituted,

i.e., {x 2}. The final step is to press the keystroke combination: ‚¦.

The required keystrokes are the following:

³~„x+~„xQ2`

„ä~„x#2` ‚¦`

In ALG mode, substitution of more than one variable is possible as illustrated in

the following example (shown before and after pressing `)

In RPN mode, it is also possible to substitute more than one variable at a time,

as illustrated in the example below. Recall that RPN mode uses a list of variable

names and values for substitution.

Page 5-6

A different approach to substitution consists in defining the substitution

expressions in calculator variables and placing the name of the variables in the

original expression. For example, in ALG mode, store the following variables:

Then, enter the expression A+B:

The last expression entered is automatically evaluated after pressing the `

key, producing the result shown above.

Operations with transcendental functions

The calculator offers a number of functions that can be used to replace

expressions containing logarithmic, exponential, trigonometric, and hyperbolic

functions in terms of trigonometric identities or in terms of exponential functions.

The menus containing functions to replace trigonometric functions can be

obtained directly from the keyboard by pressing the right-shift key followed by

the 8 key, i.e., ‚Ñ. The combination of this key with the left-shift key, i.e.,

‚ Ð, produces a menu that lets you replace expressions in terms of

exponential or natural logarithm functions. In the next sections we cover those

menus in more detail.

Expansion and factoring using log-exp functions

The „Ð produces the following menu:

Information and examples on these commands are available in the help facility

of the calculator. Some of the command listed in the EXP&LN menu, i.e., LIN,

Page 5-7

LNCOLLECT, and TEXPAND are also contained in the ALG menu presented

earlier. Functions LNP1 and EXPM were introduced in menu HYPERBOLIC,

under the MTH menu (See Chapter 2). The only remaining function is EXPLN.

Its description is shown in the left-hand side, the example from the help facility is

shown to the right:

Expansion and factoring using trigonometric functions

The TRIG menu, triggered by using ‚Ñ, shows the following functions:

These functions allow to simplify expressions by replacing some category of

trigonometric functions for another one. For example, the function ACOS2S

allows to replace the function arccosine (acos(x)) with its expression in terms of

arcsine (asin(x)).

Description of these commands and examples of their applications are

available in the calculator’s help facility (IL@HELP). The user is invited to

explore this facility to find information on the commands in the TRIG menu.

Notice that the first command in the TRIG menu is the HYPERBOLIC menu,

whose functions were introduced in Chapter 2.

Page 5-8

Functions in the ARITHMETIC menu

The ARITHMETIC menu contains a number of sub-menus for specific

applications in number theory (integers, polynomials, etc.), as well as a number

of functions that apply to general arithmetic operations. The ARITHMETIC

menu is triggered through the keystroke combination „Þ (associated with

the 1 key). With system flag 117 set to CHOOSE boxes, „Þ shows

the following menu:

Out of this menu list, options 5 through 9 (DIVIS, FACTORS, LGCD, PROPFRAC,

SIMP2) correspond to common functions that apply to integer numbers or to

polynomials. The remaining options (1. INTEGER, 2. POLYNOMIAL, 3.

MODULO, and 4. PERMUTATION) are actually sub-menus of functions that

apply to specific mathematical objects. This distinction between sub-menus

(options 1 through 4) and plain functions (options 5 through 9) is made clear

when system flag 117 is set to SOFT menus. Activating the ARITHMETIC menu

(„Þ ), under these circumstances, produces:

Following, we present the help facility entries for the functions of options 5

through 9 in the ARITHMETIC menu (IL@HELP):

DIVIS:

FACTORS:

Page 5-9

LGCD (Greatest Common Denominator): PROPFRAC (proper fraction)

SIMP2:

The functions associated with the ARITHMETIC submenus: INTEGER,

POLYNOMIAL, MODULO, and PERMUTATION, are the following:

INTEGER menu

EULER

Number of integers < n, co -prime with n

Solves au + bv = c, with a,b,c = integers

n-th Bernoulli number

IABCUV

IBERNOULLI

ICHINREM

IDIV2

IEGCD

IQUOT

Chinese reminder for integers

Euclidean division of two integers

Returns u,v, such that au + bv = gcd(a,b)

Euclidean quotient of two integers

IREMAINDER Euclidean remainder of two integers

ISPRIME?

NEXTPRIME

PA2B2

Test if an integer number is prime

Next prime for a given integer number

Prime number as square norm of a complex number

Previous prime for a given integer number

PREVPRIME

POLYNOMIAL menu

ABCUV

CHINREM

Bézout polynomial equation (au+bv=c)

Chinese remainder for polynomials

CYCLOTOMIC n-th cyclotomic polynomial

DIV2

EGDC

Euclidean division of two polynomials

Returns u,v, from au+bv=gcd(a,b)

Page 5-10

FACTOR

FCOEF

FROOTS

GCD

HERMITE

HORNER

LAGRANGE

LCM

LEGENDRE

PARTFRAC

PCOEF

Factorizes an integer number or a polynomial

Generates fraction given roots and multiplicity

Returns roots and multiplicity given a fraction

Greatest common divisor of 2 numbers or polynomials

n-th degree Hermite polynomial

Horner evaluation of a polynomial

Lagrange polynomial interpolation

Lowest common multiple of 2 numbers or polynomials

n-th degree Legendre polynomial

Partial-fraction decomposition of a given fraction

(help-facility entry missing)

PTAYL

QUOT

Returns Q(x-a) in Q(x-a) = P(x), Taylor polynomial

Euclidean quotient of two polynomials

RESULTANT

REMAINDER

STURM

Determinant of the Sylvester matrix of 2 polynomials

Euclidean reminder of two polynomials

Sturm sequences for a polynomial

STURMAB

Sign at low bound and number of zeros between bounds

MODULO menu

ADDTMOD

DIVMOD

Add two expressions modulo current modulus

Divides 2 polynomials modulo current modulus

Euclidean division of 2 polynomials with modular

coefficients

DIV2MOD

EXPANDMOD

FACTORMOD

GCDMOD

INVMOD

MOD

MODSTO

MULTMOD

Expands/simplify polynomial modulo current modulus

Factorize a polynomial modulo current modulus

GCD of 2 polynomials modulo current modulus

inverse of integer modulo current modulus

(not entry available in the help facility)

Changes modulo setting to specified value

Multiplication of two polynomials modulo current

modulus

POWMOD

SUBTMOD

Raises polynomial to a power modulo current modulus

Subtraction of 2 polynomials modulo current modulus

Page 5-11

Applications of the ARITHMETIC menu

This section is intended to present some of the background necessary for

application of the ARITHMETIC menu functions. Definitions are presented next

regarding the subjects of polynomials, polynomial fractions and modular

arithmetic. The examples presented below are presented independently of the

calculator setting (ALG or RPN)

Modular arithmetic

Consider a counting system of integer numbers that periodically cycles back on

itself and starts again, such as the hours in a clock. Such counting system is

called a ring. Because the number of integers used in a ring is finite, the

arithmetic in this ring is called finite arithmetic. Let our system of finite integer

numbers consists of the numbers 0, 1, 2, 3, …, n-1, n. We can also refer to the

arithmetic of this counting system as modular arithmetic of modulus n. In the

case of the hours of a clock, the modulus is 12. (If working with modular

arithmetic using the hours in a clock, however, we would have to use the integer

numbers 0, 1, 2, 3, …, 10, 11, rather than 1, 2, 3,…,11, 12).

Operations in modular arithmetic

Addition in modular arithmetic of modulus n, which is a positive integer, follow

the rules that if j and k are any two nonnegative integer numbers, both smaller

than n, if j+k≥ n, then j+k is defined as j+k-n. For example, in the case of the

clock, i.e., for n = 12, 6+9 “=” 3. To distinguish this ‘equality’ from infinite

arithmetic equalities, the symbol ≡ is used in place of the equal sign, and the

relationship between the numbers is referred to as a congruence rather than an

equality. Thus, for the previous example we would write 6+9 ≡ 3 (mod 12),

and read this expression as “six plus nine is congruent to three, modulus

twelve.” If the numbers represent the hours since midnight, for example, the

congruence 6+9 ≡ 3 (mod 12), can be interpreted as saying that “six hours

past the ninth hour after midnight will be three hours past noon.” Other sums

that can be defined in modulus 12 arithmetic are: 2+5 ≡ 7 (mod 12); 2+10 ≡

0 (mod 12); 7+5 ≡ 0 (mod 12); etcetera.

The rule for subtraction will be such that if j – k < 0, then j-k is defined as j-k+n.

Therefore, 8-10 ≡ 2 (mod 12), is read “eight minus ten is congruent to two,

modulus twelve.” Other examples of subtraction in modulus 12 arithmetic

would be 10-5 ≡ 5 (mod 12); 6-9 ≡ 9 (mod 12); 5 – 8 ≡ 9 (mod 12); 5 –10 ≡

7 (mod 12); etcetera.

Multiplication follows the rule that if j⋅k > n, so that j⋅k = m⋅n + r, where m and r

are nonnegative integers, both less than n, then j⋅k ≡ r (mod n). The result of

Page 5-12

multiplying j times k in modulus n arithmetic is, in essence, the integer

remainder of j⋅k/n in infinite arithmetic, if j⋅k>n. For example, in modulus 12

arithmetic we have 7⋅3 = 21 = 12 + 9, (or, 7⋅3/12 = 21/12 = 1 + 9/12, i.e.,

the integer reminder of 21/12 is 9). We can now write 7⋅3 ≡ 9 (mod 12), and

read the latter result as “seven times three is congruent to nine, modulus twelve.”

The operation of division can be defined in terms of multiplication as follows, r/

k ≡ j (mod n), if, j⋅k ≡ r (mod n). This means that r must be the remainder of

j⋅k/n. For example, 9/7 ≡ 3 (mod 12), because 7⋅3 ≡ 9 (mod 12). Some

divisions are not permitted in modular arithmetic. For example, in modulus 12

arithmetic you cannot define 5/6 (mod 12) because the multiplication table of

6 does not show the result 5 in modulus 12 arithmetic. This multiplication table

is shown below:

6*0 (mod 12)

6*1 (mod 12)

6*2 (mod 12)

6*3 (mod 12)

6*4 (mod 12)

6*5 (mod 12)

6*6 (mod 12)

6*7 (mod 12)

6*8 (mod 12)

6*9 (mod 12)

6*10 (mod 12)

6*11 (mod 12)

Formal definition of a finite arithmetic ring

The expression a ≡ b (mod n) is interpreted as “a is congruent to b, modulo n,”

and holds if (b-a) is a multiple of n. With this definition the rules of arithmetic

simplify to the following:

a ≡ b (mod n) and c ≡ d (mod n),

then

a+c ≡ b+d (mod n),

a-c ≡ b - d (mod n),

a×c ≡ b×d (mod n).

For division, follow the rules presented earlier. For example, 17 ≡ 5 (mod 6),

and 21 ≡ 3 (mod 6). Using these rules, we can write:

17 + 21 ≡ 5 + 3 (mod 6) => 38 ≡ 8 (mod 6) => 38 ≡ 2 (mod 6)

17 – 21 ≡ 5 - 3 (mod 6) =>

-4 ≡ 2 (mod 6)

17 × 21 ≡ 5 × 3 (mod 6) => 357 ≡ 15 (mod 6) => 357 ≡ 3 (mod 6)

Page 5-13

Notice that, whenever a result in the right-hand side of the “congruence” symbol

produces a result that is larger than the modulo (in this case, n = 6), you can

always subtract a multiple of the modulo from that result and simplify it to a

number smaller than the modulo. Thus, the results in the first case 8 (mod 6)

simplifies to 2 (mod 6), and the result of the third case, 15 (mod 6) simplifies to

3 (mod 6). Confusing? Well, not if you let the calculator handle those

operations. Thus, read the following section to understand how finite arithmetic

rings are operated upon in your calculator.

Finite arithmetic rings in the calculator

All along we have defined our finite arithmetic operation so that the results are

always positive. The modular arithmetic system in the calculator is set so that

the ring of modulus n includes the numbers -n/2+1, …,-1, 0, 1,…,n/2-1, n/2,

if n is even, and –(n-1)/2, -(n-3)/2,…,-1,0,1,…,(n-3)/2, (n-1)/2, if n is odd.

For example, for n = 8 (even), the finite arithmetic ring in the calculator includes

the numbers: (-3,-2,-1,0,1,3,4), while for n = 7 (odd), the corresponding

calculator’s finite arithmetic ring is given by (-3,-2,-1,0,1,2,3).

Modular arithmetic in the calculator

To launch the modular arithmetic menu in the calculator select the MODULO

sub-menu within the ARITHMETIC menu („Þ). The available menu

includes functions: ADDTMOD, DIVMOD, DIV2MOD, EXPANDMOD,

FACTORMOD, GCDMOD, INVMOD, MOD, MODSTO, MULTMOD,

POWMOD, and SUBTMOD. Brief descriptions of these functions were provided

in an earlier section. Next we present some applications of these functions.

Setting the modulus (or MODULO)

The calculator contains a variable called MODULO that is placed in the

{HOME CASDIR} directory and will store the magnitude of the modulus to be

used in modular arithmetic.

The default value of MODULO is 13. To change the value of MODULO, you

can either store the new value directly in the variable MODULO in the sub-

directory {HOME CASDIR} Alternatively, you can store a new MODULO value

by using function MODSTO.

Modular arithmetic operations with numbers

To add, subtract, multiply, divide, and raise to a power using modular

arithmetic you will use the functions ADDTMOD, SUBTMOD, MULTMOD,

DIV2MOD and DIVMOD (for division), and POWMOD. In RPN mode you

need to enter the two numbers to operate upon, separated by an [ENTER] or an

Page 5-14

[SPC] entry, and then press the corresponding modular arithmetic function. For

example, using a modulus of 12, try the following operations:

ADDTMOD examples

6+5 ≡ -1 (mod 12)

6+6 ≡ 0 (mod 12)

6+7 ≡ 1 (mod 12)

5 –10 ≡ -5 (mod 12)

3⋅2 ≡ 6 (mod 12)

11+5 ≡ 4 (mod 12)

8+10 ≡ -6 (mod 12)

SUBTMOD examples

5 - 7 ≡ -2 (mod 12)

8 – 4 ≡ 4 (mod 12)

11 – 8 ≡ 3 (mod 12) 8 - 12 ≡ -4 (mod 12)

MULTMOD examples

6⋅8 ≡ 0 (mod 12)

9⋅8 ≡ 0 (mod 12)

11⋅3 ≡ -3 (mod 12)

5⋅6 ≡ 6 (mod 12)

DIVMOD examples

12/3 ≡ 4 (mod 12)

25/5 ≡ 5 (mod 12)

66/6 ≡ -1 (mod 12)

12/8 (mod 12) does not exist

64/13 ≡ 4 (mod 12)

DIV2MOD examples

2/3 (mod 12) does not exist

26/12 (mod 12) does not exist

125/17 (mod 12) ½ 1 with remainder = 0

68/7 ½ -4 (mod 12) with remainder = 0

7/5 ½ -1 (mod 12) with remainder = 0

Note: DIVMOD provides the quotient of the modular division j/k (mod n),

while DIMV2MOD provides no only the quotient but also the remainder of the

modular division j/k (mod n).

POWMOD examples

2 ≡ -4 (mod 12)

3 ≡ 3 (mod 12)

5¹⁰≡ 1 (mod 12)

9 ≡ -3 (mod 12)

11 ≡ 1 (mod 12)

6 ≡ 0 (mod 12)

In the examples of modular arithmetic operations shown above, we have used

numbers that not necessarily belong to the ring, i.e., numbers such as 66, 125,

17, etc. The calculator will convert those numbers to ring numbers before

Page 5-15

operating on them. You can also convert any number into a ring number by

using the function EXPANDMOD. For example,

EXPANDMOD(125) ≡ 5 (mod 12)

EXPANDMOD(17) ≡ 5 (mod 12)

EXPANDMOD(6) ≡ 6 (mod 12)

The modular inverse of a number

Let a number k belong to a finite arithmetic ring of modulus n, then the modular

inverse of k, i.e., 1/k (mod n), is a number j, such that j⋅k ≡ 1 (mod n). The

modular inverse of a number can be obtained by using the function INVMOD

in the MODULO sub-menu of the ARITHMETIC menu. For example, in modulus

12 arithmetic:

1/6 (mod 12) does not exist.

1/7 ≡ -5 (mod 12)

1/11 ≡ -1 (mod 12)

1/5 ≡ 5 (mod 12)

1/3 (mod 12) does not exist.

The MOD operator

The MOD operator is used to obtain the ring number of a given modulus

corresponding to a given integer number. On paper this operation is written

as m mod n = p, and is read as “m modulo n is equal to p”. For example, to

calculate 15 mod 8, enter:

ꢀ ALG mode:

15 MOD 8`

ꢀ RPN mode:

15`8` MOD

The result is 7, i.e., 15 mod 8 = 7. Try the following exercises:

18 mod 11 = 7

23 mod 17 = 6

23 mod 2 =1

34 mod 6 = 4

40 mod 13 = 1

One practical application of the MOD function for programming purposes is to

determine when an integer number is odd or even, since n mod 2 = 0, if n is

even, and n mode 2 = 1, if n is odd. It can also be used to determine when an

integer m is a multiple of another integer n, for if that is the case m mod n = 0.

Page 5-16

Note: Refer to the help facility in the calculator for description and examples

on other modular arithmetic. Many of these functions are applicable to

polynomials. For information on modular arithmetic with polynomials please

refer to a textbook on number theory.

Polynomials

Polynomials are algebraic expressions consisting of one or more terms

containing decreasing powers of a given variable. For example, ‘X^3+2*X^2-

3*X+2’ is a third-order polynomial in X, while ‘SIN(X)^2-2’ is a second-order

polynomial in SIN(X). A listing of polynomial-related functions in the

ARITHMETIC menu was presented earlier. Some general definitions on

polynomials are provided next. In these definitions A(X), B(X), C(X), P(X), Q(X),

U(X), V(X), etc., are polynomials.

ꢀ Polynomial fraction: a fraction whose numerator and denominator are

polynomials, say, C(X) = A(X)/B(X)

ꢀ Roots, or zeros, of a polynomial: values of X for which P(X) = 0

ꢀ Poles of a fraction: roots of the denominator

ꢀ Multiplicity of roots or poles: the number of times a root shows up, e.g., P(X)

= (X+1) (X-3) has roots {-1, 3} with multiplicities {2,1}

ꢀ Cyclotomic polynomial (P (X)): a polynomial of order EULER(n) whose roots

are the primitive n-th roots of unity, e.g., P (X) = X+1, P (X) = X +1

ꢀ Bézout’s polynomial equation: A(X) U(X) + B(X)V(X) = C(X)

Specific examples of polynomial applications are provided next.

Modular arithmetic with polynomials

The same way that we defined a finite-arithmetic ring for numbers in a previous

section, we can define a finite-arithmetic ring for polynomials with a given

polynomial as modulo. For example, we can write a certain polynomial P(X) as

P(X) = X (mod X ), or another polynomial Q(X) = X + 1 (mod X-2).

A polynomial, P(X) belongs to a finite arithmetic ring of polynomial modulus

M(X), if there exists a third polynomial Q(X), such that (P(X) – Q(X)) is a multiple

of M(X). We then would write: P(X) ≡ Q(X) (mod M(X)). The later expression is

interpreted as “P(X) is congruent to Q(X), modulo M(X)”.

The CHINREM function

CHINREM stands for CHINese REMainder. The operation coded in this

command solves a system of two congruences using the Chinese Remainder

Theorem. This command can be used with polynomials, as well as with integer

Page 5-17

numbers (function ICHINREM). The input consists of two vectors [expression_1,

modulo_1] and [expression_2, modulo_2]. The output is a vector containing

[expression_3, modulo_3], where modulo_3 is related to the product

(modulo_1)⋅(modulo_2). Example: CHINREM([X+1, X^2-1],[X+1,X^2]) =

[X+1,-(X^4-X^2)]

Statement of the Chinese Remainder Theorem for integers

If m , m ,…,m are natural numbers every pair of which are relatively prime,

and a , a , …, a are any integers, then there is an integer x that

simultaneously satisfies the congruences: x ≡ a (mod m ), x ≡ a (mod m ),

…, x ≡ a (mod m ). Additionally, if x = a is any solution then all other solutions

are congruent to a modulo equal to the product m ⋅m ⋅ … m .

The EGCD function

EGCD stands for Extended Greatest Common Divisor. Given two polynomials,

A(X) and B(X), function EGCD produces the polynomials C(X), U(X), and V(X),

so that C(X) = U(X)*A(X) + V(X)*B(X). For example, for A(X) = X^2+1, B(X) =

X^2-1, EGCD(A(X),B(X)) = {2, 1, -1}. i.e., 2 = 1*( X^2+1’)-1*( X^2-1). Also,

EGCD(‘X^3-2*X+5’,’X’) = { 5,1,-(X^2-2)}, i.e., 5 = – (X^2-2)*X + 1*(X^3-

2*X+5).

The GCD function

The function GCD (Greatest Common Denominator) can be used to obtain the

greatest common denominator of two polynomials or of two lists of polynomials

of the same length. The two polynomials or lists of polynomials will be placed

in stack levels 2 and 1 before using GCD. The results will be a polynomial or a

list representing the greatest common denominator of the two polynomials or of

each list of polynomials. Examples, in RPN mode, follow (calculator set to Exact

mode):

‘X^3-1’`’X^2-1’`GCD Results in: ‘X-1’

{‘X^2+2*X+1’,’X^3+X^2’} `{'X^3+1','X^2+1'}[[`GCD results in {'X+1'

The HERMITE function

The function HERMITE [HERMI] uses as argument an integer number, k, and

returns the Hermite polynomial of k-th degree. A Hermite polynomial, He (x) is

defined as

dⁿ

dxⁿ

/ 2

He₀=1, He_n(x) = (−1)ⁿe^x

(e^{−x / 2}), n =1,2,...

Page 5-18

An alternate definition of the Hermite polynomials is

dⁿ

dxⁿ

H₀* =1, H_n*(x) = (−1)ⁿe^x

(e^−x), n =1,2,...

where d /dx = n-th derivative with respect to x. This is the definition used in

the calculator.

Examples: The Hermite polynomials of orders 3 and 5 are given by:

HERMITE(3) = ‘8*X^3-12*X’,

And

HERMITE(5) = ‘32*x^5-160*X^3+120*X’.

The HORNER function

The function HORNER produces the Horner division, or synthetic division, of a

polynomial P(X) by the factor (X-a). The input to the function are the polynomial

P(X) and the number a. The function returns the quotient polynomial Q(X) that

results from dividing P(X) by (X-a), the value of a, and the value of P(a), in that

order. In other words, P(X) = Q(X)(X-a)+P(a). For example,

HORNER(‘X^3+2*X^2-3*X+1’,2) = {‘X^2+4*X+5’, 2, 11}. We could,

therefore, write X +2X -3X+1 = (X +4X+5)(X-2)+11. A second example:

HORNER(‘X^6-1’,-5)= {’X^5-5*X^4+25*X^3-125*X^2+625*X-3125’,-5, 15624}

i.e.,

X -1 = (X -5*X +25X -125X +625X-3125)(X+5)+15624.

The variable VX

A variable called VX exists in the calculator’s {HOME CASDIR} directory that

takes, by default, the value of ‘X’. This is the name of the preferred

independent variable for algebraic and calculus applications. Avoid using the

variable VX in your programs or equations, so as to not get it confused with the

CAS’ VX. If you need to refer to the x-component of velocity, for example, you

can use vx or Vx. For additional information on the CAS variable see

Appendix C.

The LAGRANGE function

The function LAGRANGE requires as input a matrix having two rows and n

columns. The matrix stores data points of the form [[x ,x , …, x ] [y , y , …,

y ]]. Application of the function LAGRANGE produces the polynomial

expanded from

Page 5-19

(x − x_k)

∏

k =1,k ≠ j

p (x) =

⋅ y_j.

∑

n−1

j=1

(x_j− x_k)

∏

k =1,k ≠ j

For example, for n = 2, we will write:

x − x₂

x₁− x₂

x − x₁

x₂− x₁

(y₁− y₂) ⋅ x + (y₂⋅ x₁− y₁⋅ x₂)

p₁(x) =

⋅ y₁+

⋅ y₂=

x₁− x₂

Check this result with your calculator:

LAGRANGE([[ x1,x2],[y1,y2]]) = ‘((y1-y2)*X+(y2*x1-y1*x2))/(x1-x2)’.

Other examples: LAGRANGE([[1, 2, 3][2, 8, 15]]) = ‘(X^2+9*X-6)/2’

LAGRANGE([[0.5,1.5,2.5,3.5,4.5][12.2,13.5,19.2,27.3,32.5]]) =

‘-(.1375*X^4+ -.7666666666667*X^3+ - .74375*X^2 +

1.991666666667*X-12.92265625)’.

Note: Matrices are introduced in Chapter 10.

The LCM function

The function LCM (Least Common Multiple) obtains the least common multiple

of two polynomials or of lists of polynomials of the same length. Examples:

LCM(‘2*X^2+4*X+2’ ,‘X^2-1’ ) = ‘(2*X^2+4*X+2)*(X-1)’.

LCM(‘X^3-1’,‘X^2+2*X’) = ‘(X^3-1)*( X^2+2*X)’

The LEGENDRE function

A Legendre polynomial of order n is a polynomial function that solves the

d²y

dx²

(1− x²)⋅

− 2⋅ x ⋅ + n ⋅(n +1)⋅ y = 0

differential equation

To obtain the n-th order Legendre polynomial, use LEGENDRE(n), e.g.,

LEGENDRE(3) = ‘(5*X^3-3*X)/2’

LEGENDRE(5) = ‘(63*X ^5-70*X^3+15*X)/8’

Page 5-20

The PCOEF function

Given an array containing the roots of a polynomial, the function PCOEF

generates an array containing the coefficients of the corresponding polynomial.

The coefficients correspond to decreasing order of the independent variable.

For example: PCOEF([-2,–1,0,1,1,2]) = [1. –1. –5. 5. 4. –4. 0.], which

represents the polynomial X -X -5X +5X +4X -4X.

The PROOT function

Given an array containing the coefficients of a polynomial, in decreasing order,

the function PROOT provides the roots of the polynomial. Example, from

X +5X-6 =0, PROOT([1, –5, 6]) = [2. 3.].

The PTAYL function

Given a polynomial P(X) and a number a, the function PTAYL is used to obtain

an expression Q(X-a) = P(X), i.e., to develop a polynomial in powers of (X- a).

This is also known as a Taylor polynomial, from which the name of the function,

Polynomial & TAYLor, follow:

For example, PTAYL(‘X^3-2*X+2’,2) = ‘X^3+6*X^2+10*X+6’.

In actuality, you should interpret this result to mean

‘(X-2) ^3+6*(X-2) ^2+10*(X-2) +6’.

Let’s check by using the substitution: ‘X = x – 2’. We recover the original

polynomial, but in terms of lower-case x rather than upper-case x.

The QUOT and REMAINDER functions

The functions QUOT and REMAINDER provide, respectively, the quotient Q(X)

and the remainder R(X), resulting from dividing two polynomials, P (X) and

P (X). In other words, they provide the values of Q(X) and R(X) from P (X)/P (X)

= Q(X) + R(X)/P (X). For example,

QUOT(X^3-2*X+2, X-1) = X^2+X-1

REMAINDER(X^3-2*X+2, X-1) = 1.

Thus, we can write: (X -2X+2)/(X-1) = X +X-1 + 1/(X-1).

Page 5-21

Note: you could get the latter result by using PROPFRAC:

PROPFRAC(‘(X^3-2*X+2)/(X-1)’) = ‘X^2+X-1 + 1/(X-1)’.

The EPSX0 function and the CAS variable EPS

The variable ε (epsilon) is typically used in mathematical textbooks to represent

a very small number. The calculator’s CAS creates a variable EPS, with default

-10

value 0.0000000001 = 10 , when you use the EPSX0 function. You can

change this value, once created, if you prefer a different value for EPS. The

function EPSX0, when applied to a polynomial, will replace all coefficients

whose absolute value is less than EPS with a zero. Function EPSX0 is not

available in the ARITHMETIC menu, it must be accessed from the function

catalog (N). Example:

EPSX0(‘X^3-1.2E-12*X^2+1.2E-6*X+6.2E-11)=

‘X^3-0*X^2+.0000012*X+0’.

With μ:

‘X^3+.0000012*X’.

The PEVAL function

The functions PEVAL (Polynomial EVALuation) can be used to evaluate a

n-1

polynomial p(x) = a ⋅x +a ⋅x + …+ a ⋅x +a ⋅x+ a , given an array of

n-1

coefficients [a , a , … a , a , a ] and a value of x . The result is the

n-1

evaluation p(x ). Function PEVAL is not available in the ARITHMETIC menu, it

must be accessed from the function catalog (‚N). Example:

PEVAL([1,5,6,1],5) = 281.

The TCHEBYCHEFF function

The function TCHEBYCHEFF(n) generates the Tchebycheff (or Chebyshev)

polynomial of the first kind, order n, defined as T (X) = cos(n⋅arccos(X)). If the

integer n is negative (n < 0), the function TCHEBYCHEFF(n) generates the

Tchebycheff polynomial of the second kind, order n, defined as T (X) =

sin(n⋅arccos(X))/sin(arccos(X)). Examples:

TCHEBYCHEFF(3) = 4*X^3-3*X

TCHEBYCHEFF(-3) = 4*X^2-1

Page 5-22

Fractions

Fractions can be expanded and factored by using functions EXPAND and

FACTOR, from the ALG menu (‚×). For example:

EXPAND(‘(1+X)^3/((X-1)*(X+3))’) = ‘(X^3+3*X^2+3*X+1)/(X^2+2*X-3)’

EXPAND(‘(X^2)*(X+Y)/(2*X-X^2)^2)’) = ‘(X+Y)/(X^2-4*X+4)’

EXPAND(‘X*(X+Y)/(X^2-1)’) = ‘(X^2+Y*X)/(X^2-1)’

EXPAND(‘4+2*(X-1)+3/((X-2)*(X+3))-5/X^2’) =

[[[‘(2*X^5+4*X^4-10*X^3-14*X^2-5*X+30)/(X^4+X^3-6*X^2)’

FACTOR(‘(3*X^3-2*X^2)/(X^2-5*X+6)’) = ‘X^2*(3*X-2)/((X-2)*(X-3))’

FACTOR(‘(X^3-9*X)/(X^2-5*X+6)’ ) = ‘X*(X+3)/(X-2)’

FACTOR(‘(X^2-1)/(X^3*Y-Y)’) = ‘(X+1)/((X^2+X+1)*Y)’

The SIMP2 function

Functions SIMP2 and PROPFRAC are used to simplify a fraction and to produce

a proper fraction, respectively. Function SIMP2 takes as arguments two

numbers or polynomials, representing the numerator and denominator of a

rational fraction, and returns the simplified numerator and denominator. For

example: SIMP2(‘X^3-1’,’X^2-4*X+3’) = { ‘X^2+X+1’,‘X-3’}.

The PROPFRAC function

The function PROPFRAC converts a rational fraction into a “proper” fraction,

i.e., an integer part added to a fractional part, if such decomposition is

possible. For example:

PROPFRAC(‘5/4’) = ‘1+1/4’

PROPFRAC(‘(x^2+1)/x^2’) = ‘1+1/x^2’

The PARTFRAC function

The function PARTFRAC decomposes a rational fraction into the partial fractions

that produce the original fraction. For example:

PARTFRAC(‘(2*X^6-14*X^5+29*X^4-37*X^3+41*X^2-16*X+5)/(X^5-

7*X^4+11*X^3-7*X^2+10*X)’) =

‘2*X+(1/2/(X-2)+5/(X-5)+1/2/X+X/(X^2+1))’

This technique is useful in calculating integrals (see chapter on calculus) of

rational fractions.

Page 5-23

If you have the Complex mode active, the result will be:

‘2*X+(1/2/(X+i)+1/2/(X-2)+5/(X-5)+1/2/X+1/2/(X-i))’

The FCOEF function

The function FCOEF is used to obtain a rational fraction, given the roots and

poles of the fraction.

Note: If a rational fraction is given as F(X) = N(X)/D(X), the roots of the

fraction result from solving the equation N(X) = 0, while the poles result from

solving the equation D(X) = 0.

The input for the function is a vector listing the roots followed by their multiplicity

(i.e., how many times a given root is repeated), and the poles followed by their

multiplicity represented as a negative number. For example, if we want to

create a fraction having roots 2 with multiplicity 1, 0 with multiplicity 3, and -5

with multiplicity 2, and poles 1 with multiplicity 2 and –3 with multiplicity 5,

use:

FCOEF([2, 1, 0, 3, –5, 2, 1, –2, –3, –5]) = ‘(X--5)^2*X^3*(X-2)/(X+3)^5*(X-

1)^2’

If you press μ„î` (or, simply μ, in RPN mode) you will get:

‘(X^6+8*X^5+5*X^4-50*X^3)/(X^7+13*X^6+61*X^5+105*X^4-45*X^3-

297*X^2-81*X+243)’

The FROOTS function

The function FROOTS obtains the roots and poles of a fraction. As an

example, applying function FROOTS to the result produced above, will result in:

[1 –2. –3 –5. 0 3. 2 1. –5 2.]. The result shows poles followed by their

multiplicity as a negative number, and roots followed by their multiplicity as a

positive number. In this case, the poles are (1, -3) with multiplicities (2,5)

respectively, and the roots are (0, 2, -5) with multiplicities (3, 1, 2), respectively.

Another example is: FROOTS(‘(X^2-5*X+6)/(X^5-X^2)’)= [0 –2. 1 –1. 3 1. 2

1.]. i.e., poles = 0 (2), 1(1), and roots = 3(1), 2(1). If you have had Complex

Page 5-24

mode selected, then the results would be:

[0 –2. 1 –1. – ((1+i*√3)/2) –1. – ((1–i*√3)/2) –1. 3 1. 2 1.].

Step-by-step operations with polynomials and fractions

By setting the CAS modes to Step/step the calculator will show simplifications

of fractions or operations with polynomials in a step-by-step fashion. This is

very useful to see the steps of a synthetic division. The example of dividing

X ³− 5X ²+ 3X − 2

X − 2

is shown in detail in Appendix C. The following example shows a lengthier

synthetic division:

X ⁹−1

X ²−1

Note that DIV2 is available from the ARITH/POLYNOMIAL menu.

Page 5-25

The CONVERT Menu and algebraic operations

The CONVERT menu is activated by using „Ú key (the 6 key). This

menu summarizes all conversion menus in the calculator. The list of these menus

is shown next:

The functions available in each of the sub-menus are shown next.

UNITS convert menu (Option 1)

This menu is the same as the UNITS menu obtained by using ‚Û. The

applications of this menu are discussed in detail in Chapter 3.

Page 5-26

BASE convert menu (Option 2)

This menu is the same as the UNITS menu obtained by using ‚ã. The

applications of this menu are discussed in detail in Chapter 19.

TRIGONOMETRIC convert menu (Option 3)

This menu is the same as the TRIG menu obtained by using ‚Ñ. The

applications of this menu are discussed in detail in this Chapter.

MATRICES convert menu (Option 5)

This menu contains the following functions:

These functions are discussed in detail in Chapter 10.

REWRITE convert menu (Option 4)

This menu contains the following functions:

Functions IꢀR and RꢀI are used to convert a number from integer (I) to real (R),

or vice versa. Integer numbers are shown without trailing decimal points, while

real numbers representing integers will have a trailing decimal point, e.g.,

Page 5-27

Function ꢀNUM has the same effect as the keystroke combination ‚ï

(associated with the ` key). Function ꢀNUM converts a symbolic result into

its floating-point value. Function ꢀQ converts a floating-point value into a

fraction. Function ꢀQπ converts a floating-point value into a fraction of π, if a

fraction of π can be found for the number; otherwise, it converts the number to a

fraction. Examples are of these three functions are shown next.

Out of the functions in the REWRITE menu, functions DISTRIB, EXPLN,

EXP2POW, FDISTRIB, LIN, LNCOLLECT, POWEREXPAND, and SIMPLIFY apply

to algebraic expressions. Many of these functions are presented in this

Chapter. However, for the sake of completeness we present here the help-facility

entries for these functions.

DISTRIB

EXPLN

EXP2POW

FDISTRIB

Page 5-28

LIN

LNCOLLECT

SIMPLIFY

POWEREXPAND

Page 5-29

Chapter 6

Solution to single equations

In this chapter we feature those functions that the calculator provides for solving

single equations of the form f(X) = 0. Associated with the 7 key there are

two menus of equation-solving functions, the Symbolic SOLVer („Î), and

the NUMerical SoLVer (‚Ï). Following, we present some of the functions

contained in these menus. Change CAS mode to Complex for these exercises

(see Chapter 2).

Symbolic solution of algebraic equations

Here we describe some of the functions from the Symbolic Solver menu.

Activate the menu by using the keystroke combination . With system flag 117

set to CHOOSE boxes, the following menu lists will be available:

Functions DESOLVE and LDEC are used for the solution of differential equations,

the subject of a different chapter, and therefore will not be presented here.

Similarly, function LINSOLVE relates to the solution of multiple linear equations,

and it will be presented in a different chapter. Functions ISOL and SOLVE can

be used to solve for any unknown in a polynomial equation. Function SOLVEVX

solves a polynomial equation where the unknown is the default CAS variable

VX (typically set to ‘X’). Finally, function ZEROS provides the zeros, or roots, of

a polynomial. Entries for all the functions in the S.SLV menu, except ISOL, are

available through the CAS help facility

(IL@HELP ).

Function ISOL

Function ISOL(Equation, variable) will produce the solution(s) to Equation by

isolating variable. For example, with the calculator set to ALG mode, to solve

for t in the equation at -bt = 0 we can use the following:

Page 6-1

Using the RPN mode, the solution is accomplished by entering the equation in

the stack, followed by the variable, before entering function ISOL. Right before

the execution of ISOL, the RPN stack should look as in the figure to the left.

After applying ISOL, the result is shown in the figure to the right:

The first argument in ISOL can be an expression, as shown above, or an

equation. For example, in ALG mode, try:

Note: To type the equal sign (=) in an equation, use ‚Å (associated

with the \ key).

The same problem can be solved in RPN mode as illustrated below (figures

show the RPN stack before and after the application of function ISOL):

Function SOLVE

Function SOLVE has the same syntax as function ISOL, except that SOLVE can

also be used to solve a set of polynomial equations. The help-facility entry for

function SOLVE, with the solution to equation X^4 – 1 = 3 , is shown next:

The following examples show the use of function SOLVE in ALG and RPN

modes:

Page 6-2

The screen shot shown above displays two solutions. In the first one, β⁴-5β

=125, SOLVE produces no solutions { }. In the second one, β⁴- 5β = 6, SOLVE

produces four solutions, shown in the last output line. The very last solution is

not visible because the result occupies more characters than the width of the

calculator’s screen. However, you can still see all the solutions by using the

down arrow key (˜), which triggers the line editor (this operation can be

used to access any output line that is wider than the calculator’s screen):

The corresponding RPN screens for these two examples, before and after the

application of function SOLVE, are shown next:

Use of the down-arrow key (˜) in this mode will launch the line editor:

Function SOLVEVX

The function SOLVEVX solves an equation for the default CAS variable

contained in the reserved variable name VX. By default, this variable is set to

‘X’. Examples, using the ALG mode with VX = ‘X’, are shown below:

Page 6-3

In the first case SOLVEVX could not find a solution. In the second case,

SOLVEVX found a single solution, X = 2.

The following screens show the RPN stack for solving the two examples shown

above (before and after application of SOLVEVX):

The equation used as argument for function SOLVEVX must be reducible to a

rational expression. For example, the following equation will not processed by

SOLVEVX:

Function ZEROS

The function ZEROS finds the solutions of a polynomial equation, without

showing their multiplicity. The function requires having as input the expression

for the equation and the name of the variable to solve for. Examples in ALG

mode are shown next:

To use function ZEROS in RPN mode, enter first the polynomial expression, then

the variable to solve for, and then function ZEROS. The following screen shots

show the RPN stack before and after the application of ZEROS to the two

examples above:

Page 6-4

The Symbolic Solver functions presented above produce solutions to rational

equations (mainly, polynomial equations). If the equation to be solved for has

all numerical coefficients, a numerical solution is possible through the use of the

Numerical Solver features of the calculator.

Numerical solver menu

The calculator provides a very powerful environment for the solution of single

algebraic or transcendental equations. To access this environment we start the

numerical solver (NUM.SLV) by using ‚Ï. This produces a drop-down

menu that includes the following options:

Item 2. Solve diff eq.. is to be discussed in a later chapter on differential

equations. Item 4. Solve lin sys.. will be discussed in a later Chapter on

matrices. Item 6. MSLV (Multiple equation SoLVer) will be presented in the next

chapter. Following, we present applications of items 3. Solve poly.., 5. Solve

finance, and 1. Solve equation.., in that order. Appendix 1-A, at the end of

Chapter 1, contains instructions on how to use input forms with examples for the

numerical solver applications.

Notes:

1. Whenever you solve for a value in the NUM.SLV applications, the value

solved for will be placed in the stack. This is useful if you need to keep that

value available for other operations.

2. There will be one or more variables created whenever you activate some of

the applications in the NUM.SLV menu.

Page 6-5

Polynomial Equations

Using the Solve poly… option in the calculator’s SOLVE environment you can:

(1) find the solutions to a polynomial equation;

(2) obtain the coefficients of the polynomial having a number of given roots;

(3) obtain an algebraic expression for the polynomial as a function of X.

Finding the solutions to a polynomial equation

A polynomial equation is an equation of the form: a x + a

n-1

+ …+ a x +

n-1

a = 0. The fundamental theorem of algebra indicates that there are n solutions

to any polynomial equation of order n. Some of the solutions could be complex

numbers, nevertheless. As an example, solve the equation: 3s + 2s - s + 1 =

We want to place the coefficients of the equation in a vector [a ,a ,a a ].

n-1

For this example, let's use the vector [3,2,0,-1,1]. To solve for this polynomial

equation using the calculator, try the following:

‚Ï˜˜@@OK@@

Select solve poly...

„Ô3‚í2‚í 0

‚í 1\‚í1@@OK@@

@SOLVE@

Enter vector of coefficients

Solve equation

The screen will show the solution as follows:

Press ` to return to stack. The stack will show the following results in ALG

mode (the same result would be shown in RPN mode):

To see all the solutions, press the down-arrow key (˜) to trigger the line

editor:

Page 6-6

All the solutions are complex numbers: (0.432,-0.389), (0.432,0.389), (-0.766,

0.632), (-0.766, -0.632).

Note: Recall that complex numbers in the calculator are represented as

ordered pairs, with the first number in the pair being the real part, and the

second number, the imaginary part. For example, the number (0.432,-0.389),

a complex number, will be written normally as 0.432 - 0.389i, where i is the

imaginary unit, i.e., i = -1.

Note: The fundamental theorem of algebra indicates that there are n solutions

for any polynomial equation of order n. There is another theorem of algebra

that indicates that if one of the solutions to a polynomial equation with real

coefficients is a complex number, then the conjugate of that number is also a

solution. In other words, complex solutions to a polynomial equation with real

coefficients come in pairs. That means that polynomial equations with real

coefficients of odd order will have at least one real solution.

Generating polynomial coefficients given the polynomial's roots

Suppose you want to generate the polynomial whose roots are the numbers [1,

5, -2, 4]. To use the calculator for this purpose, follow these steps:

‚Ï˜˜@@OK@@

˜„Ô1‚í5

‚í2\‚í 4@@OK@@

@SOLVE@

Select solve poly...

Enter vector of roots

Solve for coefficients

Press ` to return to stack, the coefficients will be shown in the stack.

Page 6-7

Press ˜ to trigger the line editor to see all the coefficients.

Note: If you want to get a polynomial with real coefficients, but having com-

plex roots, you must include the complex roots in pairs of conjugate numbers.

To illustrate the point, generate a polynomial having the roots [1 (1,2) (1,-

2)]. Verify that the resulting polynomial has only real coefficients. Also, try

generating a polynomial with roots [1 (1,2) (-1,2)], and verify that the result-

ing polynomial will have complex coefficients.

Generating an algebraic expression for the polynomial

You can use the calculator to generate an algebraic expression for a

polynomial given the coefficients or the roots of the polynomial. The resulting

expression will be given in terms of the default CAS variable X. (The examples

below shows how you can replace X with any other variable by using the

function |.)

To generate the algebraic expression using the coefficients, try the following

example. Assume that the polynomial coefficients are [1,5,-2,4]. Use the

following keystrokes:

‚Ï˜˜@@OK@@

Select Solve poly…

„Ô1‚í5

‚í2\‚í 4@@OK@@

—@SYMB@

Enter vector of coefficients

Generate symbolic expression

Return to stack.

The expression thus generated is shown in the stack as:

'X^3+5*X^2+-2*X+4'.

To generate the algebraic expression using the roots, try the following example.

Assume that the polynomial roots are [1,3,-2,1]. Use the following keystrokes:

‚Ï˜˜@@OK@@

Select Solve poly…

Enter vector of roots

˜„Ô1‚í3

‚í2\‚í 1@@OK@@

˜@SYMB@

Generate symbolic expression

Return to stack.

The expression thus generated is shown in the stack as:'(X-1)*(X-3)*(X+2)*(X-1)'.

Page 6-8

To expand the products, you can use the EXPAND command. The resulting

expression is: 'X^4+-3*X^3+ -3*X^2+11*X-6'.

A different approach to obtaining an expression for the polynomial is to

generate the coefficients first, then generate the algebraic expression with the

coefficients highlighted. For example, for this case try:

‚Ï˜˜@@OK@@

Select Solve poly…

Enter vector of roots

˜„Ô1‚í3

‚í2\‚í 1@@OK@@

@SOLVE@

Solve for coefficients

Generate symbolic expression

Return to stack.

˜@SYMB@

The expression thus generated is shown in the stack as: 'X^4+-3*X^3+ -

3*X^2+11*X+-6*X^0'. The coefficients are listed in stack level 2.

Financial calculations

The calculations in item 5. Solve finance.. in the Numerical Solver (NUM.SLV)

are used for calculations of time value of money of interest in the discipline of

engineering economics and other financial applications. This application can

also be started by using the keystroke combination „sÒ (associated with

the 9 key). Before discussing in detail the operation of this solving

environment, we present some definitions needed to understand financial

operations in the calculator.

Definitions

Often, to develop projects, it is necessary to borrow money from a financial

institution or from public funds. The amount of money borrowed is referred to

as the Present Value (PV). This money is to be repaid through n periods

(typically multiples or sub-multiples of a month) subject to an annual interest rate

of I%YR. The number of periods per year (P/YR) is an integer number of

periods in which the year will be divided for the purpose of repaying the loan

money. Typical values of P/YR are 12 (one payment per month), 24 (payment

twice a month), or 52 (weekly payments). The payment(PMT) is the amount

that the borrower must pay to the lender at the beginning or end of each of the

n periods of the loan. The future value of the money (FV) is the value that the

borrowed amount of money will be worth at the end of n periods. Typically

payment occurs at the end of each period, so that the borrower starts paying at

the end of the first period, and pays the same fixed amount at the end of the

second, third, etc., up to the end of the n-th period.

Page 6-9

Example 1 – Calculating payment on a loan

If $2 million are borrowed at an annual interest rate of 6.5% to be repaid in 60

monthly payments, what should be the monthly payment? For the debt to be

totally repaid in 60 months, the future values of the loan should be zero. So, for

the purpose of using the financial calculation feature of the calculator we will

use the following values: n = 60, I%YR = 6.5, PV = 2000000, FV = 0, P/YR =

12. To enter the data and solve for the payment, PMT, use:

„Ò

Start the financial calculation input form

Enter n = 60

60 @@OK@@

6.5 @@OK@@

2000000 @@OK@@

Enter I%YR = 6.5 %

Enter PV = 2,000,000 US$

Skip PMT, since we will be solving for it

Enter FV = 0, the option End is highlighted

Highlight PMT and solve for it

0 @@OK@@

— š @@SOLVE!

The solution screen will look like this:

The screen now shows the value of PMT as –39,132.30, i.e., the borrower must

pay the lender US $ 39,132.30 at the end of each month for the next 60

months to repay the entire amount. The reason why the value of PMT turned

out to be negative is because the calculator is looking at the money amounts

from the point of view of the borrower. The borrower has + US $

2,000,000.00 at time period t = 0, then he starts paying, i.e., adding -US $

39132.30 at times t = 1, 2, …, 60. At t = 60, the net value in the hands of the

borrower is zero. Now, if you take the value US $ 39,132.30 and multiply it

by the 60 payments, the total paid back by the borrower is US $ 2,347,937.79.

Thus, the lender makes a net profit of $ 347,937.79 in the 5 years that his

money is used to finance the borrower’s project.

Example 2 – Calculating amortization of a loan

The same solution to the problem in Example 1 can be found by pressing

@@AMOR!!, which is stands for AMORTIZATION. This option is used to calculate

how much of the loan has been amortized at the end of a certain number of

Page 6-10

payments. Suppose that we use 24 periods in the first line of the amortization

screen, i.e., 24@@OK@@. Then, press @@AMOR@@. You will get the following

result:

This screen is interpreted as indicating that after 24 months of paying back the

debt, the borrower has paid up US $ 723,211.43 into the principal amount

borrowed, and US $ 215,963.68 of interest. The borrower still has to pay a

balance of US $1,276,788.57 in the next 36 months.

Check what happens if you replace 60 in the Payments: entry in the

amortization screen, then press @@OK@@ @@AMOR@@. The screen now looks like this:

This means that at the end of 60 months the US $ 2,000,000.00 principal

amount has been paid, together with US $ 347,937.79 of interest, with the

balance being that the lender owes the borrower US $ 0.000316. Of course,

the balance should be zero. The value shown in the screen above is simply

round-off error resulting from the numerical solution.

Press $or `, twice, to return to normal calculator display.

Example 3 – Calculating payment with payments at beginning of period

Let’s solve the same problem as in Examples 1 and 2, but using the option that

payment occurs at the beginning of the payment period. Use:

„Ò

Start the financial calculation input form

Enter n = 60

60 @@OK@@

6.5 @@OK@@

Enter I%YR = 6.5 %

2000000 @@OK@@

Enter PV = 2,000,000 US$

Page 6-11

Skip PMT, since we will be solving for it

Enter FV = 0, the option End is highlighted

Change payment option to Begin

Highlight PMT and solve for it

0 @@OK@@

@@CHOOS !—@@OK@@

— š @@SOLVE!

The screen now shows the value of PMT as –38,921.47, i.e., the borrower must

pay the lender US $ 38,921.48 at the beginning of each month for the next 60

months to repay the entire amount. Notice that the amount the borrower pays

monthly, if paying at the beginning of each payment period, is slightly smaller

than that paid at the end of each payment period. The reason for that

difference is that the lender gets interest earnings from the payments from the

beginning of the period, thus alleviating the burden on the lender.

Notes:

1. The financial calculator environment allows you to solve for any of the terms

involved, i.e., n, I%YR, PV, FV, P/Y, given the remaining terms in the loan

calculation. Just highlight the value you want to solve for, and press @@SOLVE!.

The result will be shown in the highlighted field.

2. The values calculated in the financial calculator environment are copied to

the stack with their corresponding tag (identifying label).

Deleting the variables

When you use the financial calculator environment for the first time within the

HOME directory, or any sub-directory, it will generate the variables @@@N@@ @I©YR@

@@PV@@ @@PMT@@ @@PYR@@ @@FV@@ to store the corresponding terms in the calculations.. You

can see the contents of these variables by using:

‚@@ @n@@ ‚@I©YR@ ‚@@PV@@ ‚@@PMT@@ ‚@@PYR@@ ‚@@FV@@.

You can either keep these variables for future use, or use the PURGE function to

erase them from your directory. To erase all of the variables at once, if using

ALG mode, try the following:

I@PURGE J „ä

³‚@@@n@@

™ ‚í

Enter PURGE, prepare list of variables

Enter name of variable N

Enter a comma

Enter name of variable I%YR

Enter a comma

Enter name of variable PV

Enter a comma

Enter name of variable PMT

³ ‚@I©YR@

™ ‚í

³ ‚@@PV@@

™ ‚í

³ ‚@@PMT@@

Page 6-12

™ ‚í

³ ‚@@PYR@@

™ ‚í

³ ‚@@FV@@.

Enter a comma

Enter name of variable PYR

Enter a comma

Enter name of variable FV

Execute PURGE command

The following two screen shots show the PURGE command for purging all the

variables in the directory, and the result after executing the command.

In RPN mode, the command is executed by using:

J „ä

@@@n@@

@I©YR@

@@PV@@

@@PMT@@

@@PYR@@

@@FV@@

Prepare a list of variables to be purged

Enter name of variable N

Enter name of variable I%YR

Enter name of variable PV

Enter name of variable PMT

Enter name of variable PYR

Enter name of variable FV

Enter list of variables in stack

Purge variables in list

I@PURGE

Before the command PURGE is entered, the RPN stack will look like this:

Solving equations with one unknown through NUM.SLV

The calculator's NUM.SLV menu provides item 1. Solve equation.. solve different

types of equations in a single variable, including non-linear algebraic and

transcendental equations. For example, let's solve the equation: e -sin(πx/3) =

Simply enter the expression as an algebraic object and store it into variable

EQ. The required keystrokes in ALG mode are the following:

Page 6-13

³„¸~„x™-S„ì

*~„x/3™‚Å 0™

K~e~q`

Function STEQ

Function STEQ, available through the command catalog, ‚N, will store its

argument into variable EQ, e.g., in ALG mode:

In RPN mode, enter the equation between apostrophes and activate command

STEQ. Thus, function STEQ can be used as a shortcut to store an expression

into variable EQ.

Press J to see the newly created EQ variable:

Then, enter the SOLVE environment and select Solve equation…, by using:

‚Ï@@OK@@. The corresponding screen will be shown as:

The equation we stored in variable EQ is already loaded in the Eq field in the

SOLVE EQUATION input form. Also, a field labeled x is provided. To solve the

equation all you need to do is highlight the field in front of X: by using ˜,

and press @SOLVE@. The solution shown is X: 4.5006E-2:

Page 6-14

This, however, is not the only possible solution for this equation. To obtain a

negative solution, for example, enter a negative number in the X: field before

solving the equation. Try 3\@@@OK@@˜@SOLVE@. The solution is now X: -

3.045.

Solution procedure for Equation Solve...

The numerical solver for single-unknown equations works as follows:

ꢀ It lets the user type in or @CHOOS an equation to solve.

ꢀ It creates an input form with input fields corresponding to all variables

involved in equation stored in variable EQ.

ꢀ The user needs to enter values for all variables involved, except one.

ꢀ The user then highlights the field corresponding to the unknown for

which to solve the equation, and presses @SOLVE@

ꢀ The user may force a solution by providing an initial guess for the

solution in the appropriate input field before solving the equation.

The calculator uses a search algorithm to pinpoint an interval for which the

function changes sign, which indicates the existence of a root or solution. It

then utilizes a numerical method to converge into the solution.

The solution the calculator seeks is determined by the initial value present in the

unknown input field. If no value is present, the calculator uses a default value of

zero. Thus, you can search for more than one solution to an equation by

changing the initial value in the unknown input field. Examples of the equations

solutions are shown following.

Example 1 – Hooke’s law for stress and strain

The equation to use is Hooke’s law for the normal strain in the x-direction for a

solid particle subjected to a state of stress given by

⎡

⎤

⎥

σ _xxσ _xyσ _xz

⎢

σ _yxσ _yyσ _yz

⎢

⎥

σ _zxσ _zyσ _zz

⎣

⎦

Page 6-15

The equation is

here e is the unit

e_xx= [σ _xx− n ⋅(σ _yy+σ _zz)]+α ⋅ ΔT,

strain in the x-direction, σ , σ , and σ , are the normal stresses on the particle

xx yy

in the directions of the x-, y-, and z-axes, E is Young’s modulus or modulus of

elasticity of the material, n is the Poisson ratio of the material, α is the thermal

expansion coefficient of the material, and ΔT is a temperature increase.

Suppose that you are given the following data: σ = 2500 psi, σ =1200 psi,

and σ = 500 psi, E = 1200000 psi, n = 0.15, α = 0.00001/ F, ΔT = 60 F.

To calculate the strain e use the following:

‚Ï@@OK@@

‚O

Access numerical solver to solve equations

Access the equation writer to enter equation

At this point follow the instructions from Chapter 2 on how to use the Equation

Writer to build an equation. The equation to enter in the Eq field should look

like this (notice that we use only one sub-index to refer to the variables, i.e., e

is translated as ex, etc. -- this is done to save typing time):

Use the following shortcuts for special characters:

σ:

~‚s

α:

~‚a

Δ:

~‚c

and recall that lower-case letters are entered by using ~„ before the letter

key, thus, x is typed as ~„x.

Press `to return to the solver screen. Enter the values proposed above into

the corresponding fields, so that the solver screen looks like this:

Page 6-16

With the ex: field highlighted, press @SOLVE@ to solve for ex:

The solution can be seen from within the SOLVE EQUATION input form by

pressing @EDIT while the ex: field is highlighted. The resulting value is

2.470833333333E-3. Press @@@OK@@ to exit the EDIT feature.

Suppose that you now, want to determine the Young’s modulus that will produce

a strain of e = 0.005 under the same state of stress, neglecting thermal

expansion. In this case, you should enter a value of 0.005 in the ex: field, and

a zero in the ΔT: field (with ΔT = 0, no thermal effects are included). To solve

for E, highlight the E: field and press @SOLVE@. The result, seeing with the @EDIT

feature is, E = 449000 psi. Press @SOLVE@ ` to return to normal display.

Notice that the results of the calculations performed within the numerical solver

environment have been copied to the stack:

Also, you will see in your soft-menu key labels variables corresponding to those

variables in the equation stored in EQ (press L to see all variables in your

directory), i.e., variables ex, ΔT, α, σz, σy, n, σx, and E.

Example 2 – Specific energy in open channel flow

Page 6-17

Specific energy in an open channel is defined as the energy per unit weight

measured with respect to the channel bottom. Let E = specific energy, y =

channel depth, V = flow velocity, g = acceleration of gravity, then we write

V ²

E = y +

The flow velocity, in turn, is given by V = Q/A, where Q = water discharge, A

= cross-sectional area. The area depends on the cross-section used, for

example, for a trapezoidal cross-section, as shown in the figure below, A =

(b+m⋅y) ⋅y, where b = bottom width, and m = side slope of cross section.

ꢁ

ꢀ

ꢁ

We can type in the equation for E as shown above and use auxiliary variables

for A and V, so that the resulting input form will have fields for the fundamental

variables y, Q, g, m, and b, as follows:

ꢀ First, create a sub-directory called SPEN (SPecific ENergy) and work

within that sub-directory.

ꢀ Next, define the following variables:

ꢀ Launch the numerical solver for solving equations: ‚Ï@@OK@@.

Notice that the input form contains entries for the variables y, Q, b, m,

and g:

ꢀ Try the following input data: E = 10 ft, Q = 10 cfs (cubic feet per

second), b = 2.5 ft, m = 1.0, g = 32.2 ft/s :

Page 6-18

ꢀ Solve for y.

The result is 0.149836.., i.e., y = 0.149836.

ꢀ It is known, however, that there are actually two solutions available for

y in the specific energy equation. The solution we just found

corresponds to a numerical solution with an initial value of 0 (the

default value for y, i.e., whenever the solution field is empty, the initial

value is zero). To find the other solution, we need to enter a larger

value of y, say 15, highlight the y input field and solve for y once

The result is now 9.99990, i.e., y = 9.99990 ft.

This example illustrates the use of auxiliary variables to write complicated

equations. When NUM.SLV is activated, the substitutions implied by the

auxiliary variables are implemented, and the input screen for the equation

provides input field for the primitive or fundamental variables resulting from the

substitutions. The example also illustrates an equation that has more than one

solution, and how choosing the initial guess for the solution may produce those

different solutions.

Page 6-19

In the next example we will use the DARCY function for finding friction factors in

pipelines. Thus, we define the function in the following frame.

Special function for pipe flow: DARCY (ε/D,Re)

The Darcy-Weisbach equation is used to calculate the energy loss (per unit

weight), h , in a pipe flow through a pipe of diameter D, absolute roughness ε,

and length L, when the flow velocity in the pipe is V. The equation is written as

L V ²

h_f= f ⋅

⋅

. The quantity f is known as the friction factor of the flow and

D 2g

it has been found to be a function of the relative roughness of the pipe, ε/D,

and a (dimensionless) Reynolds number, Re. The Reynolds number is defined as

Re = ρVD/μ = VD/ν, where ρ and μ are the density and dynamic viscosity of

the fluid, respectively, and ν = μ/ρ is the kinematic viscosity of the fluid.

The calculator provides a function called DARCY that uses as input the relative

roughness ε/D and the Reynolds number, in that order, to calculate the friction

factor f. The function DARCY can be found through the command catalog:

For example, for ε/D = 0.0001, Re = 1000000, you can find the friction factor

by using: DARCY(0.0001,1000000). In the following screen, the function

ꢀNUM () was used to obtain a numerical value of the function:

The result is f = DARCY(0.0001,1000000) = 0.01341…

The function FANNING(ε/D,Re)

In aerodynamics applications a different friction factor, the Fanning friction

factor, is used. The Fanning friction factor, f , is defined as 4 times the Darcy-

Weisbach friction factor, f. The calculator also provides a function called

FANNING that uses the same input as DARCY, i.e., ε/D and Re, and provides

the FANNING friction factor. Check that FANNING(0.0001,1000000) =

0.0033603589181s.

Page 6-20

Example 3 – Flow in a pipe

You may want to create a separate sub-directory (PIPES) to try this example.

The main equation governing flow in a pipe is, of course, the Darcy-Weisbach

equation. Thus, type in the following equation into EQ:

Also, enter the following variables (f, A, V, Re):

In this case we stored the main equation (Darcy-Weisbach equation) into EQ,

and then replaced several of its variables by other expressions through the

definition of variables f, A, V, and Re. To see the combined equation, use

EVAL(EQ). In this example we changed the display setting so that we can see

the entire equation in the screen:

Thus, the equation we are solving, after combining the different variables in the

directory, is:

Page 6-21

πD²/ 4

⎛

⎜

⎞

⎟

⎠

8Q²L

h _f=

⋅ DARCY

π gD⁵

⎜ D

⎜

⎝

The combined equation has primitive variables: h, Q, L, g, D, ε, and Nu.

Launch the numerical solver (‚Ï@@OK@@) to see the primitive variables listed

in the SOLVE EQUATION input form:

Suppose that we use the values hf = 2 m, ε = 0.00001 m, Q = 0.05 m /s, Nu

= 0.000001 m /s, L = 20 m, and g = 9.806 m/s , find the diameter D. Enter

the input values, and solve for D, The solution is: 0.12, i.e., D = 0.12 m.

If the equation is dimensionally consistent, you can add units to the input

values, as shown in the figure below. However, you must add those units to the

initial guess in the solution. Thus, in the example below we place 0_m in the D:

field before solving the problem. The solution is shown in the screen to the right:

Press ` to return to normal calculator display. The solution for D will be listed

in the stack.

Page 6-22

Example 4 – Universal gravitation

Newton’s law of universal gravitation indicates that the magnitude of the

attractive force between two bodies of masses m and m separated by a

M₁⋅ M₂

r²

distance r is given by the equation

F = G ⋅

Here, G is the universal gravitational constant, whose value can be obtained

through the use of the function CONST in the calculator by using:

We can solve for any term in the equation (except G) by entering the equation

as:

This equation is then stored in EQ:

Launching the numerical solver for this equation results in an input form

containing input fields for F, G, m1, m2, and r.

Let’s solve this problem using units with the following values for the known

variables m1 = 1.0×10 kg, m2 = 1.0×10 kg, r = 1.0×10 m. Also, enter

a value of 0_N in field F to ensure the proper solution using units in the

calculator:

Page 6-23

Solve for F, and press to return to normal calculator display. The solution is F :

-15

6.67259E-15_N, or F = 6.67259×10 N.

Note: When using units in the numerical solver make sure that all the variables

have the proper units, that the units are compatible, and that the equation is

dimensionally homogeneous.

Different ways to enter equations into EQ

In all the examples shown above we have entered the equation to be solved

directly into variable EQ before activating the numerical solver. You can

actually type the equation to be solved directly into the solver after activating it

by editing the contents of the EQ field in the numerical solver input form. If

variable EQ has not been defined previously, when you launch the numerical

solver (‚Ï@@OK@@), the EQ field will be highlighted:

At this point you can either type a new equation by pressing @EDIT. You will be

provided with a set of apostrophes so that you can type the expression between

them:

Page 6-24

Type an equation, say X^2 - 125 = 0, directly on the stack, and press @@@OK@@@ .

At this point the equation is ready for solution.

Alternatively, you can activate the equation writer after pressing @EDIT to enter

your equation. Press ` to return to the numerical solver screen.

Another way to enter an equation into the EQ variable is to select a variable

already existing in your directory to be entered into EQ. This means that your

equation would have to have been stored in a variable name previously to

activating the numerical solver. For example, suppose that we have entered the

following equations into variables EQ1 and EQ2:

Now, launch the numerical solver (‚Ï@@OK@@, and highlight the EQ field. At

this point press the @CHOOS soft menu key. Use the up and down arrow keys (—

˜) to select, say, variable EQ1:

Press @@@OK@@@ after selecting EQ1 to load into variable EQ in the solver. The new

equation is ready to be solved.

Page 6-25

The SOLVE soft menu

The SOLVE soft menu allows access to some of the numerical solver functions

through the soft menu keys. To access this menu use in RPN mode: 74 MENU,

or in ALG mode: MENU(74). Alternatively, you can use ‚(hold) 7 to

activate the SOLVE soft menu. The sub-menus provided by the SOLVE soft menu

are the following:

The ROOT sub-menu

The ROOT sub-menu include the following functions and sub-menus:

Function ROOT

Function ROOT is used to solve an equation for a given variable with a starting

guess value. In RPN mode the equation will be in stack level 3, while the

variable name will be located in level 2, and the initial guess in level 1. The

following figure shows the RPN stack before and after activating function @ROOT:

In ALG mode, you would use ROOT(‘TAN(θ)=θ’,’θ’,5) to activate function ROOT:

Variable EQ

The soft menu key @@EQ@@ in this sub-menu is used as a reference to the variable

EQ. Pressing this soft menu key is equivalent to using function RCEQ (ReCall

EQ).

The SOLVR sub-menu

The SOLVR sub-menu activates the soft-menu solver for the equation currently

stored in EQ. Some examples are shown next:

Page 6-26

Example 1 - Solving the equation t -5t = -4

For example, if you store the equation ‘t^2-5*t=-4’ into EQ, and press @SOLVR, it

will activate the following menu:

This result indicates that you can solve for a value of t for the equation listed at

the top of the display. If you try, for example, „[ t ], it will give you the

result t: 1., after briefly flashing the message “Solving for t.” There is a second

root to this equation, which can be found by changing the value of t, before

solving for it again. Do the following: 10 [ t ], then press „[ t ]. The

result is now, t: 4.0000000003. To verify this result, press the soft menu key

labeled @EXPR=, which evaluates the expression in EQ for the current value of t.

The results in this case are:

To exit the SOLVR environment, press J. The access to the SOLVE menu is

lost at this point, so you have to activate it once more as indicated earlier, to

continue with the exercises below.

Example 2 - Solving the equation Q = at +bt

It is possible to store in EQ, an equation involving more than one variable, say,

‘Q = at^2 + bt’. In this case, after activating the SOLVE soft menu, and

pressing @ROOT @SOLVR, you will get the following screen:

Within this SOLVR environment you can provide values for any of the variables

listed by entering the value in the stack and pressing the corresponding soft-

menu keys. For example, say you enter the values Q = 14, a = 2, and b = 3.

You would use: 14 [ Q ], 2 [ a ], 3 [ b ].

As variables Q, a, and b, get assigned numerical values, the assignments are

listed in the upper left corner of the display. At this point we can solve for t, by

using „[ t ]. The result is t: 2. Pressing @EXPR= shows the results:

Example 3 - Solving two simultaneous equations, one at a time

Page 6-27

You can also solve more than one equation by solving one equation at a time,

and repeating the process until a solution is found. For example, if you enter

the following list of equations into variable EQ: { ‘a*X+b*Y = c’, ‘k*X*Y=s’},

the keystroke sequence @ROOT @SOLVR, within the SOLVE soft menu, will produce

the following screen:

The first equation, namely, a*X + b*Y = c, will be listed in the top part of the

display. You can enter values for the variables a, b, and c, say:

2 [ a ] 5 [ b ] 19 [ c ]. Also, since we can only solve one equation at

a time, let’s enter a guess value for Y, say, 0 [ Y ], and solve for X, by using

„[ X ]. This gives the value, X: 9.4999…. To check the value of the

equation at this point, press @EXPR=. The results are: Left: 19, Right: 19. To

solve the next equation, press L@NEXQ. The screen shows the soft menu keys

as:

Say we enter the values k = 2, s = 12. Then solve for Y, and press

@EXPR=. The results are now, Y:

We then continue moving from the first to the second equation, back and forth,

solving the first equation for X and the second for Y, until the values of X and Y

converge to a solution. To move from equation to equation use @NEXQ.

To solve for X and Y use „[ X ], and „[ Y ], respectively. The following

sequence of solutions is produced:

After solving the two equations, one at a time, we notice that, up to the third

decimal, X is converging to a value of 7.500, while Y is converging to a value o

0.799.

Page 6-28

Using units with the SOLVR sub-menu

These are some rules on the use of units with the SOLVR sub-menu:

ꢀ Entering a guess with units for a given variable, will introduce the use

of those units in the solution.

ꢀ If a new guess is given without units, the units previously saved for that

particular variable are used.

ꢀ To remove units enter a number without units in a list as the new guess,

i.e., use the format { number }.

ꢀ A list of numbers can be given as a guess for a variable. In this case,

the units takes the units used belong to the last number in the list. For

example, entering { 1.41_ft 1_cm 1_m } indicates that meters (m) will

be used for that variable.

ꢀ The expression used in the solution must have consistent units, or an

error will result when trying to solve for a value.

The DIFFE sub-menu

The DIFFE sub-menu provides a number of functions for the numerical solution of

differential equations. The functions provided are the following:

These functions are presented in detail in Chapter 16.

The POLY sub-menu

The POLY sub-menu performs operations on polynomials. The functions

included are the following:

Function PROOT

This function is used to find the roots of a polynomial given a vector containing

the polynomial coefficients in decreasing order of the powers of the

n-1

independent variable. In other words, if the polynomial is a x + a

n-1

+ …

+ a x + a x + a , the vector of coefficients should be entered as [a , a , …

n-1

, a , a , a ]. For example, the roots of the polynomial whose coefficients are

[1, -5, 6] are [2, 3].

Function PCOEF

Page 6-29

This function produces the coefficients [a , a , … , a , a , a ] of a

n-1

polynomial a x + a

+ … + a x + a x + a , given a vector of its roots

n-1

[r , r , …, r ]. For example, a vector whose roots are given by

[-1, 2, 2, 1, 0], will produce the following coefficients: [1, -4, 3, 4, -4, 0]. The

polynomial is x - 4x + 3x + 4x - 4x.

Function PEVAL

This function evaluates a polynomial, given a vector of its coefficients, [a , a

n-1

… , a , a , a ], and a value x , i.e., PEVAL calculates a x + a

+ …

n 0

n-1 0

+ a x + a x + a . For example, for coefficients [2, 3, -1, 2] and a value of

2 0

1 0

2, PEVAL returns the value 28.

The SYS sub-menu

The SYS sub-menu contains a listing of functions used to solve linear systems.

The functions listed in this sub-menu are:

These functions are presented in detail in Chapter 11.

The TVM sub-menu

The TVM sub-menu contains functions for calculating Time Value of Money. This

is an alternative way to solve FINANCE problems (see Chapter 6). The

functions available are shown next:

The SOLVR sub-menu

The SOLVR sub-menu in the TVM sub-menu will launch the solver for solving

TVM problems. For example, pressing @SOLVR, at this point, will trigger the

following screen:

As an exercise, try using the values n = 10, I%YR = 5.6, PV = 10000, and FV

= 0, and enter „[ PMT ] to find PMT = -1021.08…. Pressing L, produces

the following screen:

Page 6-30

Press J to exit the SOLVR environment. Find your way back to the TVM sub-

menu within the SOLVE sub-menu to try the other functions available.

Function TVMROOT

This function requires as argument the name of one of the variables in the TVM

problem. The function returns the solution for that variable, given that the other

variables exist and have values stored previously. For example, having solved a

TVM problem above, we can solve for, say, ‘N’, as follows: [ ‘ ] ~n`

@TVMRO. The result is 10.

Function AMORT

This function takes a value representing a period of payment (between 0 and n)

and returns the principal, interest, and balance for the values currently stored in

the TVM variables. For example, with the data used earlier, if we activate

function AMORT for a value of 10, we get:

Function BEG

If selected, the TMV calculations use payments at the beginning of each period.

If deselected, the TMV calculations use payments at the end of each period.

Page 6-31

Chapter 7

Solving multiple equations

Many problems of science and engineering require the simultaneous solutions

of more than one equation. The calculator provides several procedures for

solving multiple equations as presented below. Please notice that no discussion

of solving systems of linear equations is presented in this chapter. Linear

systems solutions will be discussed in detail in subsequent chapters on matrices

and linear algebra.

Rational equation systems

Equations that can be re-written as polynomials or rational algebraic

expressions can be solved directly by the calculator by using the function

SOLVE. You need to provide the list of equations as elements of a vector. The

list of variables to solve for must also be provided as a vector. Make sure that

the CAS is set to mode Exact before attempting a solution using this procedure.

Also, the more complicated the expressions, the longer the CAS takes in solving

a particular system of equations. Examples of this application follow:

Example 1 – Projectile motion

Use function SOLVE with the following vector arguments, the first being the list of

equations: [‘x = x0 + v0*COS(θ0)*t’ ‘y =y0+v0*SIN(θ0)*t – g*t^2/2’]`,

and the second being the variables to solve for, say t and y0, i.e., [‘t’ ‘y0’].

The solution in this case will be provided using the RPN mode. The only reason

being that we can build the solution step by step. The solution in the ALG

mode is very similar. First, we store the first vector (equations) into variable A2,

and the vector of variables into variable A1. The following screen shows the

RPN stack before saving the variables.

At this point, we need only press K twice to store these variables.

To solve, first change CAS mode to Exact, then, list the contents of A2 and A1,

in that order: @@@A2@@@ @@@A1@@@ .

Page 7-1

Use command SOLVE at this point (from the S.SLV menu: „Î) After about

40 seconds, maybe more, you get as result a list:

{ ‘t = (x-x0)/(COS(θ0)*v0)’

‘y0 = (2*COS(θ0)^2*v0^2*y+(g*x^2(2*x0*g+2*SIN(θ0))*COS(θ0)*v0^2)*x+

(x0^2*g+2*SIN(θ0)*COS(θ0)*v0^2*x0)))/(2*COS(θ0)^2*v0^2)’]}

Press μ to remove the vector from the list, then use command OBJꢀ, to get

the equations listed separately in the stack.

Note: This method worked fine in this example because the unknowns t and

y0 were algebraic terms in the equations. This method would not work for

solving for θ0, since θ0 belongs to a transcendental term.

Example 2 – Stresses in a thick wall cylinder

Consider a thick-wall cylinder for inner and outer radius a and b, respectively,

subject to an inner pressure P and outer pressure P . At any radial distance r

from the cylinder’s axis the normal stresses in the radial and transverse

directions, σ and σ , respectively, are given by

θθ

a²⋅ P − b²⋅ P a²⋅b²⋅(P − P )

σ_θθ

b²− a²

r²⋅(b²−a²)

a²⋅ P − b²⋅ P a²⋅b²⋅(P − P )

σ_rr=

−

b²− a²

r²⋅(b²−a²)

Notice that the right-hand sides of the two equations differ only in the sign

between the two terms. Therefore, to write these equations in the calculator, I

suggest you type the first term and store in a variable T1, then the second term,

and store it in T2. Writing the equations afterwards will be matter of recalling

Page 7-2

the contents of T1 and T2 to the stack and adding and subtracting them. Here

is how to do it with the equation writer:

Enter and store term T1:

Enter and store term T2:

Notice that we are using the RPN mode in this example, however, the

procedure in the ALG mode should be very similar. Create the equation for

σ : J@@@T1@@@ @@T2#@@ + ~‚s ~‚t ` ™ ‚Å

θθ

Create the equation for σ : J@@@T1@@@ @@T2#@@ - ~‚s ~„r `

™ ‚Å

Put together a vector with the two equations, using function ꢀARRY (find it

using the command catalog ‚N) after typing a 2:

Now, suppose that we want to solve for P and P , given a, b, r, σ , and σ .

θθ

We enter a vector with the unknowns:

To solve for P and P , use the command SOLVE from the S.SLV menu („Î),

it may take the calculator a minute to produce the result:

{[‘Pi=-(((σθ-σr)*r^2-(σθ+σr)*a^2)/(2*a^2))’

‘Po=-(((σθ-σr)*r^2-(σθ+σr)*b^2)/(2*b^2))’ ] }, i.e.,

Page 7-3

Notice that the result includes a vector [ ] contained within a list { }. To remove

the list symbol, use μ. Finally, to decompose the vector, use function OBJꢀ.

The result is:

These two examples constitute systems of linear equations that can be handled

equally well with function LINSOLVE (see Chapter 11). The following example

shows function SOLVE applied to a system of polynomial equations.

Example 3 - System of polynomial equations

The following screen shot shows the solution of the system X +XY=10, X -Y =-5,

using function SOLVE:

Solution to simultaneous equations with MSLV

Function MSLV is available as the last option in the ‚Ï menu:

The help-facility entry for function MSLV is shown next:

Page 7-4

Example 1 - Example from the help facility

As with all function entries in the help facility, there is an example attached to

the MSLV entry as shown above. Notice that function MSLV requires three

arguments:

1. A vector containing the equations, i.e., ‘[SIN(X)+Y,X+SIN(Y)=1]’

2. A vector containing the variables to solve for, i.e., ‘[X,Y]’

3. A vector containing initial values for the solution, i.e., the initial values

of both X and Y are zero for this example.

In ALG mode, press @ECHO to copy the example to the stack, press ` to run the

example. To see all the elements in the solution you need to activate the line

editor by pressing the down arrow key (˜):

In RPN mode, the solution for this example is produced by using:

Activating function MSLV results in the following screen.

You may have noticed that, while producing the solution, the screen shows

intermediate information on the upper left corner. Since the solution provided

by MSLV is numerical, the information in the upper left corner shows the results

of the iterative process used to obtain a solution. The final solution is X =

1.8238, Y = -0.9681.

Example 2 - Entrance from a lake into an open channel

This particular problem in open channel flow requires the simultaneous solution

V ²

of two equations, the equation of energy:

, and Manning’s

H_o= y +

Cu A^{5/ 3}

equation:

. In these equations, H represents the energy

Q =

⋅

⋅ S_o

P^{2 / 3}

head (m, or ft) available for a flow at the entrance to a channel, y is the flow

depth (m or ft), V = Q/A is the flow velocity (m/s or ft/s), Q is the volumetric

Page 7-5

discharge (m /s or ft /s), A is the cross-sectional area (m or ft ), C is a

coefficient that depends on the system of units (C = 1.0 for the SI, C = 1.486

for the English system of units), n is the Manning’s coefficient, a measure of the

channel surface roughness (e.g., for concrete, n = 0.012), P is the wetted

perimeter of the cross section (m or ft), S is the slope of the channel bed

expressed as a decimal fraction. For a trapezoidal channel, as shown below,

the area is given by

, while the wetted perimeter is given by

A = (b + my)y

P = b + 2y 1+ m²

, where b is the bottom width (m or ft), and m is the side

slope (1V:mH) of the cross section.

Typically, one has to solve the equations of energy and Manning’s

simultaneously for y and Q. Once these equations are written in terms of the

primitive variables b, m, y, g, S , n, Cu, Q, and H , we are left with a system of

equations of the form f (y,Q) = 0, f (y,Q) = 0. We can build these two

equations as follows.

We assume that we will be using the ALG and Exact modes in the calculator,

although defining the equations and solving them with MSLV is very similar in

the RPN mode. Create a sub-directory, say CHANL (for open CHANneL), and

within that sub-directory define the following variables:

To see the original equations, EQ1 and EQ2, in terms of the primitive variables

listed above, we can use function EVAL applied to each of the equations, i.e.,

Page 7-6

μ@@@EQ1@@ μ @@@EQ2@@. The equations are listed in the stack as follows (small

font option selected):

We can see that these equations are indeed given in terms of the primitive

variables b, m, y, g, S , n, Cu, Q, and H .

In order to solve for y and Q we need to give values to the other variables.

Suppose we use H = 5 ft, b = 1.5 ft, m = 1, n = 0.012, S = 0.00001, g =

32.2, and Cu = 1.486. Before being able to use MSLV for the solution, we

need to enter these values into the corresponding variable names. This can be

accomplished as follows:

Now, we are ready to solve the equation. First, we need to put the two

equations together into a vector. We can do this by actually storing the vector

into a variable that we will call EQS (EQuationS):

As initial values for the variables y and Q we will use y = 5 (equal to the value

of H , which is the maximum value that y can take) and Q = 10 (this is a

guess). To obtain the solution we select function MSLV from the NUM.SLV

menu, e.g., ‚Ï6@@@OK@@@, to place the command in the screen:

Page 7-7

Next, we’ll enter variable EQS: LL@@EQS@ , followed by vector [y,Q]:

‚í„Ô~„y‚í~q™

and by the initial guesses ‚í„Ô5‚í 10.

Before pressing `, the screen will look like this:

Press ` to solve the system of equations. You may, if your angular measure

is not set to radians, get the following request:

Press @@OK@@ and allow the solution to proceed. An intermediate solution step may

look like this:

The vector at the top representing the current value of [y,Q] as the solution

progresses, and the value .358822986286 representing the criteria for

convergence of the numerical method used in the solution. If the system is well

posed, this value will diminish until reaching a value close to zero. At that point

a numerical solution would have been found. The screen, after MSLV finds a

solution will look like this:

Page 7-8

The result is a list of three vectors. The first vector in the list will be the equations

solved. The second vector is the list of unknowns. The third vector represents the

solution. To be able to see these vectors, press the down-arrow key ˜ to

activate the line editor. The solution will be shown as follows:

The solution suggested is [4.9936.., 20.661…]. This means, y = 4.99 ft, and

Q = 20.661… ft /s. You can use the arrow keys (š™—˜) to see the

solution in detail.

Using the Multiple Equation Solver (MES)

The multiple equation solver is an environment where you can solve a system of

multiple equations by solving for one unknown from one equation at a time. It

is not really a solver to simultaneous solutions, rather, it is a one-by-one solver of

a number of related equations. To illustrate the use of the MES for solving

multiple equations we present an application related to trigonometry in the next

section. The examples shown here are developed in the RPN mode.

Application 1 - Solution of triangles

In this section we use one important application of trigonometric functions:

calculating the dimensions of a triangle. The solution is implemented in the

calculator using the Multiple Equation Solver, or MES.

Consider the triangle ABC shown in the figure below.

The sum of the interior angles of any triangle is always 180 , i.e., α + β + γ =

180 . The sine law indicates that:

sinα sin β sinγ

Page 7-9

The cosine law indicates that:

a = b + c – 2⋅b⋅c⋅cos α,

b = a + c – 2⋅a⋅c⋅cos β,

c = a + b – 2⋅a⋅b⋅cos γ.

In order to solve any triangle, you need to know at least three of the following

six variables: a, b, c, α, β, γ. Then, you can use the equations of the sine law,

cosine law, and sum of interior angles of a triangle, to solve for the other three

variables.

If the three sides are known, the area of the triangle can be calculated with

Heron’s formula

,where s is known as the

A = s ⋅(s − a)⋅(s − b)⋅(s − c)

a + b + c

semi-perimeter of the triangle, i.e.,

s =

Triangle solution using the Multiple Equation Solver (MES)

The Multiple Equation Solver (MES) is a feature that can be used to solve two or

more coupled equations. It must be pointed out, however, that the MES does

not solve the equations simultaneously. Rather, it takes the known variables,

and then searches in a list of equations until it finds one that can be solved for

one of the unknown variables. Then, it searches for another equation that can

be solved for the next unknowns, and so on, until all unknowns have been

solved for.

Creating a working directory

We will use the MES to solve for triangles by creating a list of equations

corresponding to the sine and cosine laws, the law of the sum of interior angles,

and Heron’s formula for the area. First, create a sub-directory within HOME

that we will call TRIANG, and move into that directory. See Chapter 2 for

instructions on how to create a new sub-directory.

Entering the list of equations

Within TRIANG, enter the following list of equations either by typing them

directly on the stack or by using the equation writer. (Recall that ~‚a

produces the character α, and ~‚b produces the character β. The

character γ needs to be @ECHOed from ‚

Page 7-10

‘SIN(α)/a = SIN(β)/b’

‘SIN(α)/a = SIN(γ)/c’

‘SIN(β)/b = SIN(γ)/c’

‘c^2 = a^2+b^2-2*a*b*COS(γ)’

‘b^2 = a^2+c^2-2*a*c*COS(β)’

‘a^2 = b^2+c^2-2*b*c*COS(α)’

‘α+β+γ = 180’

‘s = (a+b+c)/2’

‘A = √ (s*(s-a)*(s-b)*(s-c))’

Then, enter the number 9, and create a list of equations by using: function

ꢀLIST (use the command catalog ‚N). Store this list in the variable EQ.

The variable EQ contains the list of equations that will be scanned by the MES

when trying to solve for the unknowns.

Entering a window title

Next, we will create a string variable to be called TITLE to contain the string

“Triangle Solution”, as follows:

‚Õ

Open double quotes in stack

Locks keyboard into lower-case alpha.

Enter text: Triangle_

~~„~

„triangle#

„solution

Enter text: Solution

Enter string “Triangle Solution” in stack

Open single quotes in stack

Enter variable name ‘TITLE’

Store string into ‘TITLE’

~~title`

Creating a list of variables

Next, create a list of variable names in the stack that will look like this:

{ a b c α β γ A s }

and store it in variable LVARI (List of VARIables). The list of variables represents

the order in which the variables will be listed when the MES gets started. It must

include all the variables in the equations, or it will not work with function MITM

(see below). Here is the sequence of keystrokes to use to prepare and store this

list:

Page 7-11

Press J, if needed, to get your variables menu. Your menu should show the

variables @LVARI! !@TITLE @@EQ@@ .

Preparing to run the MES

The next step is to activate the MES and try one sample solution. Before we do

that, however, we want to set the angular units to DEGrees, if they are not

already set to that, by typing ~~deg`.

Next, we want to keep in the stack the contents of TITLE and LVARI, by using:

!@TITLE @LVARI!

We will use the following MES functions

ꢀ MINIT: MES INITialization: initializes the variables in the equations

stored in EQ.

ꢀ MITM: MES’ Menu Item: Takes a title from stack level 2 and the list of

variables from stack level 1 and places the title atop of the MES

window, and the list of variables as soft menu keys in the order

indicated by the list. In the present exercise, we already have a title

(“Triangle Solution”) and a list of variables ({ a b c α β γ A s }) in

stack levels 2 and 1, respectively, ready to activate MITM.

ꢀ MSOLVR: MES SOLVER; activates the Multiple Equation Solver (MES)

and waits for input by the user.

Running the MES interactively

To get the MES started, with the variables TITLE and LVARI listed in the stack,

activate command MINIT, then MITM, and finally, MSOLVR (find these functions

in the catalog ‚N).

The MES is launched with the following list of variables available (Press L to

see the next list of variables):

Press L to see the third list of variables. You should see:

Press L once more to recover the first variable menu.

Page 7-12

Let’s try a simple solution of Case I, using a = 5, b = 3, c = 5. Use the

following entries:

5[ a ]

3[ b ]

5[ c ]

a:5 is listed in the top left corner of the display.

b:3 is listed in the top left corner of the display.

c:5 is listed in the top left corner of the display.

To solve for the angles use:

„[ α ] Calculator reports Solving for α, and shows the result α:

72.5423968763.

Note: If you get a value that is larger than 180, try the following:

10[ α ]

Re-initialize a to a smaller value.

„ [ α ]

Calculator reports Solving for α

Next, we calculate the other two values:

„ [ β ] The result is β: 34.9152062474.

„ [ The result is γ: 72.5423968763.

]

You should have the values of the three angles listed in stack levels 3 through

1. Press + twice to check that they add indeed to 180 .

Press L to move to the next variables menu. To calculate the area use: „[

A ]. The calculator first solves for all the other variables, and then finds the

area as A: 7.15454401063.

Note: When a solution is found, the calculator reports the conditions for the

solution as either Zero, or Sign Reversal. Other messages may occur if the

calculator has difficulties finding a solution.

Page 7-13

Pressing „@@ALL@@ will solve for all the variables, temporarily showing the

intermediate results. Press ‚@@ALL@@ to see the solutions:

When done, press $ to return to the MES environment. Press J to exit the

MES environment and return to the normal calculator display.

Organizing the variables in the sub directory

Your variable menu will now contain the variables (press L to see the second

set of variables):

Variables corresponding to all the variables in the equations in EQ have been

created. There is also a new variable called Mpar (MES parameters), which

contains information regarding the setting up of the MES for this particular set

of equations. If you use ‚@Mpar to see the contents of the variable Mpar. You

will get the cryptic message: Library Data. The meaning of this is that the MES

parameters are coded in a binary file, which cannot be accessed by the editor.

Next, we want to place them in the menu labels in a different order than the

one listed above, by following these steps:

1. Create a list containing { EQ Mpar LVARI TITLE }, by using:

„ä @@@EQ@@@ @Mpar! !@LVARI @@TITLE `

2. Place contents of LVARI in the stack, by using: @LVARI.

3. Join the two lists by pressing +.

Use function ORDER (use the command catalog ‚N) to order the variables

as shown in the list in stack level 1.

4. Press J to recover your variables list. It should now look like this:

5. Press L to recover the first variable menu.

Page 7-14

Programming the MES triangle solution using User RPL

To facilitate activating the MES for future solutions, we will create a program

that will load the MES with a single keystroke. The program should look like

this: << DEG MINIT TITLE LVARI MITM MSOLVR >>, and can be typed in by

using:

‚å

deg#

minit#

Opens the program symbol

Locks alphanumeric keyboard

Type in DEG (angular units set to DEGrees)

Type in MINIT_

Unlocks alphanumeric keyboard

List the name TITLE in the program

List the name LVARI in the program

Locks alphanumeric keyboard

Type in MITM_

@TITLE

@LVARI

mitm#

msolvr

Type in MSOLVR

Enter program in stack

Store the program in a variable called TRISOL, for TRIangle SOLution, by using:

³~~trisol` K

Press J, if needed, to recover your list of variables. A soft key label @TRISO

should be available in your menu.

Running the program – solution examples

To run the program, press the @TRISO soft menu key. You will now have the MES

menu corresponding to the triangle solution. Let’s try examples of the three

cases listed earlier for triangle solution.

Example 1 – Right triangle

Use a = 3, b = 4, c = 5. Here is the solution sequence:

3[ a ] 4 [ b ] 5[ c ] To enter data

„[ α ]

„[ β ]

The result is α: 36.8698976458

The result is β: 53.1301023541.

The result is γ: 90.

To move to the next variables menu.

The result is A: 6.

„[

]

[ꢀ][ A ]

L L

To move to the next variables menu.

Example 2 - Any type of triangle

Page 7-15

Use a = 3, b = 4, c = 6. The solution procedure used here consists of solving

for all variables at once, and then recalling the solutions to the stack:

J@TRISO

To clear up data and re-start MES

3[ a ] 4 [ b ] 6[ c ] To enter data

To move to the next variables menu.

„ @ALL!

‚ @ALL!

Solve for all the unknowns.

Show the solution:

The solution is:

At the bottom of the screen, you will have the soft menu keys:

@VALUꢁ @EQNS! @PRINT %%%% %%%% @EXIT

The square dot in @VALUꢁ indicates that the values of the variables, rather than

the equations from which they were solved, are shown in the display. To see the

equations used in the solution of each variable, press the @EQNS! soft menu key.

The display will now look like this:

The soft menu key @PRINT is used to print the screen in a printer, if available.

And @EXIT returns you to the MES environment for a new solution, if needed. To

return to normal calculator display, press J.

The following table of triangle solutions shows the data input in bold face and

the solution in italics. Try running the program with these inputs to verify the

solutions. Please remember to press J@TRISO at the end of each solution to

clear up variables and start the MES solution again. Otherwise, you may carry

over information from the previous solution that may wreck havoc with your

current calculations.

Page 7-16

α( )

2.5 6.9837 7.2 20.229

7.2 8.5 14.26 22.616 27 130.38 23.309

21.92 17.5

41.92 23

10.27 3.26

17 25

β( )

γ( )

75 84.771 8.6933

13.2

29.6

10.5

52.98 37.03 115.5

328.81

16.66

31.79 50.78 97.44 210.71

Adding an INFO button to your directory

An information button can be useful for your directory to help you remember the

operation of the functions in the directory. In this directory, all we need to

remember is to press @TRISO to get a triangle solution started. You may want to

type in the following program: <<“Press [TRISO] to start.“ MSGBOX >>, and store it

in a variable called INFO. As a result, the first variable in your directory will be

the @INFO button.

Application 2 - Velocity and acceleration in polar coordinates

Two-dimensional particle motion in polar coordinates often involves determining

the radial and transverse components of the velocity and acceleration of the

particle given r, r’ = dr/dt, r” = d r/dt , θ, θ’ = d θ /dt, and, θ” = d θ/dt .

The following equations are used:

v_r= r

a_r= r − rθ

v_θ= rθ a_θ= rθ + 2rθ

Create a subdirectory called POLC (POLar Coordinates), which we will use to

calculate velocities and accelerations in polar coordinates. Within that

subdirectory, enter the following variables:

Program or value

Store into variable:

<< PEQ STEQ MINIT NAME LIST MITM MSOLVR >>

"vel. & acc. polar coord."

{ r rD rDD θD θDD vr vθ v ar aθ a }

{ 'vr = rD' 'vθ = r*θD' 'v = √(vr^2 + vθ^2)'

'ar = rDD − r*θD^2' 'aθ = r*θDD + 2*rD*θD'

'a = √(ar^2 + aθ^2)' }

SOLVEP

NAME

LIST

PEQ

Page 7-17

An explanation of the variables follows:

SOLVEP = a program that triggers the multiple equation solver for the

particular set of equations stored in variable PEQ;

NAME = a variable storing the name of the multiple equation solver, namely,

"vel. & acc. polar coord.";

LIST =

a list of the variable used in the calculations, placed in the order we

want them to show up in the multiple equation solver environment;

PEQ =

list of equations to be solved, corresponding to the radial and

transverse components of velocity (vr, vθ) and acceleration (ar, aθ)

in polar coordinates, as well as equations to calculate the

magnitude of the velocity (v) and the acceleration (a) when the

polar components are known.

r, rD, rDD = r (radial coordinate), r-dot (first derivative of r), r-double dot

(second derivative of r).

θD, θDD = θ-dot (first derivative of θ), θ-double dot (second derivative of θ).

________________________________________________________________

Suppose you are given the following information: r = 2.5, rD = 0.5, rDD = -

1.5, θD = 2.3, θDD = -6.5, and you are asked to find vr, vθ, ar, aθ, v, and a.

Start the multiple equation solver by pressing J@SOLVE. The calculator

produces a screen labeled , "vel. & acc. polar coord.", that looks as follows:

To enter the values of the known variables, just type the value and press the

button corresponding to the variable to be entered. Use the following

keystrokes: 2.5 [ r ] 0.5 [ rD ] 1.5 \ [ rDD ] 2.3 [ θD ] 6.5 \ [ θDD ].

Page 7-18

Notice that after you enter a particular value, the calculator displays the

variable and its value in the upper left corner of the display. We have now

entered the known variables. To calculate the unknowns we can proceed in two

ways:

a). Solve for individual variables, for example, „[ vr ] gives vr: 0.500.

Press L„[ vθ ] to get vθ : 5.750 , and so on. The remaining results

are v: 5.77169819031; ar: -14.725; aθ: -13.95; and a:

20.2836911089.; or,

b). Solve for all variables at once, by pressing „@ALL!. The calculator will

flash the solutions as it finds them. When the calculator stops, you can

press ‚@ALL! to list all results. For this case we have:

Pressing the soft-menu key @EQNS will let you know the equations used to

solve for each of the values in the screen:

To use a new set of values press, either @EXIT @@ALL@ LL, or J @SOLVE.

Let's try another example using r = 2.5, vr = rD = -0.5, rDD = 1.5, v = 3.0, a =

25.0. Find, θD, θDD, vθ, ar, and aθ. You should get the following results:

Page 7-19

Page 7-20

Chapter 8

Operations with lists

Lists are a type of calculator’s object that can be useful for data processing and

in programming. This Chapter presents examples of operations with lists.

Definitions

A list, within the context of the calculator, is a series of objects enclosed

between braces and separated by spaces (#), in the RPN mode, or commas

(‚í), in both modes. Objects that can be included in a list are numbers,

letters, character strings, variable names, and/or operators. Lists are useful for

manipulating data sets and in some programming applications. Some

examples of lists are:

{ t 1 }, {"BET " h2 4}, {1 1.5 2.0},

{a a a a}, { {1 2 3} {3 2 1} {1 2 3}}

In the examples shown below we will limit ourselves to numerical lists.

Creating and storing lists

To create a list in ALG mode, first enter the braces key „ä (associated

with the + key), then type or enter the elements of the list, separating them

with commas (‚í). The following keystrokes will enter the list {1 2 3 4}

and store it into variable L1.

„ä 1 ‚í 2 ‚í 3 ‚í 4

™K~l1`

The screen will show the following:

The figure to the left shows the screen before pressing `, while the one to the

right shows the screen after storing the list into L1. Notice that before pressing

` the list shows the commas separating its elements. However, after pressing

`, the commas are replaced with spaces.

Entering the same list in RPN mode requires the following keystrokes:

„ä 1 # 2 # 3 # 4 `

~l1`K

Page 8-1

The figure below shows the RPN stack before pressing the K key:

Composing and decomposing lists

Composing and decomposing lists makes sense in RPN mode only. Under such

operating mode, decomposing a list is achieved by using function OBJꢀ. With

this function, a list in the RPN stack is decomposed into its elements, with stack

level 1: showing the number of elements in the list. The next two screen shots

show the stack with a small list before and after application of function OBJꢀ:

Notice that, after applying OBJꢀ, the elements of the list occupy levels 4:

through 2:, while level 1: shows the number of elements in the list.

To compose a list in RPN mode, place the elements of the list in the stack, enter

the list size, and apply function ꢀLIST (select it from the function catalog, as

follows: ‚N‚é, then use the up and down arrow keys (—˜) to

locate function ꢀLIST). The following screen shots show the elements of a list of

size 4 before and after application of function ꢀLIST:

Note: Function OBJꢀ applied to a list in ALG mode simply reproduces the list,

adding to it the list size:

Operations with lists of numbers

To demonstrate operations with lists of numbers, we will create a couple of

other lists, besides list L1 created above: L2={-3,2,1,5}, L3={-6,5,3,1,0,3,-4},

L4={3,-2,1,5,3,2,1}. In ALG mode, the screen will look like this after entering

lists L2, L3, L4:

Page 8-2

In RPN mode, the following screen shows the three lists and their names ready

to be stored. To store the lists in this case you need to press K three times.

Changing sign

The sign-change key (\) , when applied to a list of numbers, will change the

sign of all elements in the list. For example:

Addition, subtraction, multiplication, division

Multiplication and division of a list by a single number is distributed across the

list, for example:

Subtraction of a single number from a list will subtract the same number from

each element in the list, for example:

Addition of a single number to a list produces a list augmented by the number,

and not an addition of the single number to each element in the list. For

example:

Page 8-3

Subtraction, multiplication, and division of lists of numbers of the same length

produce a list of the same length with term-by-term operations. Examples:

The division L4/L3 will produce an infinity entry because one of the elements in

L3 is zero:

If the lists involved in the operation have different lengths, an error message is

produced (Error: Invalid Dimension).

The plus sign (+), when applied to lists, acts a concatenation operator,

putting together the two lists, rather than adding them term-by-term. For

example:

In order to produce term-by-term addition of two lists of the same length, we

need to use operator ADD. This operator can be loaded by using the function

catalog (‚N). The screen below shows an application of ADD to add lists

L1 and L2, term-by-term:

Real number functions from the keyboard

Real number functions from the keyboard (ABS, e , LN, 10 , LOG, SIN, x , √,

COS, TAN, ASIN, ACOS, ATAN, y ) can be used on lists. Here are some

examples:

Page 8-4

ABS

EXP and LN

LOG and ANTILOG

SQ and square root

COS, ACOS

SIN, ASIN

TAN, ATAN

INVERSE (1/x)

Real number functions from the MTH menu

Functions of interest from the MTH menu include, from the HYPERBOLIC menu:

SINH, ASINH, COSH, ACOSH, TANH, ATANH, and from the REAL menu: %,

%CH, %T, MIN, MAX, MOD, SIGN, MANT, XPON, IP, FP, RND, TRNC,

FLOOR, CEIL, DꢀR, RꢀD. Some of the functions that take a single argument

are illustrated below applied to lists of real numbers:

SINH, ASINH

COSH, ACOSH

Page 8-5

TANH, ATANH

SIGN, MANT, XPON

IP, FP

FLOOR, CEIL

DꢀR, RꢀD

Examples of functions that use two arguments

The screen shots below show applications of the function % to list arguments.

Function % requires two arguments. The first two examples show cases in which

only one of the two arguments is a list.

The results are lists with the function % distributed according to the list

argument. For example,

%({10, 20, 30},1) = {%(10,1),%(20,1),%(30,1)},

while

%(5,{10,20,30}) = {%(5,10),%(5,20),%(5,30)}

In the following example, both arguments of function % are lists of the same

size. In this case, a term-by-term distribution of the arguments is performed, i.e.,

Page 8-6

%({10,20,30},{1,2,3}) = {%(10,1),%(20,2),%(30,3)}

This description of function % for list arguments shows the general pattern of

evaluation of any function with two arguments when one or both arguments are

lists. Examples of applications of function RND are shown next:

Lists of complex numbers

The following exercise shows how to create a list of complex numbers given two

lists of the same length, one representing the real parts and one the imaginary

parts of the complex numbers. Use L1 ADD i*L2.

Functions such as LN, EXP, SQ, etc., can also be applied to a list of complex

numbers, e.g.,

Page 8-7

The following example shows applications of the functions RE(Real part),

IM(imaginary part), ABS(magnitude), and ARG(argument) of complex

numbers. The results are lists of real numbers:

Lists of algebraic objects

The following are examples of lists of algebraic objects with the function SIN

applied to them:

The MTH/LIST menu

The MTH menu provides a number of functions that exclusively to lists. With

flag 117 set to CHOOSE boxes:

Next, with system flag 117 set to SOFT menus:

Page 8-8

This menu contains the following functions:

ΔLIST

ΣLIST

ΠLIST

SORT

: Calculate increment among consecutive elements in list

: Calculate summation of elements in the list

: Calculate product of elements in the list

: Sorts elements in increasing order

REVLIST : Reverses order of list

ADD

: Operator for term-by-term addition of two lists of the same length

(examples of this operator were shown above)

Examples of application of these functions in ALG mode are shown next:

SORT and REVLIST can be combined to sort a list in decreasing order:

If you are working in RPN mode, enter the list onto the stack and then select the

operation you want. For example, to calculate the increment between

consecutive elements in list L3, press:

l3`!´˜˜ #OK# #OK#

This places L3 onto the stack and then selects the ΔLIST operation from the MTH

menu.

Page 8-9

Manipulating elements of a list

The PRG (programming) menu includes a LIST sub-menu with a number of

functions to manipulate elements of a list. With system flag 117 set to

CHOOSE boxes:

Item 1. ELEMENTS.. contains the following functions that can be used for the

manipulation of elements in lists:

List size

Function SIZE, from the PRG/LIST/ELEMENTS sub-menu, can be used to obtain

the size (also known as length) of the list, e.g.,

Extracting and inserting elements in a list

To extract elements of a list we use function GET, available in the PRG/LIST/

ELEMENTS sub-menu. The arguments of function GET are the list and the

number of the element you want to extract. To insert an element into a list use

function PUT (also available in the PRG/LST/ELEMENTS sub-menu). The

arguments of function PUT are the list, the position that one wants to replace,

and the value that will be replaced. Examples of applications of functions GET

and PUT are shown in the following screen:

Page 8-10

Functions GETI and PUTI, also available in sub-menu PRG/ ELEMENTS/, can

also be used to extract and place elements in a list. These two functions,

however, are useful mainly in programming. Function GETI uses the same

arguments as GET and returns the list, the element location plus one, and the

element at the location requested. Function PUTI uses the same arguments as

GET and returns the list and the list size.

Element position in the list

To determine the position of an element in a list use function POS having the list

and the element of interest as arguments. For example,

HEAD and TAIL functions

The HEAD function extracts the first element in the list. The TAIL function

removes the first element of a list, returning the remaining list. Some examples

are shown next:

The SEQ function

Item 2. PROCEDURES.. in the PRG/LIST menu contains the following functions

that can be used to operate on lists.

Functions REVLIST and SORT were introduced earlier as part of the MTH/LIST

menu. Functions DOLIST, DOSUBS, NSUB, ENDSUB, and STREAM, are

designed as programming functions for operating lists in RPN mode. Function

Page 8-11

SEQ is useful to produce a list of values given a particular expression and is

described in more detail here.

The SEQ function takes as arguments an expression in terms of an index, the

name of the index, and starting, ending, and increment values for the index,

and returns a list consisting of the evaluation of the expression for all possible

values of the index. The general form of the function is SEQ(expression, index,

start, end, increment).

In the following example, in ALG mode, we identify expression = n , index =

n, start = 1, end = 4, and increment = 1:

The list produced corresponds to the values {1 , 2 , 3 , 4 }. In RPN mode, you

can list the different arguments of the function as follows:

before applying function SEQ.

The MAP function

The MAP function, available through the command catalog (‚N), takes as

arguments a list of numbers and a function f(X) or a program of the form << ꢀ

a … >>, and produces a list consisting of the application of that function or

program to the list of numbers. For example, the following call to function MAP

applies the function SIN(X) to the list {1,2,3}:

In ALG mode, the syntax is:

~~map~!Ü!ä1@í2@í3™@

í S~X`

In RPN mode, the syntax is:

!ä1@í2@í3`³S~X`~~m

ap`

Page 8-12

In both cases, you can either type out the MAP command (as in the examples

above) or select the command from the CAT menu.

The following call to function MAP uses a program instead of a function as

second argument:

Defining functions that use lists

In Chapter 3 we introduced the use of the DEFINE function ( „à) to

create functions of real numbers with one or more arguments. A function

defined with DEF can also be used with list arguments, except that, any function

incorporating an addition must use the ADD operator rather than the plus sign

(+). For example, if we define the function F(X,Y) = (X-5)*(Y-2), shown here

in ALG mode:

we can use lists (e.g., variables L1 and L2, defined earlier in this Chapter) to

evaluate the function, resulting in:

Since the function statement includes no additions, the application of the

function to list arguments is straightforward. However, if we define the function

G(X,Y) = (X+3)*Y, an attempt to evaluate this function with list arguments (L1,

L2) will fail:

To fix this problem we can edit the contents of variable @@@G@@@ , which we can list

in the stack by using …@@@G@@@,

Page 8-13

to replace the plus sign (+) with ADD:

Next, we store the edited expression into variable @@@G@@@:

Evaluating G(L1,L2) now produces the following result:

As an alternative, you can define the function with ADD rather than the plus

sign (+), from the start, i.e., use DEFINE('G(X,Y)=(X DD 3)*Y') :

You can also define the function as G(X,Y) = (X--3)*Y.

Page 8-14

Applications of lists

This section shows a couple of applications of lists to the calculation of statistics

of a sample. By a sample we understand a list of values, say, {s , s , …, s }.

Suppose that the sample of interest is the list

{1, 5, 3, 1, 2, 1, 3, 4, 2, 1}

and that we store it into a variable called S (The screen shot below shows this

action in ALG mode, however, the procedure in RPN mode is very similar. Just

keep in mind that in RPN mode you place the arguments of functions in the

stack before activating the function):

Harmonic mean of a list

This is a small enough sample that we can count on the screen the number of

elements (n=10). For a larger list, we can use function SIZE to obtain that

number, e.g.,

Suppose that we want to calculate the harmonic mean of the sample, defined

s_h=

⎛

⎞

1 1

⎜

⎟

+L +

∑

_k=1s_n

n s₁s₂

s_n

⎝

⎠

To calculate this value we can follow this procedure:

1. Apply function INV () to list S:

2. Apply function ΣLIST() to the resulting list in1.

Page 8-15

3. Divide the result above by n = 10:

4. Apply the INV() function to the latest result:

Thus, the harmonic mean of list S is s = 1.6348…

Geometric mean of a list

The geometric mean of a sample is defined as

x_g=

x_k= x₁⋅ x₂L x_n

∏

k=1

To find the geometric mean of the list stored in S, we can use the following

procedure:

1. Apply function ΠLIST() to list S:

2. Apply function XROOT(x,y), i.e., keystrokes ‚», to the result in 1:

Page 8-16

Thus, the geometric mean of list S is s = 1.003203…

Weighted average

Suppose that the data in list S, defined above, namely:

S = {1,5,3,1,2,1,3,4,2,1}

is affected by the weights,

W = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

If we define the weight list as W = {w ,w ,…,w }, we notice that the k -th

element in list W, above, can be defined by w = k. Thus we can use function

SEQ to generate this list, and then store it into variable @@@W@@@ as follows:

Given the data list {s , s , …, s }, and the weight list {w , w , …, w }, the

weighted average of the data in S is defined as

w ⋅ s

∑

k=1

s_w=

∑

k=1

To calculate the weighted average of the data in list S with the weights in list W,

we can use th e following steps:

1. Multiply lists S and W:

2. Use function ΣLIST in this result to calculate the numerator of s :

Page 8-17

3. Use function ΣLIST, once more, to calculate the denominator of s :

4. Use the expression ANS(2)/ANS(1) to calculate the weighted average:

Thus, the weighted average of list S with weights in list W is s = 2.2.

Note: ANS(1) refers to the most recent result (55), while ANS(2) refers to

the previous to last result (121).

Statistics of grouped data

Grouped data is typically given by a table showing the frequency (w) of data in

data classes or bins. Each class or bin is represented by a class mark (s),

typically the midpoint of the class. An example of grouped data is shown next:

Class

mark

Frequency

count

Class

boundaries

0 - 2

2 - 4

4 - 6

6 - 8

8 -10

Page 8-18

The class mark data can be stored in variable S, while the frequency count can

be stored in variable W, as follows:

Given the list of class marks S = {s , s , …, s }, and the list of frequency counts

W = {w , w , …, w }, the weighted average of the data in S with weights W

represents the mean value of the grouped data, that we call ⎯s, in this context:

w ⋅ s

∑

k=1

s =

∑

k=1

where

represents the total frequency count.

N =

∑

k=1

The mean value for the data in lists S and W, therefore, can be calculated using

the procedure outlined above for the weighted average, i.e.,

We’ll store this value into a variable called XBAR:

The variance of this grouped data is defined as

Page 8-19

w ⋅(s − s)²

∑

k=1

V =

∑

k=1

To calculate this last result, we can use the following:

The standard deviation of the grouped data is the square root of the variance:

Page 8-20

Chapter 9

Vectors

This Chapter provides examples of entering and operating with vectors, both

mathematical vectors of many elements, as well as physical vectors of 2 and 3

components.

Definitions

From a mathematical point of view, a vector is an array of 2 or more elements

arranged into a row or a column. These will be referred to as row and column

vectors. Examples are shown below:

−1

⎡

⎢

⎤

⎥

v = 3 , u = [1,− 3,5,2]

⎥

⎢

⎥

⎣

⎦

Physical vectors have two or three components and can be used to represent

physical quantities such as position, velocity, acceleration, forces, moments,

linear and angular momentum, angular velocity and acceleration, etc.

Referring to a Cartesian coordinate system (x,y,z), there exists unit vectors i, j, k

associated with each coordinate direction, such that a physical vector A can be

written in terms of its components A , A , A , as A = A i + A j + A k.

Alternative notation for this vector are: A = [A , A , A ], A = (A , A , A ), or A

= < A , A , A >. A two dimensional version of this vector will be written as A

= A i + A j, A = [A , A ], A = (A , A ), or A = < A , A >. Since in the

calculator vectors are written between brackets [ ], we will choose the notation

A = [A , A , A ] or A = [A , A , A ], to refer to two- and three-dimensional

vectors from now on. The magnitude of a vector A is defined as |A| =

A_x²+ A_y²+ A_z²

. A unit vector in the direction of vector A, is defined as e =

A/|A|. Vectors can be multiplied by a scalar, e.g., kA = [kA , kA , kA ].

Physically, the vector kA is parallel to vector A, if k>0, or anti-parallel to vector

A, if k<0. The negative of a vector is defined as –A = (–1)A = [–A , –A , –A ].

Division by as scalar can be interpreted as a multiplication, i.e., A/k = (1/k)⋅A.

Addition and subtraction of vectors are defined as A B = [A B , A B , A

B ], where B is the vector B = [B , B , B ].

There are two definitions of products of physical vectors, a scalar or internal

product (the dot product) and a vector or external product (the cross product).

The dot product produces a scalar value defined as A•B = |A||B|cos(θ),

Page 9-1

where θ is the angle between the two vectors. The cross product produces a

vector A×B whose magnitude is |A×B| = |A||B|sin(θ), and its direction is

given by the so-called right-hand rule (consult a textbook on Math, Physics, or

Mechanics to see this operation illustrated graphically). In terms of Cartesian

components, A•B = A B +A B +A B , and A×B = [A B -A B ,A B -A B ,A B -

x x

y y

z z

y z z y z x x z x y

A B ]. The angle between two vectors can be found from the definition of the

y x

dot product as cos(θ) = A•B/|A||B|= e •e . Thus, if two vectors A and B

rad

are perpendicular (θ = 90 = π/2 ), A•B = 0.

Entering vectors

In the calculator, vectors are represented by a sequence of numbers enclosed

between brackets, and typically entered as row vectors. The brackets are

generated in the calculator by the keystroke combination „Ô , associated

with the * key. The following are examples of vectors in the calculator:

[3.5, 2.2, -1.3, 5.6, 2.3]

[1.5,-2.2]

A general row vector

A 2-D vector

[3,-1,2]

A 3-D vector

['t','t^2','SIN(t)']

A vector of algebraics

Typing vectors in the stack

With the calculator in ALG mode, a vector is typed into the stack by opening a

set of brackets („Ô) and typing the components or elements of the vector

separated by commas (‚í). The screen shots below show the entering of

a numerical vector followed by an algebraic vector. The figure to the left shows

the algebraic vector before pressing „. The figure to the right shows the

calculator’s screen after entering the algebraic vector:

In RPN mode, you can enter a vector in the stack by opening a set of brackets

and typing the vector components or elements separated by either commas

(‚í) or spaces (#). Notice that after pressing ` , in either mode,

the calculator shows the vector elements separated by spaces.

Page 9-2

Storing vectors into variables

Vectors can be stored into variables. The screen shots below show the vectors

u = [1, 2], u = [-3, 2, -2], v = [3,-1], v = [1, -5, 2]

stored into variables @@@u2@@, @@@u3@@, @@@v2@@, and @@@v3@@, respectively. First, in ALG

mode:

Then, in RPN mode (before pressing K, repeatedly):

Using the Matrix Writer (MTRW) to enter vectors

Vectors can also be entered by using the Matrix Writer „²(third key in the

fourth row of keys from the top of the keyboard). This command generates a

species of spreadsheet corresponding to rows and columns of a matrix (Details

on using the Matrix Writer to enter matrices will be presented in a subsequent

chapter). For a vector we are interested in filling only elements in the top row.

By default, the cell in the top row and first column is selected. At the bottom of

the spreadsheet you will find the following soft menu keys:

@EDIT! @VECꢀ ←WID @WID→ @GO→ꢀ @GO↓

The @EDIT key is used to edit the contents of a selected cell in the Matrix

Writer.

The @VEC@@ key, when selected, will produce a vector, as opposite to a

matrix of one row and many columns.

Page 9-3

Vectors vs. matrices

To see the @VEC@ key in action, try the following exercises:

(1) Launch the Matrix Writer („²). With @VECꢀ and @GO→ꢀ selected, enter

3`5`2``. This produces [3. 5. 2.]. (In RPN mode,

you can use the following keystroke sequence to produce the same result:

3#5#2``).

(2) With @VEC@@ deselected and @GO→ꢀ selected,, enter

3#5#2``. This produces [[3. 5. 2.]].

Although these two results differ only in the number of brackets used, for the

calculator they represent different mathematical objects. The first one is a vector

with three elements, and the second one a matrix with one row and three

columns. There are differences in the way that mathematical operations take

place on a vector as opposite to a matrix. Therefore, for the time being, keep

the soft menu key @VECꢀ selected while using the Matrix Writer.

The ←WID key is used to decrease the width of the columns in the

spreadsheet. Press this key a couple of times to see the column width

decrease in your Matrix Writer.

The @WID→ key is used to increase the width of the columns in the

spreadsheet. Press this key a couple of times to see the column width

increase in your Matrix Writer.

The @GO→ꢀ key, when selected, automatically selects the next cell to the

right of the current cell when you press `. This option is selected by

default.

The @GO↓ key, when selected, automatically selects the next cell below

the current cell when you press `.

Moving to the right vs. moving down in the Matrix Writer

Activate the Matrix Writer and enter 3`5`2`` with the

@GO→ꢀ key selected (default). Next, enter the same sequence of numbers with

the @GO↓ꢀ key selected to see the difference. In the first case you entered a

vector of three elements. In the second case you entered a matrix of three rows

and one column.

Activate the Matrix Writer again by using „², and press L to check out

the second soft key menu at the bottom of the display. It will show the keys:

@+ROW@ @-ROW @+COL@ @-COL@ @→STK@@ @GOTO@

Page 9-4

The @+ROW@ key will add a row full of zeros at the location of the selected

cell of the spreadsheet.

The @-ROW key will delete the row corresponding to the selected cell of

the spreadsheet.

The @+COL@ key will add a column full of zeros at the location of the

selected cell of the spreadsheet.

The @-COL@ key will delete the column corresponding to the selected cell

of the spreadsheet.

The ^@→STK@@ key will place the contents of the selected cell on the stack.

The @GOTO@ key, when pressed, will request that the user indicate the

number of the row and column where he or she wants to position the

cursor.

Pressing L once more produces the last menu, which contains only one

function @@DEL@ (delete).

The function @@DEL@ will delete the contents of the selected cell and

replace it with a zero.

To see these keys in action try the following exercise:

(1) Activate the Matrix Writer by using „². Make sure the @VECꢀ and

@GO→ꢀ keys are selected.

(2) Enter the following:

1`2`3`

L @GOTO@ 2@@OK@@ 1 @@OK@@ @@OK@@

2`1`5`

4`5`6`

7`8`9`

(3) Move the cursor up two positions by using ———. Then press @-ROW.

The second row will disappear.

(4) Press @+ROW@. A row of three zeroes appears in the second row.

(5) Press @-COL@. The first column will disappear.

(6) Press @+COL@. A row of two zeroes appears in the first row.

(7) Press @GOTO@ 3@@OK@@ 3@@OK@@ @@OK@@ to move to position (3,3).

(8) Press ^@→STK@@. This will place the contents of cell (3,3) on the stack, although

you will not be able to see it yet.

(9) Press ` to return to normal display. Element (3,3) and the full matrix will

be available in the screen.

Page 9-5

Summary of Matrix Writer use for entering vectors

In summary, to enter a vector using the Matrix Writer, simply activate the writer

(„²), and place the elements of the vector, pressing ` after each of

them. Then, press ``. Make sure that the @VECꢀ and @GO→ꢀ@ keys are

selected.

Example: „²³~„xQ2`2`5\``

produces:

[‘x^2‘ 2 –5 ]

Building a vector with ꢀARRY

The function →ARRY, available in the function catalog (‚N‚é, use

—˜ to locate the function), can also be used to build a vector or array in

the following way. In ALG mode, enter ꢀARRY(vector elements, number of

elements), e.g.,

In RPN mode:

(1) Enter the n elements of the array in the order you want them to appear in

the array (when read from left to right) into the RPN stack.

(2) Enter n as the last entry.

(3) Use function ꢀARRY.

The following screen shots show the RPN stack before and after applying

function ꢀARRY:

Page 9-6

In RPN mode, the function [→ARRY] takes the objects from stack levels n+1, n,

n-1, …, down to stack levels 3 and 2, and converts them into a vector of n

elements. The object originally at stack level n+1 becomes the first element, the

object originally at level n becomes the second element, and so on.

Note: Function ꢀARRY is also available in the PRG/TYPE menu („°)

Identifying, extracting, and inserting vector elements

If you store a vector into a variable name, say A, you can identify elements of

the vector by using A(i), where i is an integer number less than or equal to the

vector size. For example, create the following array and store it in variable A:

[-1, -2, -3, -4, -5]:

To recall the third element of A, for example, you could type in A(3) into the

calculator. In ALG mode, simply type A(3). In RPN mode, type ‘A(3)’ `μ.

You can operate with elements of the array by writing and evaluating algebraic

expressions such as:

More complicated expressions involving elements of A can also be written. For

example, using the Equation Writer (‚O), we can write the following

summation of the elements of A:

Page 9-7

Highlighting the entire expression and using the @EVAL@ soft menu key, we get the

result: -15.

Note: The vector A can also be referred to as an indexed variable because

the name A represents not one, but many values identified by a sub-index.

To replace an element in an array use function PUT (you can find it in the

function catalog ‚N, or in the PRG/LIST/ELEMENTS sub-menu – the later

was introduced in Chapter 8). In ALG mode, you need to use function PUT with

the following arguments: PUT(array, location to be replaced, new value). For

example, to change the contents of A(3) to 4.5, use:

In RPN mode, you can change the value of an element of A, by storing a new

value in that particular element. For example, if we want to change the

contents of A(3) to read 4.5 instead of its current value of –3., use:

4.5`³~a„Ü3`K

To verify that the change took place use: ‚@@@@A@@ . The result now shown is: [-

1 -2 4.5 -4 -5 ].

Note: This approach for changing the value of an array element is not

allowed in ALG mode, if you try to store 4.5 into A(3) in this mode you get the

following error message: Invalid Syntax.

To find the length of a vector you can use the function SIZE, available through

the command catalog (N) or through the PRG/LIST/ELEMENTS sub-menu.

Some examples, based on the arrays or vectors stored previously, are shown

below:

Page 9-8

Simple operations with vectors

To illustrate operations with vectors we will use the vectors A, u2, u3, v2, and

v3, stored in an earlier exercise.

Changing sign

To change the sign of a vector use the key \, e.g.,

Addition, subtraction

Addition and subtraction of vectors require that the two vector operands have

the same length:

Attempting to add or subtract vectors of different length produces an error

message (Invalid Dimension), e.g., v2+v3, u2+u3, A+v3, etc.

Multiplication by a scalar, and division by a scalar

Multiplication by a scalar or division by a scalar is straightforward:

Page 9-9

Absolute value function

The absolute value function (ABS), when applied to a vector, produces the

magnitude of the vector. For a vector A = [A ,A ,…,A ], the magnitude is

defined as

. In the ALG mode, enter the function

| A |= A_x+ A_y+L + A_z

name followed by the vector argument. For example: BS([1,-2,6]),

BS( ), BS(u3), will show in the screen as follows:

The MTH/VECTOR menu

The MTH menu („´) contains a menu of functions that specifically to

vector objects:

The VECTOR menu contains the following functions (system flag 117 set to

CHOOSE boxes):

Magnitude

The magnitude of a vector, as discussed earlier, can be found with function

ABS. This function is also available from the keyboard („Ê). Examples of

application of function ABS were shown above.

Page 9-10

Dot product

Function DOT is used to calculate the dot product of two vectors of the same

length. Some examples of application of function DOT, using the vectors A, u2,

u3, v2, and v3, stored earlier, are shown next in ALG mode. Attempts to

calculate the dot product of two vectors of different length produce an error

message:

Cross product

Function CROSS is used to calculate the cross product of two 2-D vectors, of two

3-D vectors, or of one 2-D and one 3-D vector. For the purpose of calculating a

cross product, a 2-D vector of the form [A , A ], is treated as the 3-D vector [A ,

A ,0]. Examples in ALG mode are shown next for two 2-D and two 3-D vectors.

Notice that the cross product of two 2-D vectors will produce a vector in the z-

direction only, i.e., a vector of the form [0, 0, C ]:

Examples of cross products of one 3-D vector with one 2-D vector, or vice versa,

are presented next:

Attempts to calculate a cross product of vectors of length other than 2 or 3,

produce an error message (Invalid Dimension), e.g., CROSS(v3,A), etc.

Decomposing a vector

Function Vꢀ is used to decompose a vector into its elements or components. If

used in the ALG mode, Vꢀ will provide the elements of the vector in a list, e.g.,

Page 9-11

In the RPN mode, application of function Vꢀ will list the components of a

vector in the stack, e.g., Vꢀ(A) will produce the following output in the RPN

stack (vector A is listed in stack level 6:).

Building a two-dimensional vector

Function ꢀV2 is used in the RPN mode to build a vector with the values in stack

levels 1: and 2:. The following screen shots show the stack before and after

applying function ꢀV2:

Building a three-dimensional vector

Function ꢀV3 is used in the RPN mode to build a vector with the values in stack

levels 1: , 2:, and 3:. The following screen shots show the stack before and

after applying function ꢀV2:

Changing coordinate system

Functions RECT, CYLIN, and SPHERE are used to change the current coordinate

system to rectangular (Cartesian), cylindrical (polar), or spherical coordinates.

The current system is shown highlighted in the corresponding CHOOSE box

(system flag 117 unset), or selected in the corresponding SOFT menu label

(system flag 117 set). In the following figure the RECTangular coordinate

system is shown as selected in these two formats:

Page 9-12

When the rectangular, or Cartesian, coordinate system is selected, the top line

of the display will show an XYZ field, and any 2-D or 3-D vector entered in the

calculator is reproduced as the (x,y,z) components of the vector. Thus, to enter

the vector A = 3i+2j-5k, we use [3,2,-5], and the vector is shown as:

If instead of entering Cartesian components of a vector we enter cylindrical

(polar) components, we need to provide the magnitude, r, of the projection of

the vector on the x-y plane, an angle θ (in the current angular measure)

representing the inclination of r with respect to the positive x-axis , and a z-

component of the vector. The angle θ must be entered preceded by the angle

character (∠), generated by using ~‚6. For example, suppose that we

have a vector with r = 5, θ = 25 (DEG should be selected as the angular

measure), and z = 2.3, we can enter this vector in the following way:

„Ô5 ‚í ~‚6 25 ‚í 2.3

Before pressing `, the screen will look as in the left-hand side of the

following figure. After pressing `, the screen will look as in the right-hand

side of the figure (For this example, the numerical format was changed to Fix,

with three decimals).

Notice that the vector is displayed in Cartesian coordinates , with components

x = r cos(θ), y = r sin(θ), z = z, even though we entered it in polar coordinates.

This is because the vector display will default to the current coordinate system.

For this case, we have x = 4.532, y = 2.112, and z = 2.300.

Suppose that we now enter a vector in spherical coordinates (i.e., in the form

(ρ,θ,φ), where ρ is the length of the vector, θ is the angle that the xy projection

of the vector forms with the positive side of the x-axis, and φ is the angle that ρ

forms with the positive side of the z axis), with ρ = 5, θ = 25 , and φ = 45 .

We will use:„Ô5 ‚í ~‚6 25 í

~‚6 45

Page 9-13

The figure below shows the transformation of the vector from spherical to

Cartesian coordinates, with x = ρ sin(φ) cos(θ), y = ρ sin (φ) cos (θ), z = ρ

cos(φ). For this case, x = 3.204, y = 1.494, and z = 3.536.

If the CYLINdrical system is selected, the top line of the display will show an R∠

Z field, and a vector entered in cylindrical coordinates will be shown in its

cylindrical (or polar) coordinate form (r,θ,z). To see this in action, change the

coordinate system to CYLINdrical and watch how the vector displayed in the

last screen changes to its cylindrical (polar) coordinate form. The second

component is shown with the angle character in front to emphasize its angular

nature.

The conversion from Cartesian to cylindrical coordinates is such that r =

2 1/2

-1

(x +y ) , θ = tan (y/x), and z = z. For the case shown above the

transformation was such that (x,y,z) = (3.204, 2.112, 2.300), produced (r,θ,z)

= (3.536,25 ,3.536).

At this point, change the angular measure to Radians. If we now enter a vector

of integers in Cartesian form, even if the CYLINdrical coordinate system is

active, it will be shown in Cartesian coordinates, e.g.,

This is because the integer numbers are intended for use with the CAS and,

therefore, the components of this vector are kept in Cartesian form. To force the

conversion to polar coordinates enter the vector components as real numbers

(i.e., add a decimal point), e.g., [2., 3., 5.].

With the cylindrical coordinate system selected, if we enter a vector in spherical

coordinates it will be automatically transformed to its cylindrical (polar)

Page 9-14

equivalent (r,θ,z) with r = ρ sin φ, θ = θ, z = ρ cos φ. For example, the following

figure shows the vector entered in spherical coordinates, and transformed to

polar coordinates. For this case, ρ = 5, θ = 25 , and φ = 45 , while the

transformation shows that r = 3.563, and z = 3.536. (Change to DEG):

Next, let’s change the coordinate system to spherical coordinates by using

function SPHERE from the VECTOR sub-menu in the MTH menu. When this

coordinate system is selected, the display will show the R∠∠ format in the top

line. The last screen will change to show the following:

Notice that the vectors that were written in cylindrical polar coordinates have

now been changed to the spherical coordinate system. The transformation is

2 1/2

-1

such that ρ = (r +z ) , θ = θ, and φ = tan (r/z). However, the vector that

originally was set to Cartesian coordinates remains in that form.

Application of vector operations

This section contains some examples of vector operations that you may

encounter in Physics or Mechanics applications.

Resultant of forces

Suppose that a particle is subject to the following forces (in N): F = 3i+5j+2k,

F = -2i+3j-5k, and F = 2i-3k. To determine the resultant, i.e., the sum, of all

these forces, you can use the following approach in ALG mode:

Thus, the resultant is R = F + F + F = (3i+8j-6k)N. RPN mode use:

[3,5,2] ` [-2,3,-5] ` [2,0,3] ` + +

Angle between vectors

The angle between two vectors A, B, can be found as θ =cos (A•B/|A||B|)

-1

Page 9-15

Suppose that you want to find the angle between vectors A = 3i-5j+6k, B =

2i+j-3k, you could try the following operation (angular measure set to degrees)

in ALG mode:

1 - Enter vectors [3,-5,6], press `, [2,1,-3], press `.

2 - DOT(ANS(1),ANS(2)) calculates the dot product

3 - ABS(ANS(3))*ABS((ANS(2)) calculates product of magnitudes

4 - ANS(2)/ANS(1) calculates cos(θ)

5 - ACOS(ANS(1)), followed by ,ꢀNUM(ANS(1)), calculates θ

The steps are shown in the following screens (ALG mode, of course):

ꢀꢀꢀ

Thus, the result is θ = 122.891 . In RPN mode use the following:

[3,-5,6] ` [2,1,-3] ` DOT

[3,-5,6] ` BS [2,1,-3] ` BS *

COS ꢀNUM

Moment of a force

The moment exerted by a force F about a point O is defined as the cross-

product M = r×F, where r, also known as the arm of the force, is the position

vector based at O and pointing towards the point of application of the force.

Suppose that a force F = (2i+5j-6k) N has an arm r = (3i-5j+4k)m. To

determine the moment exerted by the force with that arm, we use function

CROSS as shown next:

Page 9-16

Thus, M = (10i+26j+25k) m⋅N. We know that the magnitude of M is such that

|M| = |r||F|sin(θ), where θ is the angle between r and F. We can find this

-1

angle as, θ = sin (|M| /|r||F|) by the following operations:

1 – ABS(ANS(1))/(ABS(ANS(2))*ABS(ANS(3)) calculates sin(θ)

2 – ASIN(ANS(1)), followed by ꢀNUM(ANS(1)) calculates θ

These operations are shown, in ALG mode, in the following screens:

Thus the angle between vectors r and F is θ = 41.038 . RPN mode, we can

use: [3,-5,4] ` [2,5,-6] ` CROSS BS [3,-5,4] `

BS [2,5,-6] ` BS * / SIN ꢀNUM

Equation of a plane in space

Given a point in space P (x ,y ,z ) and a vector N = N i+N j+N k normal to

a plane containing point P , the problem is to find the equation of the plane.

We can form a vector starting at point P and ending at point P(x,y,z), a

generic point in the plane. Thus, this vector r = P P = (x-x )i+ (y-y )j + (z-z )k,

is perpendicular to the normal vector N, since r is contained entirely in the

plane. We learned that for two normal vectors N and r, N•r =0. Thus, we can

use this result to determine the equation of the plane.

To illustrate the use of this approach, consider the point P (2,3,-1) and the

normal vector N = 4i+6j+2k, we can enter vector N and point P as two

vectors, as shown below. We also enter the vector [x,y,z] last:

Page 9-17

Next, we calculate vector P P = r as ANS(1) – ANS(2), i.e.,

Finally, we take the dot product of ANS(1) and ANS(4) and make it equal to

zero to complete the operation N•r =0:

We can now use function EXPAND (in the ALG menu) to expand this

expression:

Thus, the equation of the plane through point P (2,3,-1) and having normal

vector N = 4i+6j+2k, is 4x + 6y + 2z – 24 = 0. In RPN mode, use:

[2,3,-1] ` ['x','y','z'] ` - [4,6,2] DOT EXP ND

Row vectors, column vectors, and lists

The vectors presented in this chapter are all row vectors. In some instances, it is

necessary to create a column vector (e.g., to use the pre-defined statistical

functions in the calculator). The simplest way to enter a column vector is by

enclosing each vector element within brackets, all contained within an external

set of brackets. For example, enter:

[[1.2],[2.5],[3.2],[4.5],[6.2]] `

This is represented as the following column vector:

Page 9-18

In this section we will showing you ways to transform: a column vector into a

row vector, a row vector into a column vector, a list into a vector, and a vector

(or matrix) into a list.

We first demonstrate these transformations using the RPN mode. In this mode,

we will use functions OBJꢀ, ꢀLIST, ꢀARRY and DROP to perform the

transformation. To facilitate accessing these functions we will set system flag

117 to SOFT menus (see Chapter 1). With this flag set, functions OBJꢀ,

ꢀARRY, and ꢀLIST will be accessible by using „° @TYPE!. Functions

OBJꢀ, ꢀARRY, and ꢀLIST will be available in soft menu keys A, B, and

C. Function DROP is available by using „°@STACK @DROP.

Following we introduce the operation of functions OBJꢀ, ꢀLIST, ꢀARRY, and

DROP with some examples.

Function OBJꢀ

This function decomposes an object into its components. If the argument is a

list, function OBJꢀ will list the list elements in the stack, with the number of

elements in stack level 1, for example: {1,2,3} ` „°@TYPE! @OBJꢀ@

results in:

When function OBJꢀ is applied to a vector, it will list the elements of the vector

in the stack, with the number of elements in level 1: enclosed in braces (a list).

The following example illustrates this application: [1,2,3] `

„°@TYPE! @OBJꢀ@ results in:

Page 9-19

If we now apply function OBJꢀ once more, the list in stack level 1:, {3.}, will be

decomposed as follows:

Function ꢀLIST

This function is used to create a list given the elements of the list and the list

length or size. In RPN mode, the list size, say, n, should be placed in stack level

1:. The elements of the list should be located in stack levels 2:, 3:, …, n+1:.

For example, to create the list {1, 2, 3}, type: 1` 2` 3`

3` „°@TYPE! !ꢀLIST@.

Function ꢀARRY

This function is used to create a vector or a matrix. In this section, we will use it

to build a vector or a column vector (i.e., a matrix of n rows and 1 column). To

build a regular vector we enter the elements of the vector in the stack, and in

stack level 1: we enter the vector size as a list, e.g., 1` 2`

3` „ä 3` „°@TYPE! !ꢀARRY@.

To build a column vector of n elements, enter the elements of the vector in the

stack, and in stack level 1 enter the list {n 1}. For example,1` 2`

3` „ä 1‚í3` „°@TYPE! !ꢀARRY@.

Function DROP

This function has the same effect as the delete key (ƒ).

Transforming a row vector into a column vector

We illustrate the transformation with vector [1,2,3]. Enter this vector into

the RPN stack to follow the exercise. To transform a row vector into a column

vector, we need to carry on the following operations in the RPN stack:

1 - Decompose the vector with function OBJꢀ

2 - Press 1+ to transform the list in stack level 1: from {3} to {3,1}

Page 9-20

3 - Use function ꢀARRY to build the column vector

These three steps can be put together into a UserRPL program, entered as

follows (in RPN mode, still): ‚å„°@TYPE! @OBJꢀ@ 1 + !ꢀARRY@

`³~~rxc` K

A new variable, @@RXC@@, will be available in the soft menu labels after pressing

Press ‚@@RXC@@ to see the program contained in the variable RXC:

<< OBJꢀ 1 + ꢀ RRY >>

This variable, @@RXC@@, can now be used to directly transform a row vector to a

column vector. In RPN mode, enter the row vector, and then press @@RXC@@. Try,

for example: [1,2,3] ` @@RXC@@.

After having defined this variable , we can use it in ALG mode to transform a

row vector into a column vector. Thus, change your calculator’s mode to ALG

and try the following procedure: [1,2,3] ` J @@RXC@@ „ Ü „

î, resulting in:

Transforming a column vector into a row vector

To illustrate this transformation, we’ll enter the column vector

[[1],[2],[3]] in RPN mode. Then, follow the next exercise to transform

a row vector into a column vector:

1 - Use function OBJꢀ to decompose the column vector

Page 9-21

2 - Use function OBJꢀ to decompose the list in stack level 1:

3 - Press the delete key ƒ (also known as function DROP) to eliminate the

number in stack level 1:

4 - Use function ꢀLIST to create a list

5 - Use function ꢀARRY to create the row vector

These five steps can be put together into a UserRPL program, entered as follows

(in RPN mode, still):

‚å„°@TYPE! @OBJꢀ@ @OBJꢀ@

„°@STACK @DROP „°@TYPE! !ꢀLIST@ !ꢀARRY@ `

³~~cxr ` K

A new variable, @@CXR@@, will be available in the soft menu labels after pressing

Press ‚@@CXR@@ to see the program contained in the variable CXR:

<< OBJꢀ OBJꢀ DROP ꢀ RRY >>

Page 9-22

This variable, @@CXR@@, can now be used to directly transform a column vector to a

row vector. In RPN mode, enter the column vector, and then press @@CXR@@. Try,

for example:

[[1],[2],[3]] ` @@CXR@@.

After having defined variable @@CXR@@, we can use it in ALG mode to transform a

row vector into a column vector. Thus, change your calculator’s mode to ALG

and try the following procedure:

[[1],[2],[3]] ` J @@CXR@@ „Ü „î

resulting in:

Transforming a list into a vector

To illustrate this transformation, we’ll enter the list {1,2,3} in RPN mode.

Then, follow the next exercise to transform a list into a vector:

1 - Use function OBJꢀ to decompose the column vector

2 - Type a 1 and use function ꢀLIST to create a list in stack level 1:

3 - Use function ꢀARRY to create the vector

These three steps can be put together into a UserRPL program, entered as

follows (in RPN mode):

‚å„°@TYPE! @OBJꢀ@ 1 !ꢀLIST@ !ꢀARRY@ `

³~~lxv ` K

Page 9-23

A new variable, @@LXV@@, will be available in the soft menu labels after pressing

Press ‚@@LXV@@ to see the program contained in the variable LXV:

<< OBJꢀ 1 ꢀLIST ꢀ RRY >>

This variable, @@LXV@@, can now be used to directly transform a list into a vector.

In RPN mode, enter the list, and then press @@LXV@@. Try, for example: {1,2,3}

` @@LXV@@.

After having defined variable @@LXV@@, we can use it in ALG mode to transform a

list into a vector. Thus, change your calculator’s mode to ALG and try the

following procedure: {1,2,3} ` J @@LXV@@ „Ü „î, resulting

in:

Transforming a vector (or matrix) into a list

To transform a vector into a list, the calculator provides function AXL. You can

find this function through the command catalog, as follows:

‚N~~axl~@@OK@@

As an example, apply function AXL to the vector [1,2,3] in RPN mode by

using:[1,2,3] ` XL. The following screen shot shows the application of

function AXL to the same vector in ALG mode.

Page 9-24

Chapter 10ꢀ

Creating and manipulating matrices

This chapter shows a number of examples aimed at creating matrices in the

calculator and demonstrating manipulation of matrix elements.

Definitions

A matrix is simply a rectangular array of objects (e.g., numbers, algebraics)

having a number of rows and columns. A matrix A having n rows and m

columns will have, therefore, n×m elements. A generic element of the matrix is

represented by the indexed variable a , corresponding to row i and column j.

With this notation we can write matrix A as A = [a ]

. The full matrix is

ij n×m

shown next:

a₁₂L a_1m

⎡

⎢

⎣

⎤

⎥

⎦

a₂₁a₂₂L a_2m

A = [a_ij]_n×m

a_n1a_n2L a_nm

A matrix is square if m = n. The transpose of a matrix is constructed by

swapping rows for columns and vice versa. Thus, the transpose of matrix A, is

A = [(a ) ]

= [a ] . The main diagonal of a square matrix is the

ij m×n

ji m×n

collection of elements a . An identity matrix, I , is a square matrix whose

n×n

main diagonal elements are all equal to 1, and all off-diagonal elements are

zero. For example, a 3×3 identity matrix is written as

1 0 0

⎡

⎢

⎤

⎥

I = 0 1 0

⎢

⎥

0 0 1

⎣

⎦

An identity matrix can be written as I

= [δ ], where δ is a function known as

ij ij

n×n

Kronecker’s delta, and defined as

1, if i = j

0, if i ≠ j

⎧

⎨

⎩

δ =

Page 10-1

Entering matrices in the stack

In this section we present two different methods to enter matrices in the

calculator stack: (1) using the Matrix Writer, and (2) typing the matrix directly

into the stack.

Using the Matrix Writer

As with the case of vectors, discussed in Chapter 9, matrices can be entered

into the stack by using the Matrix Writer. For example, to enter the matrix:

− 2.5 4.2 2.0

⎡

⎢

⎤

⎥

0.3

1.9 2.8 ,

⎢

⎥

− 0.1 0.5

⎣

⎦

first, start the matrix writer by using „². Make sure that the option @GO→ꢀ

is selected. Then use the following keystrokes:

2.5\` 4.2` 2`˜ššš

.3` 1.9` 2.8 `

2` .1\` .5`

At this point, the Matrix Writer screen may look like this:

Press ` once more to place the matrix on the stack. The ALG mode stack is

shown next, before and after pressing `, once more:

Page 10-2

If you have selected the textbook display option (using H@DISP! and checking

off ꢀTextbook), the matrix will look like the one shown above. Otherwise, the

display will show:

The display in RPN mode will look very similar to these.

Note: Details on the use of the matrix writer were presented in Chapter 9.

Typing in the matrix directly into the stack

The same result as above can be achieved by entering the following directly

into the stack:

„Ô

„Ô 2.5\ ‚í 4.2 ‚í 2 ™

‚í

„Ô .3 ‚í 1.9 ‚í 2.8 ™

‚í

„Ô 2 ‚í .1\ ‚í .5

Thus, to enter a matrix directly into the stack open a set of brackets („Ô)

and enclose each row of the matrix with an additional set of brackets

(„Ô). Commas (‚í .) should separate the elements of each

row, as well as the brackets between rows. (Note: In RPN mode, you can omit

the inner brackets after the first set has been entered, thus, instead of typing, for

example, [[1 2 3] [4 5 6] [7 8 9]], type [[1 2 3] 4 5 6 7 8 9].)

For future exercises, let’s save this matrix under the name A. In ALG mode use

K~a. In RPN mode, use ³~a K.

Creating matrices with calculator functions

Some matrices can be created by using the calculator functions available in

either the MTH/MATRIX/MAKE sub-menu within the MTH menu („´),

Page 10-3

or in the MATRICES/CREATE menu available through „Ø:

The MTH/MATRIX/MAKE sub menu (let’s call it the MAKE menu) contains the

following functions:

while the MATRICES/CREATE sub-menu (let’s call it the CREATE menu) has the

following functions:

Page 10-4

As you can see from exploring these menus (MAKE and CREATE), they both

have the same functions GET, GETI, PUT, PUTI, SUB, REPL, RDM, RANM,

HILBERT, VANDERMONDE, IDN, CON, →DIAG, and DIAG→. The CREATE

menu includes the COLUMN and ROW sub-menus, that are also available

under the MTH/MATRIX menu. The MAKE menu includes the functions SIZE,

that the CREATE menu does not include. Basically, however, both menus,

MAKE and CREATE, provide the user with the same set of functions. In the

examples that follow, we will show how to access functions through use of the

matrix MAKE menu. At the end of this section we present a table with the

keystrokes required to obtain the same functions with the CREATE menu when

system flag 117 is set to SOFT menus.

If you have set that system flag (flag 117) to SOFT menu, the MAKE menu will

be available through the keystroke sequence: „´!MATRX !MAKE!

The functions available will be shown as soft-menu key labels as follows (press

L to move to the next set of functions):

With system flag 117 set to SOFT menus, the functions of the CREATE menu,

triggered by „Ø@CREAT ,will show as follows:

In the next sections we present applications of the matrix functions in the MAKE

and CREATE menu.

Page 10-5

Functions GET and PUT

Functions GET, GETI, PUT, and PUTI, operate with matrices in a similar manner

as with lists or vectors, i.e., you need to provide the location of the element that

you want to GET or PUT. However, while in lists and vectors only one index is

required to identify an element, in matrices we need a list of two indices {row,

column} to identify matrix elements. Examples of the use of GET and PUT

follow.

Let’s use the matrix we stored above into variable A to demonstrate the use of

the GET and PUT functions. For example, to extract element a from matrix A,

in ALG mode, can be performed as follows:

Notice that we achieve the same result by simply typing (2,3) and pressing

`. In RPN mode, this exercise is performed by entering @@@A@@@ ` 3 `

GET, or by using (2,3) `.

Suppose that we want to place the value ‘π’ into element a of the matrix.

We can use function PUT for that purpose, e.g.,

In RPN mode you can use: J @@@A@@@ {3,1} ` „ì PUT.

Alternatively, in RPN mode you can use: „ì³ (2,3)` K . To

see the contents of variable A after this operation, use @@@A@@@.

Functions GETI and PUTI

Functions PUTI and GETI are used in UserRPL programs since they keep track of

an index for repeated application of the PUT and GET functions. The index list

in matrices varies by columns first. To illustrate its use, we propose the

following exercise in RPN mode: @@@A@@@ {2,2}` GETI. Screen shots showing

the RPN stack before and after the application of function GETI are shown

below:

Page 10-6

Notice that the screen is prepared for a subsequent application of GETI or GET,

by increasing the column index of the original reference by 1, (i.e., from {2,2}

to {2,3}), while showing the extracted value, namely A(2,2) = 1.9, in stack level

Now, suppose that you want to insert the value 2 in element {3 1} using PUTI.

Still in RPN mode, try the following keystrokes: ƒ ƒ{3 1} ` 2 `

PUTI. The screen shots below show the RPN stack before and after the

application of function PUTI:

In this case, the 2 was replaced in position {3 1}, i.e., now A(3,1) = 2, and the

index list was increased by 1 (by column first), i.e., from {3,1} to {3,2}. The

matrix is in level 2, and the incremented index list is in level 1.

Function SIZE

Function SIZE provides a list showing the number of rows and columns of the

matrix in stack level 1. The following screen shows a couple of applications of

function SIZE in ALG mode:

In RPN mode, these exercises are performed by using @@@A@@@ SIZE, and

[[1,2],[3,4]] ` SIZE .

Function TRN

Function TRN is used to produce the transconjugate of a matrix, i.e., the

transpose (TRAN) followed by its complex conjugate (CONJ). For example, the

following screen shot shows the original matrix in variable A and its transpose,

shown in small font display (see Chapter 1):

Page 10-7

If the argument is a real matrix, TRN simply produces the transpose of the real

matrix. Try, for example, TRN(A), and compare it with TRAN(A).

In RPN mode, the transconjugate of matrix A is calculated by using @@@A@@@ TRN.

Note: The calculator also includes Function TRAN in the MATRICES/

OPERATIONS sub-menu:

For example, in ALG mode:

Function CON

The function takes as argument a list of two elements, corresponding to the

number of row and columns of the matrix to be generated, and a constant

value. Function CON generates a matrix with constant elements. For example,

in ALG mode, the following command creates a 4×3 matrix whose elements are

all equal to –1.5:

Page 10-8

In RPN mode this is accomplished by using {4,3} ` 1.5 \

` CON.

Function IDN

Function IDN (IDeNtity matrix) creates an identity matrix given its size. Recall

that an identity matrix has to be a square matrix, therefore, only one value is

required to describe it completely. For example, to create a 4×4 identity matrix

in ALG mode use:

You can also use an existing square matrix as the argument of function IDN,

e.g.,

The resulting identity matrix will have the same dimensions as the argument

matrix. Be aware that an attempt to use a rectangular (i.e., non-square) matrix

as the argument of IDN will produce an error.

In RPN mode, the two exercises shown above are created by using: 4`

IDN and @@@A@@@ IDN.

Function RDM

Function RDM (Re-DiMensioning) is used to re-write vectors and matrices as

matrices and vectors. The input to the function consists of the original vector or

matrix followed by a list of a single number, if converting to a vector, or two

numbers, if converting to a matrix. In the former case the number represents the

Page 10-9

vector’s dimension, in the latter the number of rows and columns of the matrix.

The following examples illustrate the use of function RDM:

Re-dimensioning a vector into a matrix

The following example shows how to re-dimension a vector of 6 elements into a

matrix of 2 rows and 3 columns in ALG mode:

In RPN mode, we can use [1,2,3,4,5,6] ` {2,3} ` RDM to

produce the matrix shown above.

Re-dimensioning a matrix into another matrix

In ALG mode, we now use the matrix created above and re-dimension it into a

matrix of 3 rows and 2 columns:

In RPN mode, we simply use {3,2}` RDM.

Re-dimensioning a matrix into a vector

To re-dimension a matrix into a vector, we use as arguments the matrix followed

by a list containing the number of elements in the matrix. For example, to

convert the matrix from the previous example into a vector of length 6, in ALG

mode, use:

Page 10-10

If using RPN mode, we assume that the matrix is in the stack and use {6} `

RDM.

Note: Function RDM provides a more direct and efficient way to transform lists

to arrays and vice versa, than that provided at the end of Chapter 9.

Function RANM

Function RANM (RANdom Matrix) will generate a matrix with random integer

elements given a list with the number of rows and columns (i.e., the dimensions

of the matrix). For example, in ALG mode, two different 2×3 matrices with

random elements are produced by using the same command, namely,

R NM({2,3}) :

In RPN mode, use {2,3} ` R NM.

Obviously, the results you will get in your calculator will most certainly be

different than those shown above. The random numbers generated are integer

numbers uniformly distributed in the range [-10,10], i.e., each one of those 21

numbers has the same probability of being selected. Function RANM is useful

for generating matrices of any size to illustrate matrix operations, or the

application of matrix functions.

Function SUB

Function SUB extracts a sub-matrix from an existing matrix, provided you

indicate the initial and final position of the sub-matrix. For example, if we want

to extract elements a , a , a , and a from the last result, as a 2×2 sub-

12 13 22

matrix, in ALG mode, use:

Page 10-11

In RPN mode, assuming that the original 2×3 matrix is already in the stack, use

{1,2} ` {2,3} ` SUB.

Function REPL

Function REPL replaces or inserts a sub-matrix into a larger one. The input for

this function is the matrix where the replacement will take place, the location

where the replacement begins, and the matrix to be inserted. For example,

keeping the matrix that we inherited from the previous example, enter the

matrix: [[1,2,3],[4,5,6],[7,8,9]] . In ALG mode, the following

screen shot to the left shows the new matrix before pressing `. The screen

shot to the right shows the application of function RPL to replace the matrix in

NS(2), the 2×2 matrix, into the 3×3 matrix currently located in NS(1),

starting at position {2,2}:

If working in the RPN mode, assuming that the 2×2 matrix was originally in the

stack, we proceed as follows:

[[1,2,3],[4,5,6],[7,8,9]]`™ (this last key swaps the

contents of stack levels 1 and 2) {1,2} ` ™ (another swapping of levels

1 and 2) REPL.

Function →DIAG

Function →DIAG takes the main diagonal of a square matrix of dimensions n×n,

and creates a vector of dimension n containing the elements of the main

diagonal. For example, for the matrix remaining from the previous exercise, we

can extract its main diagonal by using:

Page 10-12

In RPN mode, with the 3×3 matrix in the stack, we simply have to activate

function ꢀDI G to obtain the same result as above.

Function DIAG→

Function DIAG→ takes a vector and a list of matrix dimensions {rows, columns},

and creates a diagonal matrix with the main diagonal replaced with the proper

vector elements. For example, the command

DI Gꢀ([1,-1,2,3],{3,3})

produces a diagonal matrix with the first 3 elements of the vector argument:

In RPN mode, we can use [1,-1,2,3] ` {3,3}` DI Gꢀ to

obtain the same result as above.

Another example of application of the DIAG→ function follows, in ALG mode:

In RPN mode, use [1,2,3,4,5] ` {3,2}` DI Gꢀ .

In this case a 3×2 matrix was to be created using as main diagonal elements as

many elements as possible form the vector [1,2,3,4,5]. The main diagonal, for

a rectangular matrix, starts at position (1,1) and moves on to position (2,2),

(3,3), etc. until either the number of rows or columns is exhausted. In this case,

the number of columns (2) was exhausted before the number of rows (3), so the

main diagonal included only the elements in positions (1,1) and (2,2). Thus,

only the first two elements of the vector were required to form the main

diagonal.

Function VANDERMONDE

Function VANDERMONDE generates the Vandermonde matrix of dimension n

based on a given list of input data. The dimension n is, of course, the length of

the list. If the input list consists of objects {x , x ,… x }, then, a Vandermonde

matrix in the calculator is a matrix made of the following elements:

Page 10-13

1 x₁x₁²L x₁ⁿ⁻¹

⎡

⎤

⎥

⎢

1 x₂x₂²L x₂ⁿ⁻¹

⎢

1 x₃x₃²L x₃ⁿ⁻¹

⎢

1 x_nx_n²L x_nⁿ⁻¹

⎥

⎣

⎦

For example, the following command in ALG mode for the list {1,2,3,4}:

In RPN mode, enter {1,2,3,4} ` V NDERMONDE.

Function HILBERT

Function HILBERT creates the Hilbert matrix corresponding to a dimension n. By

definition, the n×n Hilbert matrix is H = [h ] , so that

jk n×n

h_jk=

j + k −1

The Hilbert matrix has application in numerical curve fitting by the method of

linear squares.

A program to build a matrix out of a number of lists

In this section we provide a couple of UserRPL programs to build a matrix out of

a number of lists of objects. The lists may represent columns of the matrix

(program @CRMC) or rows of the matrix (program @CRMR). The programs are

entered with the calculator set to RPN mode, and the instructions for the

keystrokes are given for system flag 117 set to SOFT menus. This section is

intended for you to practice accessing programming functions in the calculator.

The programs are listed below showing, in the left-hand side, the keystrokes

necessary to enter the program steps, and, in the right-hand side, the characters

Page 10-14

entered in the display as you perform those keystrokes. First, we present the

steps necessary to produce program CRMC.

Lists represent columns of the matrix

The program @CRMC allows you to put together a p×n matrix (i.e., p rows, n

columns) out of n lists of p elements each. To create the program enter the

following keystrokes:

Keystroke sequence:

‚ å

Produces:

„°@STACK! @@DUP@

‚ é # ~ „n

‚ å

DUP

ꢀ n

1„°@STACK! @SWAP

„°@BRCH! @FOR@! @FOR@

~„j

1 SWAP

FOR

„°@TYPE OBJꢀ

ꢀARRY@

OBJꢀ

ꢀARRY

„°@BRCH! @@IF@@ @@IF@@

~ „j#

~ „n

„°@TEST! @@@<@@@

„°@BRCH! @@IF@ @THEN

~ „j #1+

„°@STACK! L@ROLL

„°@BRCH! @@IF@ @END

„°@BRCH! @FOR@! @NEXT

„°@BRCH! @@IF@ @@IF@@

~ „n #1

„°@TEST! @@@>@@@

THEN

j 1 +

ROLL

END

n 1

„°@BRCH! @@IF@ @THEN

THEN

~ „n #1-

„°@BRCH! @FOR@! @FOR@

~ „j #

n 1 -

FOR

~ „j #1+

„°@STACK! L@ROLL!

„°@BRCH! @FOR@! @NEXT!

„°@BRCH! @@IF@! @END@

j 1 +

ROLL

END

Page 10-15

~„n #

„´@MATRX! @COL! @COL!ꢀ

COLꢀ

Program is displayed in level 1

To save the program:

ꢀ³~~crmc~ K

Note: if you save this program in your HOME directory it will be available

from any other sub-directory you use.

To see the contents of the program use J ‚@CRMC. The program listing is

the following:

« DUP → n « 1 SWAP FOR j OBJ→ → RRY IF j n < THEN j 1 +

ROLL END NEXT IF n 1 > THEN 1 n 1 - FOR j j 1 + ROLL

NEXT END n COL→ » »

To use this program, in RPN mode, enter the n lists in the order that you want

them as columns of the matrix, enter the value of n, and press @CRMC. As an

example, try the following exercise:

{1,2,3,4} ` {1,4,9,16} ` {1,8,27,64} ` 3 ` @CRMC

The following screen shots show the RPN stack before and after running

program @CRMC:

To use the program in ALG mode, press @CRMC followed by a set of parentheses

(„Ü). Within the parentheses type the lists of data representing the

columns of the matrix, separated by commas, and finally, a comma, and the

number of columns. The command should look like this:

CRMC({1,2,3,4}, {1,4,9,16}, {1,8,27,64}, 3)

The ALG screen showing the execution of program CRMC is shown below:

Page 10-16

Lists represent rows of the matrix

The previous program can be easily modified to create a matrix when the input

lists will become the rows of the resulting matrix. The only change to be

performed is to change COL→ for ROW→ in the program listing. To perform

this change use:

‚@CRMC

˜‚˜—ššš

ƒƒƒ

List program CRMC in stack

Move to end of program

Delete COL

~~row~`

Type in ROW, enter program

To store the program use: ³~~crmr~ K

{1,2,3,4} ` {1,4,9,16} ` {1,8,27,64} ` 3 ` @CRMR

The following screen shots show the RPN stack before and after running

program @CRMR:

These programs can be useful for statistical applications, specifically to create

the statistical matrix ΣDAT. Examples of the use of these program are shown in

a latter chapters.

Manipulating matrices by columns

The calculator provides a menu with functions for manipulating matrices by

operating in their columns. This menu is available through the MTH/MATRIX/

COL.. sequence: („´) shown in the figure below with system flag 117 set

to CHOOSE boxes:

or through the MATRICES/CREATE/COLUMN sub-menu:

Page 10-17

Both approaches will show the same functions:

When system flag 117 is set to SOFT menus, the COL menu is accessible

through „´!MATRX !@@COL@ , or through „Ø!@CREAT@ !@@COL@ . Both

approaches will show the same set of functions:

The operation of these functions is presented below.

Function →COL

Function →COL takes as argument a matrix and decomposes it into vectors

corresponding to its columns. An application of function ꢀCOL in ALG mode

is shown below. The matrix used has been stored earlier in variable A. The

matrix is shown in the figure to the left. The figure to the right shows the matrix

decomposed in columns. To see the full result, use the line editor (triggered by

pressing ˜).

In RPN mode, you need to list the matrix in the stack, and the activate function

ꢀCOL, i.e., @@@A@@@ ꢀCOL. The figure below shows the RPN stack before and

after the application of function ꢀCOL.

Page 10-18

In this result, the first column occupies the highest stack level after

decomposition, and stack level 1 is occupied by the number of columns of the

original matrix. The matrix does not survive decomposition, i.e., it is no longer

available in the stack.

Function COL→

Function COL→ has the opposite effect of Function →COL, i.e., given n vectors

of the same length, and the number n, function COLꢀ builds a matrix by

placing the input vectors as columns of the resulting matrix. Here is an example

in ALG mode. The command used was:

COLꢀ([1,2,3],[4,5,6],[7,8,9],3)

In RPN mode, place the n vectors in stack levels n+1, n, n-1,…,2, and the

number n in stack level 1. With this set up, function COLꢀ places the vectors

as columns in the resulting matrix. The following figure shows the RPN stack

before and after using function COLꢀ.

Function COL+

Function COL+ takes as argument a matrix, a vector with the same length as

the number of rows in the matrix, and an integer number n representing the

location of a column. Function COL+ inserts the vector in column n of the

matrix. For example, in ALG mode, we’ll insert the second column in matrix A

with the vector [-1,-2,-3], i.e.,

Page 10-19

In RPN mode, enter the matrix first, then the vector, and the column number,

before applying function COL+. The figure below shows the RPN stack before

and after applying function COL+.

Function COL-

Function COL- takes as argument a matrix and an integer number representing

the position of a column in the matrix. Function returns the original matrix minus

a column, as well as the extracted column shown as a vector. Here is an

example in the ALG mode using the matrix stored in A:

In RPN mode, place the matrix in the stack first, then enter the number

representing a column location before applying function COL-. The following

figure shows the RPN stack before and after applying function COL-.

Function CSWP

Function CSWP (Column SWaP) takes as arguments two indices, say, i and j,

(representing two distinct columns in a matrix), and a matrix, and produces a

new matrix with columns i and j swapped. The following example, in ALG

mode, shows an application of this function. We use the matrix stored in

variable A for the example. This matrix is listed first.

Page 10-20

In RPN mode, function CSWP lets you swap the columns of a matrix listed in

stack level 3, whose indices are listed in stack levels 1 and 2. For example, the

following figure shows the RPN stack before and after applying function CSWP

to matrix A in order to swap columns 2 and 3:

As you can see, the columns that originally occupied positions 2 and 3 have

been swapped. Swapping of columns, and of rows (see below), is commonly

used when solving systems of linear equations with matrices. Details of these

operations will be given in a subsequent Chapter.

Manipulating matrices by rows

The calculator provides a menu with functions for manipulating matrices by

operating in their rows. This menu is available through the MTH/MATRIX/

ROW.. sequence: („´) shown in the figure below with system flag 117 set

to CHOOSE boxes:

or through the MATRICES/CREATE/ROW sub-menu:

Both approaches will show the same functions:

Page 10-21

When system flag 117 is set to SOFT menus, the ROW menu is accessible

through „´!MATRX !@@ROW@ , or through „Ø!@CREAT@ !@@ROW@ . Both

approaches will show the same set of functions:

The operation of these functions is presented below.

Function →ROW

Function →ROW takes as argument a matrix and decomposes it into vectors

corresponding to its rows. An application of function ꢀROW in ALG mode is

shown below. The matrix used has been stored earlier in variable A. The

matrix is shown in the figure to the left. The figure to the right shows the matrix

decomposed in rows. To see the full result, use the line editor (triggered by

pressing ˜).

In RPN mode, you need to list the matrix in the stack, and the activate function

ꢀROW, i.e., @@@A@@@ ꢀROW. The figure below shows the RPN stack before and

after the application of function ꢀROW.

In this result, the first row occupies the highest stack level after decomposition,

and stack level 1 is occupied by the number of rows of the original matrix. The

Page 10-22

matrix does not survive decomposition, i.e., it is no longer available in the

stack.

Function ROW→

Function ROW→ has the opposite effect of the function →ROW, i.e., given n

vectors of the same length, and the number n, function ROWꢀ builds a matrix

by placing the input vectors as rows of the resulting matrix. Here is an example

in ALG mode. The command used was:

ROWꢀ([1,2,3],[4,5,6],[7,8,9],3)

In RPN mode, place the n vectors in stack levels n+1, n, n-1,…,2, and the

number n in stack level 1. With this set up, function ROWꢀ places the vectors

as rows in the resulting matrix. The following figure shows the RPN stack before

and after using function ROWꢀ.

Function ROW+

Function ROW+ takes as argument a matrix, a vector with the same length as

the number of rows in the matrix, and an integer number n representing the

location of a row. Function ROW+ inserts the vector in row n of the matrix. For

example, in ALG mode, we’ll insert the second row in matrix A with the vector [-

1,-2,-3], i.e.,

In RPN mode, enter the matrix first, then the vector, and the row number, before

applying function ROW+. The figure below shows the RPN stack before and

after applying function ROW+.

Page 10-23

Function ROW-

Function ROW- takes as argument a matrix and an integer number representing

the position of a row in the matrix. The function returns the original matrix,

minus a row, as well as the extracted row shown as a vector. Here is an

example in the ALG mode using the matrix stored in A:

In RPN mode, place the matrix in the stack first, then enter the number

representing a row location before applying function ROW-. The following

figure shows the RPN stack before and after applying function ROW-.

Function RSWP

Function RSWP (Row SWaP) takes as arguments two indices, say, i and j,

(representing two distinct rows in a matrix), and a matrix, and produces a new

matrix with rows i and j swapped. The following example, in ALG mode,

shows an application of this function. We use the matrix stored in variable A

for the example. This matrix is listed first.

In RPN mode, function RSWP lets you swap the rows of a matrix listed in stack

level 3, whose indices are listed in stack levels 1 and 2. For example, the

following figure shows the RPN stack before and after applying function RSWP

to matrix A in order to swap rows 2 and 3:

Page 10-24

As you can see, the rows that originally occupied positions 2 and 3 have been

swapped.

Function RCI

Function RCI stands for multiplying Row I by a Constant value and replace the

resulting row at the same location. The following example, written in ALG

mode, takes the matrix stored in A, and multiplies the constant value 5 into row

number 3, replacing the row with this product.

This same exercise done in RPN mode is shown in the next figure. The left-hand

side figure shows the setting up of the matrix, the factor and the row number, in

stack levels 3, 2, and 1. The right-hand side figure shows the resulting matrix

after function RCI is activated.

Function RCIJ

Function RCIJ stands for “take Row I and multiplying it by a constant C and then

add that multiplied row to row J, replacing row J with the resulting sum.” This

type of row operation is very common in the process of Gaussian or Gauss-

Jordan elimination (more details on this procedure are presented in a

subsequent Chapter). The arguments of the function are: (1) the matrix, (2) the

constant value, (3) the row to be multiplied by the constant in(2), and (4) the

row to be replaced by the resulting sum as described above. For example,

taking the matrix stored in variable A, we are going to multiply column 3 times

1.5, and add it to column 2. The following example is performed in ALG

mode:

Page 10-25

In RPN mode, enter the matrix first, followed by the constant value, then by the

row to be multiplied by the constant value, and finally enter the row that will be

replaced. The following figure shows the RPN stack before and after applying

function RCIJ under the same conditions as in the ALG example shown above:

Page 10-26

Chapter 11

Matrix Operations and Linear Algebra

In Chapter 10 we introduced the concept of a matrix and presented a number

of functions for entering, creating, or manipulating matrices. In this Chapter we

present examples of matrix operations and applications to problems of linear

algebra.

Operations with matrices

Matrices, like other mathematical objects, can be added and subtracted. They

can be multiplied by a scalar, or among themselves. They can also be raised to

a real power. An important operation for linear algebra applications is the

inverse of a matrix. Details of these operations are presented next.

To illustrate the operations we will create a number of matrices that we will store

in variables. The generic name of the matrices will be Aij and Bij, where i

represents the number of rows and j the number of columns of the matrices. The

matrices to be used are generated by using function RANM (random matrices).

If you try this exercise in your calculator you will get different matrices than the

ones listed herein, unless you store them into your calculator exactly as shown

below. Here are the matrices A22, B22, A23, B23, A32, B32, A33 and B33

created in ALG mode:

In RPN mode, the steps to follow are:

{2,2}` R NM ' 22'K {2,2}` R NM 'B22'K

{2,3}` R NM ' 23'K {2,3}` R NM 'B23'K

{3,2}` R NM ' 32'K {3,2}` R NM 'B32'K

{3,3}` R NM ' 33'K {3,3}` R NM 'B33'K

Page 11-1

Addition and subtraction

Consider a pair of matrices A = [a ]

and B = [b ] . Addition and

ij m×n

subtraction of these two matrices is only possible if they have the same number

of rows and columns. The resulting matrix, C = A B = [c ] has elements c

ij m×n

= a

b . Some examples in ALG mode are shown below using the matrices

stored above (e.g., @A22@ + @B22@)

In RPN mode, the steps to follow are:

22 ` B22`+

23 ` B23`+

32 ` B32`+

22 ` B22`-

23 ` B23`-

32 ` B32`-

Translating the ALG examples to RPN is straightforward, as illustrated here. The

remaining examples of matrix operations will be performed in ALG mode only.

Multiplication

There are numerous multiplication operations that involve matrices. These are

described next.

Multiplication by a scalar

Multiplication of the matrix A = [a ]

by a scalar k results in the matrix C =

ij m×n

kA = [c ]

= [ka ] . In particular, the negative of a matrix is defined by

ij m×n

the operation -A =(-1)A = [-a ]

. Some examples of multiplication of a

ij m×n

matrix by a scalar are shown below.

Page 11-2

By combining addition and subtraction with multiplication by a scalar we can

form linear combinations of matrices of the same dimensions, e.g.,

In a linear combination of matrices, we can multiply a matrix by an imaginary

number to obtain a matrix of complex numbers, e.g.,

Matrix-vector multiplication

Matrix-vector multiplication is possible only if the number of columns of the

matrix is equal to the length of the vector. This operation follows the rules of

matrix multiplication as shown in the next section. A couple of examples of

matrix-vector multiplication follow:

Vector-matrix multiplication, on the other hand, is not defined. This

multiplication can be performed, however, as a special case of matrix

multiplication as defined next.

Page 11-3

Matrix multiplication

Matrix multiplication is defined by C

= A

⋅B , where A = [a ] , B =

m×n

m×p p×n ij m×p

[b ] , and C = [c ] . Notice that matrix multiplication is only possible if the

ij p×n

ij m×n

number of columns in the first operand is equal to the number of rows of the

second operand. The general term in the product, c , is defined as

c = a ⋅b , for i =1,2,K,m; j =1,2,K,n.

∑

k=1

This is the same as saying that the element in the i-th row and j-th column of the

product, C, results from multiplying term-by-term the i-th row of A with the j-th

column of B, and adding the products together. Matrix multiplication is not

commutative, i.e., in general, A⋅B ≠ B⋅A. Furthermore, one of the

multiplications may not even exist.

The following screen shots show the results of multiplications of the matrices that

we stored earlier:

ꢀꢀꢀ

The matrix-vector multiplication introduced in the previous section can be

thought of as the product of a matrix m×n with a matrix n×1 (i.e., a column

vector) resulting in an m×1 matrix (i.e., another vector). To verify this assertion

check the examples presented in the previous section. Thus, the vectors defined

in Chapter 9 are basically column vectors for the purpose of matrix

multiplication.

The product of a vector with a matrix is possible if the vector is a row vector,

i.e., a 1×m matrix, which multiplied with a matrix m×n produces a 1xn matrix

Page 11-4

(another row vector). For the calculator to identify a row vector, you must use

double brackets to enter it:

Term-by-term multiplication

Term-by-term multiplication of two matrices of the same dimensions is possible

through the use of function HADAMARD. The result is, of course, another matrix

of the same dimensions. This function is available through Function catalog

(‚N), or through the MATRICES/OPERATIONS sub-menu („Ø).

Applications of function HADAMARD are presented next:

Raising a matrix to a real power

You can raise a matrix to any power as long as the power is either an

integer or a real number with no fractional part. The following example shows

the result of raising matrix B22, created earlier, to the power of 5:

You can also raise a matrix to a power without first storing it as a variable:

Page 11-5

In algebraic mode, the keystrokes are: [enter or select the matrix] Q [enter the

power] `. In RPN mode, the keystrokes are: [enter or select the matrix] †

[enter the power] Q`.

Matrices can be raised to negative powers. In this case, the result is equivalent

to 1/[matrix]^ABS(power).

The identity matrix

In Chapter 9 we introduce the identity matrix as the matrix I = [δ ] , where δ

ij n×n

is the Kronecker’s delta function. Identity matrices can be obtained by using

function IDN described in Chapter 9. The identity matrix has the property that

A⋅I = I⋅A = A. To verify this property we present the following examples using

the matrices stored earlier on:

The inverse matrix

-1

The inverse of a square matrix A is the matrix A such that A⋅A = A ⋅A = I,

where I is the identity matrix of the same dimensions as A. The inverse of a

matrix is obtained in the calculator by using the inverse function, INV (i.e., the

Y key). An example of the inverse of one of the matrices stored earlier is

presented next:

Page 11-6

To verify the properties of the inverse matrix, consider the following

multiplications:

Characterizing a matrix (The matrix NORM menu)

The matrix NORM (NORMALIZE) menu is accessed through the keystroke

sequence „´ (system flag 117 set to CHOOSE boxes):

This menu contains the following functions:

These functions are described next. Because many of these functions use

concepts of matrix theory, such as singular values, rank, etc., we will include

short descriptions of these concepts intermingled with the description of

functions.

Page 11-7

Function ABS

Function ABS calculates what is known as the Frobenius norm of a matrix. For a

matrix A = [a ]

, the Frobenius norm of the matrix is defined as

ij m×n

a²

∑∑

i=1 j=1

If the matrix under consideration in a row vector or a column vector, then the

Frobenius norm , ||A|| , is simply the vector’s magnitude. Function ABS is

accessible directly in the keyboard as „Ê.

Try the following exercises in ALG mode (using the matrices stored earlier for

matrix operations):

Function SNRM

Function SNRM calculates the Spectral NoRM of a matrix, which is defined as

the matrix’s largest singular value, also known as the Euclidean norm of the

matrix. For example,

Page 11-8

Singular value decomposition

To understand the operation of Function SNRM, we need to introduce the

concept of matrix decomposition. Basically, matrix decomposition involves the

determination of two or more matrices that, when multiplied in a certain order

(and, perhaps, with some matrix inversion or transposition thrown in), produce

the original matrix. The Singular Value Decomposition (SVD) is such that a

rectangular matrix A

is written as A

= U

⋅S

⋅V

n×n

m×n

m×m m×n

where U and V are orthogonal matrices, and S is a diagonal matrix. The

diagonal elements of S are called the singular values of A and are usually

ordered so that s ≥ s , for i = 1, 2, …, n-1. The columns [u ] of U and [v ] of

i+1

V are the corresponding singular vectors. (Orthogonal matrices are such that

U⋅ U = I. A diagonal matrix has non-zero elements only along its main

diagonal).

The rank of a matrix can be determined from its SVD by counting the number of

non-singular values. Examples of SVD will be presented in a subsequent

section.

Functions RNRM and CNRM

Function RNRM returns the Row NoRM of a matrix, while function CNRM

returns the Column NoRM of a matrix. Examples,

Row norm and column norm of a matrix

The row norm of a matrix is calculated by taking the sums of the absolute values

of all elements in each row, and then, selecting the maximum of these sums.

The column norm of a matrix is calculated by taking the sums of the absolute

values of all elements in each column, and then, selecting the maximum of these

sums.

Page 11-9

Function SRAD

Function SRAD determines the Spectral RADius of a matrix, defined as the

largest of the absolute values of its eigenvalues. For example,

Definition of eigenvalues and eigenvectors of a matrix

The eigenvalues of a square matrix result from the matrix equation A⋅x = λ⋅x.

The values of λ that satisfy the equation are known as the eigenvalues of the

matrix A. The values of x that result from the equation for each value of l are

known as the eigenvectors of the matrix. Further details on calculating

eigenvalues and eigenvectors are presented later in the chapter.

Function COND

Function COND determines the condition number of a matrix:

Condition number of a matrix

The condition number of a square non-singular matrix is defined as the product

of the matrix norm times the norm of its inverse, i.e.,

-1

cond(A) = ||A||×||A ||. We will choose as the matrix norm, ||A||, the

maximum of its row norm (RNRM) and column norm (CNRM), while the norm

-1

of the inverse, ||A ||, will be selected as the minimum of its row norm and

-1

column norm. Thus, ||A|| = max(RNRM(A),CNRM(A)), and ||A || =

-1

min(RNRM(A ), CNRM(A )).

The condition number of a singular matrix is infinity. The condition number of

a non-singular matrix is a measure of how close the matrix is to being singular.

The larger the value of the condition number, the closer it is to singularity. (A

singular matrix is one for which the inverse does not exist).

Page 11-10

Try the following exercise for matrix condition number on matrix A33. The

condition number is COND(A33) , row norm, and column norm for A33 are

shown to the left. The corresponding numbers for the inverse matrix, INV(A33),

are shown to the right:

Since RNRM(A33) > CNRM(A33), then we take ||A33|| = RNRM(A33) =

21. Also, since CNRM(INV(A33)) < RNRM(INV(A33)), then we take

||INV(A33)|| = CNRM(INV(A33)) = 0.261044... Thus, the condition

number is also calculated as CNRM(A33)*CNRM(INV(A33)) = COND(A33) =

6.7871485…

Function RANK

Function RANK determines the rank of a square matrix. Try the following

examples:

The rank of a matrix

The rank of a square matrix is the maximum number of linearly independent

rows or columns that the matrix contains. Suppose that you write a square

matrix A

as A = [c c … c ], where c (i = 1, 2, …, n) are vectors

n×n

representing the columns of the matrix A, then, if any of those columns, say c ,

c_k=

d ⋅c ,

∑

can be written as

j≠k, j∈{1,2,...,n}

Page 11-11

where the values d are constant, we say that c is linearly dependent on the

columns included in the summation. (Notice that the values of j include any

value in the set {1, 2, …, n}, in any combination, as long as j≠k.) If the

expression shown above cannot be written for any of the column vectors then

we say that all the columns are linearly independent. A similar definition for

the linear independence of rows can be developed by writing the matrix as a

column of row vectors. Thus, if we find that rank(A) = n, then the matrix has

an inverse and it is a non-singular matrix. If, on the other hand, rank(A) < n,

then the matrix is singular and no inverse exist.

For example, try finding the rank for the matrix:

You will find that the rank is 2. That is because the second row [2,4,6] is equal

to the first row [1,2,3] multiplied by 2, thus, row two is linearly dependent of

row 1 and the maximum number of linearly independent rows is 2. You can

check that the maximum number of linearly independent columns is 3. The rank

being the maximum number of linearly independent rows or columns becomes

2 for this case.

Function DET

Function DET calculates the determinant of a square matrix. For example,

Page 11-12

The determinant of a matrix

The determinant of a 2x2 and or a 3x3 matrix are represented by the same

arrangement of elements of the matrices, but enclosed between vertical lines,

i.e.,

a₁₁a₁₂a₁₃

a₁₁a₁₂

a₂₁a₂₂a₂₃

a₃₁a₃₂a₃₃

a₂₁a₂₂

A 2×2 determinant is calculated by multiplying the elements in its diagonal

and adding those products accompanied by the positive or negative sign as

indicated in the diagram shown below.

The 2×2 determinant is, therefore,

a₁₁a₁₂

= a₁₁⋅a₂₂− a₁₂⋅ a₂₁

a₂₁a₂₂

A 3×3 determinant is calculated by augmenting the determinant, an

operation that consists on copying the first two columns of the determinant,

and placing them to the right of column 3, as shown in the diagram below.

The diagram also shows the elements to be multiplied with the corresponding

sign to attach to their product, in a similar fashion as done earlier for a 2×2

determinant. After multiplication the results are added together to obtain the

determinant.

Page 11-13

For square matrices of higher order determinants can be calculated by using

smaller order determinant called cofactors. The general idea is to "expand"

a determinant of a n×n matrix (also referred to as a n×n determinant) into a

sum of the cofactors, which are (n-1)×(n-1) determinants, multiplied by the

elements of a single row or column, with alternating positive and negative

signs. This "expansion" is then carried to the next (lower) level, with

cofactors of order (n-2)×(n-2), and so on, until we are left only with a long

sum of 2×2 determinants. The 2×2 determinants are then calculated through

the method shown above.

The method of calculating a determinant by cofactor expansion is very

inefficient in the sense that it involves a number of operations that grows very

fast as the size of the determinant increases. A more efficient method, and

the one preferred in numerical applications, is to use a result from Gaussian

elimination. The method of Gaussian elimination is used to solve systems of

linear equations. Details of this method are presented in a later part of this

chapter.

To refer to the determinant of a matrix A, we write det(A). A singular matrix

has a determinant equal to zero.

Function TRACE

Function TRACE calculates the trace of square matrix, defined as the sum of the

elements in its main diagonal, or

tr(A) =

∑

i=1

Examples:

Page 11-14

Function TRAN

Function TRAN returns the transpose of a real or the conjugate transpose of a

complex matrix. TRAN is equivalent to TRN. The operation of function TRN

was presented in Chapter 10.

Additional matrix operations (The matrix OPER menu)

The matrix OPER (OPERATIONS) is available through the keystroke sequence

„Ø (system flag 117 set to CHOOSE boxes):

The OPERATIONS menu includes the following functions:

Functions ABS, CNRM, COND, DET, RANK, RNRM, SNRM, TRACE, and TRAN

are also found in the MTH/MATRIX/NORM menu (the subject of the previous

section). Function SIZE was presented in Chapter 10. Function HADAMARD

was presented earlier in the context of matrix multiplication. Functions LSQ ,

Page 11-15

MAD and RSD are related to the solution of systems of linear equations and will

be presented in a subsequent section in this Chapter. In this section we’ll

discuss only functions AXL and AXM.

Function AXL

Function AXL converts an array (matrix) into a list, and vice versa:

Note: the latter operation is similar to that of the program CRMR presented in

Chapter 10.

Function AXM

Function AXM converts an array containing integer or fraction elements into its

corresponding decimal, or approximate, form:

Function LCXM

Function LCXM can be used to generate matrices such that the element aij is a

function of i and j. The input to this function consists of two integers, n and m,

representing the number of rows and columns of the matrix to be generated,

and a program that takes i and j as input. The numbers n, m, and the program

occupy stack levels 3, 2, and 1, respectively. Function LCXM is accessible

through the command catalog ‚N.

For example, to generate a 2´3 matrix whose elements are given by a = (i+j) ,

first, store the following program into variable P1 in RPN mode. This is the way

that the RPN stack looks before pressing K.

Page 11-16

The implementation of function LCXM for this case requires you to enter:

2`3`‚@@P1@@ LCXM `

The following figure shows the RPN stack before and after applying function

LCXM:

In ALG mode, this example can be obtained by using:

The program P1 must still have been created and stored in RPN mode.

Solution of linear systems

A system of n linear equations in m variables can be written as

⋅x + a ⋅x + a ⋅x + …+ a

⋅x

+ a ⋅x

= b ,

1,m-1 m-1

1,m

⋅x + a ⋅x + a ⋅x + …+ a

⋅x

+ a ⋅x

= b ,

2,m-1 m-1

2,m

⋅x + a ⋅x + a ⋅x + …+ a

⋅x

+ a ⋅x

= b ,

3,m-1 m-1

3,m

…

ꢀꢀ .

ꢀꢀ

⋅x + a

⋅x + …+ a

⋅x

+ a

⋅x = b

n-1

n-1,1

n-1,2

n-1,3

n-1,m-1 m-1

n-1,m

⋅x + a ⋅x + a ⋅x + …+ a

⋅x

+ a ⋅x

= b .

n,m-1 m-1

n,m

This system of linear equations can be written as a matrix equation, A ⋅x

n×m m×1

= b , if we define the following matrix and vectors:

n×1

Page 11-17

a₁₂L a_1m

⎡ ⎤

⎡

⎢

⎣

⎤

⎥

⎦

⎡

⎢

⎣

⎤

⎥

⎦

⎢ ⎥

a₂₁a₂₂L a_2m

x₂

b₂

⎢ ⎥

A =

x =

b =

⎢ ⎥

a_n1a_n2L a_nm

x_m

b_n

⎣ ⎦_n×1

n×m

m×1

Using the numerical solver for linear systems

There are many ways to solve a system of linear equations with the calculator.

One possibility is through the numerical solver ‚Ï. From the numerical

solver screen, shown below (left), select the option 4. Solve lin sys.., and press

@@@OK@@@. The following input form will be provided (right):

To solve the linear system A⋅x = b, enter the matrix A, in the format [[ a , a

11 12,

… ], … [….]] in the A: field. Also, enter the vector b in the B: field. When the

X: field is highlighted, press [SOLVE]. If a solution is available, the solution

vector x will be shown in the X: field. The solution is also copied to stack level

1. Some examples follow.

A square system

The system of linear equations

2x + 3x –5x = 13,

x – 3x + 8x = -13,

2x – 2x + 4x = -6,

can be written as the matrix equation A⋅x = b, if

− 5

⎡ ⎤

⎢ ⎥

⎡

⎢

⎤

⎥

⎡

⎢

⎤

⎥

A = 1 − 3 8 , x = x₂, and b = −13 .

⎥

⎢

⎥

⎢ ⎥

⎢

⎥

2 − 2

x₃

− 6

⎣

⎦

⎣ ⎦

⎣

⎦

Page 11-18

This system has the same number of equations as of unknowns, and will be

referred to as a square system. In general, there should be a unique solution to

the system. The solution will be the point of intersection of the three planes in

the coordinate system (x , x , x ) represented by the three equations.

To enter matrix A you can activate the Matrix Writer while the A: field is

selected. The following screen shows the Matrix Writer used for entering matrix

A, as well as the input form for the numerical solver after entering matrix A

(press ` in the Matrix Writer):

Press ˜ to select the B: field. The vector b can be entered as a row vector

with a single set of brackets, i.e., [13,-13,-6] @@@OK@@@ .

After entering matrix A and vector b, and with the X: field highlighted, we can

press @SOLVE! to attempt a solution to this system of equations:

A solution was found as shown next.

To see the solution in the stack press `. The solution is x = [1,2,-1].

Page 11-19

To check that the solution is correct, enter the matrix A and multiply times this

solution vector (example in algebraic mode):

Under-determined system

The system of linear equations

2x + 3x –5x = -10,

x – 3x + 8x = 85,

can be written as the matrix equation A⋅x = b, if

⎡ ⎤

− 5

−10

⎡

⎤

⎥

⎦

⎡

⎤

⎢ ⎥

A =

, x = x₂, and b =

⎢

⎥

⎢ ⎥

1 − 3

⎣

⎦

⎢ ⎥

x₃

⎣ ⎦

This system has more unknowns than equations, therefore, it is not uniquely

determined. We can visualize the meaning of this statement by realizing that

each of the linear equations represents a plane in the three-dimensional

Cartesian coordinate system (x , x , x ). The solution to the system of

equations shown above will be the intersection of two planes in space. We

know, however, that the intersection of two (non-parallel) planes is a straight

line, and not a single point. Therefore, there is more than one point that satisfy

the system. In that sense, the system is not uniquely determined.

Let’s use the numerical solver to attempt a solution to this system of equations:

‚Ï ˜˜˜ @@OK@@ . Enter matrix A and vector b as illustrated in the

previous example, and press @SOLVE when the X: field is highlighted:

Page 11-20

To see the details of the solution vector, if needed, press the @EDIT! button. This

will activate the Matrix Writer. Within this environment, use the right- and left-

arrow keys to move about the vector:

Thus, the solution is x = [15.373, 2.4626, 9.6268].

To return to the numerical solver environment, press `.

The procedure that we describe next can be used to copy the matrix A and the

solution vector X into the stack. To check that the solution is correct, try the

following:

•

Press ——, to highlight the A: field.

Press L @CALC@ `, to copy matrix A onto the stack.

Press @@@OK@@@ to return to the numerical solver environment.

Press ˜ ˜@CALC@ `, to copy solution vector X onto the stack.

Press @@@OK@@@ to return to the numerical solver environment.

Press ` to return to the stack.

In ALG mode, the stack will now look like this:

Page 11-21

Let’s store the latest result in a variable X, and the matrix into variable A, as

follows:

Press K~x` to store the solution vector into variable X

Press ƒ ƒ ƒ to clear three levels of the stack

Press K~a` to store the matrix into variable A

Now, let’s verify the solution by using: @@@A@@@ * @@@X@@@ `, which results in

(press ˜ to see the vector elements): [-9.99999999992 85. ], close enough

to the original vector b = [-10 85].

Try also this, @@A@@@ * [15,10/3,10] ` ‚ï`, i.e.,

This result indicates that x = [15,10/3,10] is also a solution to the system,

confirming our observation that a system with more unknowns than equations is

not uniquely determined (under-determined).

How does the calculator came up with the solution x = [15.37… 2.46…

9.62…] shown earlier? Actually, the calculator minimizes the distance from a

point, which will constitute the solution, to each of the planes represented by the

equations in the linear system. The calculator uses a least-square method, i.e.,

minimizes the sum of the squares of those distances or errors.

Over-determined system

The system of linear equations

x + 3x = 15,

2x – 5x = 5,

-x + x = 22,

Page 11-22

can be written as the matrix equation A⋅x = b, if

⎡

⎢

⎤

⎥

⎡ ⎤

⎢ ⎥

, and b = 5 .

⎢ ⎥

A = 2 − 5 , x =

⎢ ⎥

x₂

⎣ ⎦

⎢

⎥

⎢ ⎥

−1

⎣

⎦

⎣ ⎦

This system has more equations than unknowns (an over-determined system).

The system does not have a single solution. Each of the linear equations in

the system presented above represents a straight line in a two-dimensional

Cartesian coordinate system (x , x ). Unless two of the three equations in the

system represent the same equation, the three lines will have more than one

intersection points. For that reason, the solution is not unique. Some numerical

algorithms can be used to force a solution to the system by minimizing the

distance from the presumptive solution point to each of the lines in the system.

Such is the approach followed by the calculator numerical solver.

Let’s use the numerical solver to attempt a solution to this system of equations:

‚Ï ˜˜˜ @@OK@@ . Enter matrix A and vector b as illustrated in the

previous example, and press @SOLVE when the X: field is highlighted:

To see the details of the solution vector, if needed, press the @EDIT! button. This

will activate the Matrix Writer. Within this environment, use the right- and left-

arrow keys to move about the vector:

Page 11-23

Press ` to return to the numerical solver environment. To check that the

solution is correct, try the following:

•

Press ——, to highlight the A: field.

Press L @CALC@ `, to copy matrix A onto the stack.

Press @@@OK@@@ to return to the numerical solver environment.

Press ˜ ˜@CALC@ `, to copy solution vector X onto the stack.

Press @@@OK@@@ to return to the numerical solver environment.

Press ` to return to the stack.

In ALG mode, the stack will now look like this:

Let’s store the latest result in a variable X, and the matrix into variable A, as

follows:

Press K~x` to store the solution vector into variable X

Press ƒ ƒ ƒ to clear three levels of the stack

Press K~a` to store the matrix into variable A

Now, let’s verify the solution by using: @@@A@@@ * @@@X@@@ `, which results in the

vector [8.6917… -3.4109… -1.1301…], which is not equal to [15 5 22], the

original vector b. The “solution” is simply the point that is closest to the three

lines represented by the three equations in the system, and not an exact

solution.

Least-square solution (function LSQ)

The LSQ function returns the minimum-norm least-square solution of a linear

system Ax = b, according to the following criteria:

Page 11-24

•

If A is a square matrix and A is non-singular (i.e., it’s inverse matrix

exist, or its determinant is non-zero), LSQ returns the exact solution to

the linear system.

If A has less than full row rank (underdetermined system of equations),

LSQ returns the solution with the minimum Euclidean length out of an

infinity number of solutions.

If A has less than full column rank (over-determined system of

equations), LSQ returns the "solution" with the minimum residual value

e = A⋅x – b. The system of equations may not have a solution,

therefore, the value returned is not a real solution to the system, just the

one with the smallest residual.

Function LSQ takes as input vector b and matrix A, in that order. Function LSQ

can be found in Function catalog (‚N). Next, we use function LSQ to

repeat the solutions found earlier with the numerical solver:

Square system

Consider the system

2x + 3x –5x = 13,

x – 3x + 8x = -13,

2x – 2x + 4x = -6,

with

− 5

⎢ ⎥

⎡

⎢

⎤

⎥

⎡ ⎤

⎡

⎢

⎤

⎥

A = 1 − 3 8 , x = x₂, and b = −13 .

⎥

⎢

⎥

⎢ ⎥

⎢

⎥

2 − 2

x₃

− 6

⎣

⎦

⎣ ⎦

⎣

⎦

The solution using LSQ is shown next:

Page 11-25

Under-determined system

Consider the system

2x + 3x –5x = -10,

x – 3x + 8x = 85,

with

⎡ ⎤

− 5

−10

⎡

⎤

⎥

⎦

⎡

⎤

⎢ ⎥

A =

, x = x₂, and b =

⎢

⎥

⎢ ⎥

1 − 3

⎣

⎦

⎢ ⎥

x₃

⎣ ⎦

The solution using LSQ is shown next:

Over-determined system

Consider the system

x + 3x = 15,

2x – 5x = 5,

-x + x = 22,

with

⎡

⎢

⎤

⎥

⎡ ⎤

⎢ ⎥

, and b = 5 .

⎢ ⎥

A = 2 − 5 , x =

⎢ ⎥

x₂

⎣ ⎦

⎢

⎥

⎢ ⎥

−1

⎣

⎦

⎣ ⎦

The solution using LSQ is shown next:

Page 11-26

Compare these three solutions with the ones calculated with the numerical

solver.

Solution with the inverse matrix

The solution to the system A⋅x = b, where A is a square matrix is x = A ⋅ b.

-1

This results from multiplying the first equation by A , i.e., A ⋅A⋅x = A ⋅b. By

-1

definition, A ⋅A = I, thus we write I⋅x = A ⋅b. Also, I⋅x = x, thus, we have, x

-1

= A ⋅ b.

For the example used earlier, namely,

2x + 3x –5x = 13,

x – 3x + 8x = -13,

2x – 2x + 4x = -6,

we can find the solution in the calculator as follows:

which is the same result found earlier.

Solution by “division” of matrices

While the operation of division is not defined for matrices, we can use the

calculator’s / key to “divide” vector b by matrix A to solve for x in the matrix

equation A⋅x = b. This is an arbitrary extension of the algebraic division

operation to matrices, i.e., from A⋅x = b, we dare to write x = b/A

(Mathematicians would cringe if they see this!) This, of course is interpreted as

-1

(1/A)⋅b = A ⋅b, which is the same as using the inverse of A as in the previous

section.

Page 11-27

The procedure for the case of “dividing” b by A is illustrated below for the case

2x + 3x –5x = 13,

x – 3x + 8x = -13,

2x – 2x + 4x = -6,

The procedure is shown in the following screen shots:

The same solution as found above with the inverse matrix.

Solving multiple set of equations with the same coefficient matrix

Suppose that you want to solve the following three sets of equations:

X +2Y+3Z = 14, 2X +4Y+6Z = 9,

3X -2Y+ Z = 2, 3X -2Y+ Z = -5,

4X +2Y -Z = 5, 4X +2Y -Z = 19,

2X +4Y+6Z = -2,

3X -2Y+ Z = 2,

4X +2Y -Z = 12.

We can write the three systems of equations as a single matrix equation: A⋅X

= B, where

⎡

⎢

⎤

⎥

X₍₁₎X₍₂₎X ₍₃₎

⎡

⎢

⎤

⎥

A = 3 − 2 1 , X = Y₍₁₎Y₍₂₎Y₍₃₎

⎢

⎥

⎢

⎥

−1

Z₍₁₎Z₍₂₎Z₍₃₎

⎣

⎦

⎣

⎦

− 2

⎤

⎡

⎢

⎥

B = 2 − 5 2 .

⎥

⎢

⎥

19 12

⎣

⎦

The sub-indices in the variable names X, Y, and Z, determine to which equation

system they refer to. To solve this expanded system we use the following

procedure, in RPN mode,

Page 11-28

[[14,9,-2],[2,-5,2],[5,19,12]] `

[[1,2,3],[3,-2,1],[4,2,-1]] `/

The result of this operation is:

⎡

⎢

⎤

⎥

X = 2

1 .

⎢

⎥

3 −1 − 2

⎣

⎦

Gaussian and Gauss-Jordan elimination

Gaussian elimination is a procedure by which the square matrix of coefficients

belonging to a system of n linear equations in n unknowns is reduced to an

upper-triangular matrix (echelon form) through a series of row operations. This

procedure is known as forward elimination. The reduction of the coefficient

matrix to an upper-triangular form allows for the solution of all n unknowns,

utilizing only one equation at a time, in a procedure known as backward

substitution.

Example of Gaussian elimination using equations

To illustrate the Gaussian elimination procedure we will use the following system

of 3 equations in 3 unknowns:

2X +4Y+6Z = 14,

3X -2Y+ Z = -3,

4X +2Y -Z = -4.

We can store these equations in the calculator in variables E1, E2, and E3,

respectively, as shown below. For backup purposes, a list containing the three

equations was also created and stored into variable EQS. This way, if a

mistake is made, the equations will still be available to the user.

Page 11-29

To start the process of forward elimination, we divide the first equation (E1) by

2, and store it in E1, and show the three equations again to produce:

Next, we replace the second equation E2 by (equation 2 – 3×equation 1, i.e.,

E1-3×E2), and the third by (equation 3 – 4×equation 1), to get:

Next, divide the second equation by –8, to get:

Next, replace the third equation, E3, with (equation 3 + 6×equation 2, i.e.,

E2+6×E3), to get:

Notice that when we perform a linear combination of equations the calculator

modifies the result to an expression on the left-hand side of the equal sign, i.e.,

Page 11-30

an expression = 0. Thus, the last set of equations is interpreted to be the

following equivalent set of equations:

X +2Y+3Z = 7,

Y+ Z = 3,

-7Z = -14.

The process of backward substitution in Gaussian elimination consists in finding

the values of the unknowns, starting from the last equation and working

upwards. Thus, we solve for Z first:

Next, we substitute Z=2 into equation 2 (E2), and solve E2 for Y:

Next, we substitute Z=2 and Y = 1 into E1, and solve E1 for X:

The solution is, therefore, X = -1, Y = 1, Z = 2.

Example of Gaussian elimination using matrices

The system of equations used in the example above can be written as a matrix

equation A⋅x = b, if we use:

Page 11-31

⎢ ⎥

⎛

⎞

⎡ ⎤

⎡

⎢

⎤

⎥

⎜

⎟

A = 3 − 2 1 , x = Y , b = − 3 .

⎜

⎟

⎢ ⎥

⎜

⎟

⎢

⎥

−1

− 4

⎝

⎠

⎣ ⎦

⎣

⎦

To obtain a solution to the system matrix equation using Gaussian elimination,

we first create what is known as the augmented matrix corresponding to A, i.e.,

⎛2

6 14 ⎞

⎜

⎟

A_aug= 3 − 2 1 − 3

⎜

⎟

⎜

⎝

⎟

⎠

−1 − 4

The matrix A

is the same as the original matrix A with a new row,

aug

corresponding to the elements of the vector b, added (i.e., augmented) to the

right of the rightmost column of A.

Once the augmented matrix is put together, we can proceed to perform row

operations on it that will reduce the original A matrix into an upper-triangular

matrix. For this exercise, we will use the RPN mode (H\@@OK@@), with system

flag 117 set to SOFT menu. In your calculator, use the following keystrokes.

First, enter the augmented matrix, and make an extra copy of the same in the

stack (This step is not necessary, except as an insurance that you have an extra

copy of the augmented matrix saved in case you make a mistake in the forward

elimination procedure that we are about to undertake.):

[[2,4,6,14],[3,-2,1,-3],[4,2,-1,-4]] ``

Save augmented matrix in variable AAUG: ³~~aaug~ K

With a copy of the augmented matrix in the stack, press „´ @MATRX! @ROW! to

activate the ROW operation menu. Next, perform the following row operations

on your augmented matrix.

Multiply row 1 by ½: 2Y 1 @RCI!

Multiply row 1 by –3 add it to row 2, replacing it: 3\ # 1 #2

@RCIJ!

Multiply row 1 by –4 add it to row 3, replacing it: 4\#1#3@RCIJ!

Page 11-32

Multiply row 2 by –1/8: 8\Y2 @RCI!

Multiply row 2 by 6 add it to row 3, replacing it: 6#2#3 @RCIJ!

If you were performing these operations by hand, you would write the

following:

⎛2

6 14 ⎞ ⎛1

⎟ ⎜

7 ⎞

⎜

⎟

A_aug= 3 − 2 1 −3 ≅ 3 − 2 1 −3

⎜

⎝

⎟ ⎜

⎠ ⎝

⎟

⎠

−1 − 4

⎛1

7 ⎞ ⎛1

⎟ ⎜

7 ⎞

⎜

⎟

A_aug≅ 0 −8 −8 − 24 ≅ 0

⎜

⎝

⎟ ⎜

⎠ ⎝

⎟

⎠

0 − 6 −13 −32

⎛1 2

7 ⎞

⎜

⎟

A_aug≅ 0 1

⎜

⎟

⎠

⎜

⎝

0 0 − 7 −14

The symbol ≅ (“ is equivalent to”) indicates that what follows is equivalent to the

previous matrix with some row (or column) operations involved.

The resulting matrix is upper-triangular, and equivalent to the set of equations

X +2Y+3Z = 7,

Y+ Z = 3,

-7Z = -14,

which can now be solved, one equation at a time, by backward substitution, as

in the previous example.

Gauss-Jordan elimination using matrices

Gauss-Jordan elimination consists in continuing the row operations in the upper-

triangular matrix resulting from the forward elimination process until an identity

matrix results in place of the original A matrix. For example, for the case we

just presented, we can continue the row operations as follows:

Page 11-33

Multiply row 3 by –1/7: 7\Y 3 @RCI!

Multiply row 3 by –1, add it to row 2, replacing it: 1\ # 3

#2 @RCIJ!

Multiply row 3 by –3, add it to row 1, replacing it:

3\#3#1@RCIJ!

Multiply row 2 by –2, add it to row 1, replacing it: 2\#2#1

@RCIJ!

Writing this process by hand will result in the following steps:

⎛

⎞

⎛1 0 0 −1⎞

1 2 0 1

⎜

⎟ ⎜

⎟

A_aug≅ 0 1 0 1 ≅ 0 1 0 1 .

⎜

⎝

⎟ ⎜

⎠ ⎝

⎟

⎠

0 0 1 2

Pivoting

If you look carefully at the row operations in the examples shown above, you

will notice that many of those operations divide a row by its corresponding

element in the main diagonal. This element is called a pivot element, or simply,

a pivot. In many situations it is possible that the pivot element become zero, in

which case we cannot divide the row by its pivot. Also, to improve the

numerical solution of a system of equations using Gaussian or Gauss-Jordan

elimination, it is recommended that the pivot be the element with the largest

absolute value in a given column. In such cases, we exchange rows before

performing row operations. This exchange of rows is called partial pivoting. To

follow this recommendation is it often necessary to exchange rows in the

augmented matrix while performing a Gaussian or Gauss-Jordan elimination.

Page 11-34

While performing pivoting in a matrix elimination procedure, you can improve

the numerical solution even more by selecting as the pivot the element with the

largest absolute value in the column and row of interest. This operation may

require exchanging not only rows, but also columns, in some pivoting

operations. When row and column exchanges are allowed in pivoting, the

procedure is known as full pivoting.

When exchanging rows and columns in partial or full pivoting, it is necessary to

keep track of the exchanges because the order of the unknowns in the solution

is altered by those exchanges. One way to keep track of column exchanges in

partial or full pivoting mode, is to create a permutation matrix P = I , at the

n×n

beginning of the procedure. Any row or column exchange required in the

augmented matrix A

is also registered as a row or column exchange,

aug

respectively, in the permutation matrix. When the solution is achieved, then,

we multiply the permutation matrix by the unknown vector x to obtain the order

of the unknowns in the solution. In other words, the final solution is given by

P⋅x = b’, where b’ is the last column of the augmented matrix after the solution

has been found.

Example of Gauss-Jordan elimination with full pivoting

Let’s illustrate full pivoting with an example. Solve the following system of

equations using full pivoting and the Gauss-Jordan elimination procedure:

X + 2Y + 3Z = 2,

2X +

3Z = -1,

8X +16Y- Z = 41.

The augmented matrix and the permutation matrix are as follows:

1 0 0

⎡ ⎤

⎡

⎢

⎤

⎥

⎢

⎥

A_aug= 2

−1 , P = 0 1 0 .

⎢

⎥

⎢

⎥

8 16 −1 41

0 0 1

⎣

⎦

⎣

⎦

Store the augmented matrix in variable AAUG, then press ‚ @AAUG to get a

copy in the stack. We want to keep the CSWP (Column Swap) command

readily available, for which we use: ‚N~~cs~ (find CSWP),

@@OK@@. You’ll get an error message, press $, and ignore the message.

Next, get the ROW menu available by pressing: „Ø @CREAT @@ROW@.

Page 11-35

Now we are ready to start the Gauss-Jordan elimination with full pivoting. We

will need to keep track of the permutation matrix by hand, so take your

notebook and write the P matrix shown above.

First, we check the pivot a . We notice that the element with the largest

absolute value in the first row and first column is the value of a = 8. Since we

want this number to be the pivot, then we exchange rows 1 and 3, by using:

1#3L @RSWP. The augmented matrix and the permutation matrix

now are:

-1

0 0 1

0 1 0

1 0 0

Checking the pivot at position (1,1) we now find that 16 is a better pivot than

8, thus, we perform a column swap as follows: 1#2‚N @@OK@@

@RSWP. The augmented matrix and the permutation matrix now are:

-1

0 0 1

1 0 0

0 1 0

Now we have the largest possible value in position (1,1), i.e., we performed

full pivoting at (1,1). Next, we proceed to divide by the pivot:

16Y1L@RCI@ . The permutation matrix does not change, but the

augmented matrix is now:

1/2 -1/16 41/16

0 0 1

1 0 0

0 1 0

-1

The next step is to eliminate the 2 from position (3,2) by using:

2\#1#3@RCIJ

1/2 -1/16 41/16

-1

25/8 -25/8

Page 11-36

Having filled up with zeros the elements of column 1 below the pivot, now we

proceed to check the pivot at position (2,2). We find that the number 3 in

position (2,3) will be a better pivot, thus, we exchange columns 2 and 3 by

using: 2#3 ‚N@@@OK@@

-1/16 1/2 41/16

1 0

0 0

0 1

-1

-25/8

25/8

Checking the pivot at position (2,2), we now find that the value of 25/8, at

position (3,2), is larger than 3. Thus, we exchange rows 2 and 3 by using:

2#3 L@RSWP

-1/16 1/2 41/16

25/8

-25/8

-1

Now, we are ready to divide row 2 by the pivot 25/8, by using

³ 8/25™#2 L@RCI

-1/16 1/2 41/16

0 1 0

0 0 1

1 0 0

-1

Next, we eliminate the 3 from position (3,2) by using:

3\#2#3@RCIJ

-1/16 1/2 41/16

0 1

0 0

1 0

-1

Having filled with zeroes the position below the pivot, we proceed to check the

pivot at position (3,3). The current value of 2 is larger than ½ or 0, thus, we

keep it unchanged. We do divide the whole third row by 2 to convert the pivot

to 1, by using: 2Y3@RCI

-1/16 1/2 41/16

0 1 0

0 0 1

1 0 0

-1

Next, we proceed to eliminate the ½ in position (1,3) by using:

Page 11-37

2 Y \#3#1@RCIJ

-1/16

33/16

-1

Finally, we eliminate the –1/16 from position (1,2) by using:

16 Y # 2#1@RCIJ

-1

0 1 0

0 0 1

1 0 0

We now have an identity matrix in the portion of the augmented matrix

corresponding to the original coefficient matrix A, thus we can proceed to

obtain the solution while accounting for the row and column exchanges coded

in the permutation matrix P. We identify the unknown vector x, the modified

independent vector b’ and the permutation matrix P as:

⎡ ⎤

0 1 0

⎡ ⎤

⎡

⎤

⎢ ⎥

⎢

⎥

⎢

⎥

x = Y , b'= −1 , P = 0 0 1 .

⎢ ⎥

⎢

⎥

⎢

⎥

1 0 0

⎣ ⎦

⎣

⎦

⎣

⎦

The solution is given by P⋅x=b’, or

0 1 0

⎡

⎢

⎤ ⎡ ⎤

⎥ ⎢ ⎥

⎡

⎢

⎤

⎥

0 0 1 ⋅ Y = −1 .

1 0 0

⎥ ⎢ ⎥

⎢

⎥

⎣

⎦ ⎣ ⎦

⎣

⎦

Which results in

⎡ ⎤

⎡

⎤

⎢ ⎥

⎢

⎥

Z = −1 .

⎢ ⎥

⎢

⎥

⎣ ⎦

⎣

⎦

Step-by-step calculator procedure for solving linear systems

The example we just worked is, of course, the step-by-step, user-driven

procedure to use full pivoting for Gauss-Jordan elimination solution of linear

equation systems. You can see the step-by-step procedure used by the

calculator to solve a system of equations, without user intervention, by setting

the step-by-step option in the calculator’s CAS, as follows:

Page 11-38

Then, for this particular example, in RPN mode, use:

[2,-1,41] ` [[1,2,3],[2,0,3],[8,16,-1]] `/

The calculator shows an augmented matrix consisting of the coefficients matrix

A and the identity matrix I, while, at the same time, showing the next procedure

to calculate:

L2 = L2-2⋅L1 stands for “replace row 2 (L2) with the operation L2 – 2⋅L1. If we

had done this operation by hand, it would have corresponded to:

2\#1#1@RCIJ. Press @@@OK@@@, and follow the operations in your

calculator’s screen. You will see the following operations performed:

L3=L3-8⋅L1, L1 = 2⋅L1--1⋅L2, L1=25⋅L1--3⋅L3, L2 = 25⋅L2-3⋅L3,

and finally a message indicating “Reduction result” showing:

When you press @@@OK@@@ , the calculator returns the final result [1 2 –1].

Calculating the inverse matrix step-by-step

The calculation of an inverse matrix can be considered as calculating the

solution to the augmented system [A | I ]. For example, for the matrix A used

in the previous example, we would write this augmented matrix as

Page 11-39

⎡

⎢

3 1 0 0⎤

⎥

A_{aug(I )}= 3 − 2 1 0 1 0 .

⎥

⎢

⎥

−1 0 0 1

⎣

⎦

To see the intermediate steps in calculating and inverse, just enter the matrix A

from above, and press Y, while keeping the step-by-step option active in the

calculator’s CAS. Use the following:

[[ 1,2,3],[3,-2,1],[4,2,-1]] `Y

After going through the different steps, the solution returned is:

What the calculator showed was not exactly a Gauss-Jordan elimination with

full pivoting, but a way to calculate the inverse of a matrix by performing a

Gauss-Jordan elimination, without pivoting. This procedure for calculating the

inverse is based on the augmented matrix (A

)

= [A

|I ].

aug n×n

n×n n×n

The calculator showed you the steps up to the point in which the left-hand half

of the augmented matrix has been converted to a diagonal matrix. From there,

the final step is to divide each row by the corresponding main diagonal pivot.

In other words, the calculator has transformed (A

)

= [A

|I ], into [I

aug n×n

n×n n×n

-1

|A ].

Inverse matrices and determinants

Notice that all the elements in the inverse matrix calculated above are divided

by the value 56 or one of its factors (28, 7, 8, 4 or 1). If you calculate the

determinant of the matrix A, you get det(A) = 56.

-1

We could write, A = C/det(A), where C is the matrix

⎡

⎤

⎢

⎥

C = 7 −13 8 .

⎢

⎥

− 8

⎣

⎦

Page 11-40

-1

The result (A )

= C

/det(A

), is a general result that applies to any

n×n

non-singular matrix A. A general form for the elements of C can be written

based on the Gauss-Jordan algorithm.

-1

Based on the equation A = C/det(A), sketched above, the inverse matrix,

-1

A , is not defined if det(A) = 0. Thus, the condition det(A) = 0 defines also a

singular matrix.

Solution to linear systems using calculator functions

The simplest way to solve a system of linear equations, A⋅x = b, in the

calculator is to enter b, enter A, and then use the division function /. If the

system of linear equations is over-determined or under-determined, a “solution”

can be produced by using Function LSQ (Least-SQuares), as shown earlier. The

calculator, however, offers other possibilities for solving linear systems of

equations by using Functions included in the MATRICES’ LINEAR SYSTEMS..

menu accessible through „Ø(Set system flag 117 to CHOOSE boxes):

The functions included are LINSOLVE, REF, rref, RREF, and SYST2MAT.

Function LINSOLVE

Function LINSOLVE takes as arguments an array of equations and a vector

containing the names of the unknowns, and produces the solution to the linear

system. The following screens show the help-facility entry (see Chapter 1) for

function LINSOLVE, and the corresponding example listed in the entry. The

right-hand side screen shows the result using the line editor (press ˜ to

activate):

Here is another example in ALG mode. Enter the following:

Page 11-41

LINSOLVE([X-2*Y+Z=-8,2*X+Y-2*Z=6,5*X-2*Y+Z=-12],

[X,Y,Z])

to produce the solution: [X=-1,Y=2,Z = -3].

Function LINSOLVE works with symbolic expressions. Functions REF, rref, and

RREF, work with the augmented matrix in a Gaussian elimination approach.

Functions REF, rref, RREF

The upper triangular form to which the augmented matrix is reduced during the

forward elimination part of a Gaussian elimination procedure is known as an

"echelon" form. Function REF (Reduce to Echelon Form) produces such a matrix

given the augmented matrix in stack level 1.

Consider the augmented matrix,

⎡1 − 2

0 ⎤

⎢

⎥

A_aug= 2

− 2 − 3 .

⎢

⎥

5 − 2

⎣

⎦

Representing a linear system of equations, A⋅x = b, where

A = [[1,-2,1],[2,1,-2],[5,-2,1]],

and

b = [[0],[-3],[12]].

Enter the augmented matrix, and save it into variable AAUG, in ALG mode:

[[1,-2,1,0],[2,1,-2,-3][5,-2,1,12]] ꢀ

Application of function REF produces:

The result is the upper triangular (echelon form) matrix of coefficients resulting

from the forward elimination step in a Gaussian elimination procedure.

Page 11-42

The diagonal matrix that results from a Gauss-Jordan elimination is called a

row-reduced echelon form. Function RREF ( Row-Reduced Echelon Form) The

result of this function call is to produce the row-reduced echelon form so that the

matrix of coefficients is reduced to an identity matrix. The extra column in the

augmented matrix will contain the solution to the system of equations.

As an example, we show the result of applying function RREF to matrix AAUG

in ALG mode:

The result is final augmented matrix resulting from a Gauss-Jordan elimination

without pivoting.

A row-reduced echelon form for an augmented matrix can be obtained by

using function rref. This function produces a list of the pivots and an equivalent

matrix in row-reduced echelon form so that the matrix of coefficients is reduced

to a diagonal matrix.

For example, for matrix AAUG, function rref produces the following result:

The second screen above is obtained by activating the line editor (press ˜).

The result shows pivots of 3, 1, 4, 1, 5, and 2, and a reduced diagonal matrix.

Function SYST2MAT

This function converts a system of linear equations into its augmented matrix

equivalent. The following example is available in the help facility of the

calculator:

Page 11-43

The result is the augmented matrix corresponding to the system of equations:

X+Y = 0

X-Y =2

Residual errors in linear system solutions (Function RSD)

Function RSD calculates the ReSiDuals or errors in the solution of the matrix

equation A⋅x=b, representing a system of n linear equations in n unknowns.

We can think of solving this system as solving the matrix equation: f(x) = b -A⋅x

= 0. Suppose that, through a numerical method, we produce as a first

approximation the solution x(0). Evaluating f(x(0)) = b - A⋅x(0) = e ≠ 0. Thus,

e is a vector of residuals of Function for the vector x = x (0).

To use Function RSD you need the terms b, A, and x(0), as arguments. The

vector returned is e = b - A⋅x(0). For example, using A = [[2,-

1][0,2]], x(0) = [1.8,2.7], and b = [1,6], we can find the vector

of residuals as follows:

The result is e = b - A⋅x(0) = [ 0.1 0.6 ].

Note: If we let the vector Δx = x – x (0), represent the correction in the

values of x (0), we can write a new matrix equation for Δx, namely A⋅Δx = e.

Solving for Δx we can find the actual solution of the original system as x =

x(0) + Δx.

Page 11-44

Eigenvalues and eigenvectors

Given a square matrix A, we can write the eigenvalue equation A⋅x = λ⋅x,

where the values of λ that satisfy the equation are known as the eigenvalues of

matrix A. For each value of λ, we can find, from the same equation, values of

x that satisfy the eigenvalue equation. These values of x are known as the

eigenvectors of matrix A. The eigenvalues equation can be written also as (A

– λ⋅I)x = 0.

This equation will have a non-trivial solution only if the matrix (A – λ⋅I) is

singular, i.e., if det(A – λ⋅I) = 0.

The last equation generates an algebraic equation involving a polynomial of

order n for a square matrix A . The resulting equation is known as the

n×n

characteristic polynomial of matrix A. Solving the characteristic polynomial

produces the eigenvalues of the matrix.

The calculator provides a number of functions that provide information

regarding the eigenvalues and eigenvectors of a square matrix. Some of these

functions are located under the menu MATRICES/EIGEN activated through

„Ø.

Function PCAR

Function PCAR generates the characteristic polynomial of a square matrix using

the contents of variable VX (a CAS reserved variable, typically equal to ‘X’) as

the unknown in the polynomial. For example, enter the following matrix in ALG

mode and find the characteristic equation using PCAR:

[[1,5,-3],[2,-1,4],[3,5,2]]

Page 11-45

Using the variable λ to represent eigenvalues, this characteristic polynomial is to

be interpreted as

λ -2λ -22λ +21=0.

Function EGVL

Function EGVL (EiGenVaLues) produces the eigenvalues of a square matrix. For

example, the eigenvalues of the matrix shown below are calculated in ALG

mode using function EGVL:

The eigenvalues λ = [ -√10, √10 ].

Note: In some cases, you may not be able to find an ‘exact’ solution to the

characteristic polynomial, and you will get an empty list as a result when using

Function EGVL. If that were to happen to you, change the calculation mode to

Approx in the CAS, and repeat the calculation.

For example, in exact mode, the following exercise produces an empty list as

the solution:

Change mode to Approx and repeat the entry, to get the following eigenvalues:

[(1.38,2.22), (1.38,-2.22), (-1.76,0)].

Function EGV

Function EGV (EiGenValues and eigenvectors) produces the eigenvalues and

eigenvectors of a square matrix. The eigenvectors are returned as the columns

Page 11-46

of a matrix, while the corresponding eigenvalues are the components of a

vector.

For example, in ALG mode, the eigenvectors and eigenvalues of the matrix

listed below are found by applying function EGV:

The result shows the eigenvalues as the columns of the matrix in the result list.

To see the eigenvalues we can use: GET(ANS(1),2), i.e., get the second

element in the list in the previous result. The eigenvalues are:

In summary,

λ = 0.29, x = [ 1.00,0.79,–0.91] ,

λ = 3.16, x = [1.00,-0.51, 0.65] ,

λ = 7.54, x = [-0.03, 1.00, 0.84] .

Note: A symmetric matrix produces all real eigenvalues, and its eigenvectors

are mutually perpendicular. For the example just worked out, you can check

that x •x = 0, x •x = 0, and x •x = 0.

Function JORDAN

Function JORDAN is intended to produce the diagonalization or Jordan-cycle

decomposition of a matrix. In RPN mode, given a square matrix A, function

JORDAN produces four outputs, namely:

•

The minimum polynomial of matrix A (stack level 4)

The characteristic polynomial of matrix A (stack level 3)

Page 11-47

•

A list with the eigenvectors corresponding to each eigenvalue of matrix

A (stack level 2)

A vector with the eigenvectors of matrix A (stack level 4)

For example, try this exercise in RPN mode:

[[4,1,-2],[1,2,-1],[-2,-1,0]]

JORD N

The output is the following:

4: ‘X^3+-6*x^2+2*X+8’

3: ‘X^3+-6*x^2+2*X+8’

2: { }

1: { }

The same exercise, in ALG mode, looks as in the following screen shots:

Function MAD

This function, although not available in the EIGEN menu, also provides

information related to the eigenvalues of a matrix. Function MAD is available

through the MATRICES OPERATIONS sub-menu („Ø) and is intended to

produce the adjoint matrix of a matrix.

In RPN mode, function MAD generates a number of properties of a square

matrix, namely:

•

the determinant (stack level 4)

the formal inverse (stack level 3),

in stack level 2, the matrix coefficients of the polynomial p(x) defined

by (x⋅I-A) ⋅p(x)=m(x)⋅I,

•

the characteristic polynomial of the matrix (stack level 1)

Page 11-48

Notice that the equation (x⋅I-A)⋅p(x)=m(x)⋅I is similar, in form, to the

eigenvalue equation A⋅x = λ⋅x.

As an example, in RPN mode, try:

[[4,1,-2] [1,2,-1][-2,-1,0]] M D

The result is:

4: -8.

3: [[ 0.13 –0.25 –0.38][-0.25 0.50 –0.25][-0.38 –0.25 –0.88]]

2: {[[1 0 0][0 1 0][0 0 1]] [[ -2 1 –2][1 –4 –1][-2 –1 –6] [[-1 2 3][2 –4 2][3 2 7]]}

1: ‘X^3+-6*x^2+2*X+8’

The same exercise, in ALG mode, will look as follows:

Matrix factorization

Matrix factorization or decomposition consists of obtaining matrices that when

multiplied result in a given matrix. We present matrix decomposition through

the use of Functions contained in the matrix FACT menu. This menu is accessed

through „Ø.

Function contained in this menu are: LQ, LU, QR,SCHUR, SVD, SVL.

Page 11-49

Function LU

Function LU takes as input a square matrix A, and returns a lower-triangular

matrix L, an upper triangular matrix U, and a permutation matrix P, in stack

levels 3, 2, and 1, respectively. The results L, U, and P, satisfy the equation

P⋅A = L⋅U. When you call the LU function, the calculator performs a Crout LU

decomposition of A using partial pivoting.

For example, in RPN mode: [[-1,2,5][3,1,-2][7,6,5]] LU

produces:

3:[[7 0 0][-1 2.86 0][3 –1.57 –1]

2: [[1 0.86 0.71][0 1 2][0 0 1]]

1: [[0 0 1][1 0 0][0 1 0]]

In ALG mode, the same exercise will be shown as follows:

Orthogonal matrices and singular value decomposition

A square matrix is said to be orthogonal if its columns represent unit vectors that

are mutually orthogonal. Thus, if we let matrix U = [v v … v ] where the v,

i = 1, 2, …, n, are column vectors, and if v•v = δ , where δ is the Kronecker’s

delta function, then U will be an orthogonal matrix. This conditions also imply

that U⋅ U = I.

The Singular Value Decomposition (SVD) of a rectangular matrix

determining the matrices , and , such that

consists in

×n

⋅

⋅V

m×

m×m

n×

where U and V are orthogonal matrices, and S is a diagonal matrix. The

diagonal elements of S are called the singular values of A and are usually

ordered so that s ≥ s , for i = 1, 2, …, n-1. The columns [u ] of U and [v ] of

i+1

V are the corresponding singular vectors.

Function SVD

In RPN, function SVD (Singular Value Decomposition) takes as input a matrix

, and returns the matrices U , V

, and a vector s in stack levels 3, 2,

n×m

n×n

m×m

and 1, respectively. The dimension of vector s is equal to the minimum of the

values n and m. The matrices U and V are as defined earlier for singular value

Page 11-50

decomposition, while the vector s represents the main diagonal of the matrix S

used earlier.

For example, in RPN mode: [[5,4,-1],[2,-3,5],[7,2,8]] SVD

3: [[-0.27 0.81 –0.53][-0.37 –0.59 –0.72][-0.89 3.09E-3 0.46]]

2: [[ -0.68 –0.14 –0.72][ 0.42 0.73 –0.54][-0.60 0.67 0.44]]

1: [ 12.15 6.88 1.42]

Function SVL

Function SVL (Singular VaLues) returns the singular values of a matrix A

as a

n×m

vector s whose dimension is equal to the minimum of the values n and m. For

example, in RPN mode, [[5,4,-1],[2,-3,5],[7,2,8]] SVL

produces [ 12.15 6.88 1.42].

Function SCHUR

In RPN mode, function SCHUR produces the Schur decomposition of a square

matrix A returning matrices Q and T, in stack levels 2 and 1, respectively, such

that A = Q⋅T⋅Q , where Q is an orthogonal matrix, and T is a triangular

matrix. For example, in RPN mode,

[[2,3,-1][5,4,-2][7,5,4]] SCHUR

results in:

2: [[0.66 –0.29 –0.70][-0.73 –0.01 –0.68][ -0.19 –0.96 0.21]]

1: [[-1.03 1.02 3.86 ][ 0 5.52 8.23 ][ 0 –1.82 5.52]]

Function LQ

The LQ function produces the LQ factorization of a matrix A

returning a

n×m

lower L

trapezoidal matrix, a Q

orthogonal matrix, and a P

n×m

m×m n×n

permutation matrix, in stack levels 3, 2, and 1. The matrices A, L, Q and P

are related by P⋅A = L⋅Q. (A trapezoidal matrix out of an n×m matrix is the

equivalent of a triangular matrix out of an n×n matrix). For example,

[[ 1, -2, 1][ 2, 1, -2][ 5, -2, 1]] LQ

produces

3: [[-5.48 0 0][-1.10 –2.79 0][-1.83 1.43 0.78]]

2: [[-0.91 0.37 -0.18] [-0.36 -0.50 0.79] [-0.20 -0.78 -0.59]]

1: [[0 0 1][0 1 0][1 0 0]]

Page 11-51

Function QR

In RPN, function QR produces the QR factorization of a matrix A

returning a

n×m

orthogonal matrix, a R

upper trapezoidal matrix, and a P

n×n

n×m m×m

permutation matrix, in stack levels 3, 2, and 1. The matrices A, P, Q and R are

related by A⋅P = Q⋅R. For example, [[ 1,-2,1][ 2,1,-2][ 5,-

2,1]] QR

produces

3: [[-0.18 0.39 0.90][-0.37 –0.88 0.30][-0.91 0.28 –0.30]]

2: [[ -5.48 –0.37 1.83][ 0 2.42 –2.20][0 0 –0.90]]

1: [[1 0 0][0 0 1][0 1 0]]

Note: Examples and definitions for all functions in this menu are available

through the help facility in the calculator. Try these exercises in ALG mode to

see the results in that mode.

Matrix Quadratic Forms

A quadratic form from a square matrix A is a polynomial expression originated

from x⋅A⋅x . For example, if we use A = [[2,1,–1][5,4,2][3,5,–1]], and x =

[X Y Z] , the corresponding quadratic form is calculated as

2 1 −1

⎤ ⎡ ⎤

⎥ ⎢ ⎥

⎡

⎢

x ⋅ A ⋅ x^T=

[

X Y

]

⋅ 5 4 2 ⋅ Y

⎥ ⎢ ⎥

⎢

⎥ ⎢ ⎥

3 5 −1

⎦ ⎣ ⎦

⎣

2X + Y − Z

⋅ 5X + 4Y + 2Z

⎡

⎢

⎤

⎥

[

X Y

]

⎢

⎥

3X + 5Y − Z

⎣

⎦

Finally,

x⋅A⋅x = 2X +4Y -Z +6XY+2XZ+7ZY

The QUADF menu

The calculator provides the QUADF menu for operations related to QUADratic

Forms. The QUADF menu is accessed through „Ø.

Page 11-52

This menu includes functions AXQ, CHOLESKY, GAUSS, QXA, and SYLVESTER.

Function AXQ

In RPN mode, function AXQ produces the quadratic form corresponding to a

matrix A

in stack level 2 using the n variables in a vector placed in stack

n×n

level 1. Function returns the quadratic form in stack level 1 and the vector of

variables in stack level 1. For example,

[[2,1,-1],[5,4,2],[3,5,-1]] `

['X','Y','Z'] ` XQ

returns

2: ‘2*X^2+(6*Y+2*Z)*X+4*Y^2+7*Z*y-Z^2’

1: [‘X’ ‘Y’ ‘Z’]

Function QXA

Function QXA takes as arguments a quadratic form in stack level 2 and a vector

of variables in stack level 1, returning the square matrix A from which the

quadratic form is derived in stack level 2, and the list of variables in stack level

1. For example,

'X^2+Y^2-Z^2+4*X*Y-16*X*Z' `

['X','Y','Z'] ` QX

returns

2: [[1 2 –8][2 1 0][-8 0 –1]]

1: [‘X’ ‘Y’ ‘Z’]

Diagonal representation of a quadratic form

Given a symmetric square matrix A, it is possible to “diagonalize” the matrix A

by finding an orthogonal matrix P such that P ⋅A⋅P = D, where D is a diagonal

matrix. If Q = x⋅A⋅x is a quadratic form based on A, it is possible to write

the quadratic form Q so that it only contains square terms from a variable y,

Page 11-53

such that x = P⋅y, by using Q = x⋅A⋅x = (P⋅y)⋅A⋅ (P⋅y) = y⋅(P ⋅A⋅P)⋅y =

y⋅D⋅y

Function SYLVESTER

Function SYLVESTER takes as argument a symmetric square matrix A and returns

a vector containing the diagonal terms of a diagonal matrix D, and a matrix P,

so that P ⋅A⋅P = D. For example:

[[2,1,-1],[1,4,2],[-1,2,-1]] SYLVESTER

produces

2: [ 1/2 2/7 -23/7]

1: [[2 1 –1][0 7/2 5/2][0 0 1]]

Function GAUSS

Function GAUSS returns the diagonal representation of a quadratic form Q =

x⋅A⋅x taking as arguments the quadratic form in stack level 2 and the vector

of variables in stack level 1. The result of this function call is the following:

•

An array of coefficients representing the diagonal terms of D (stack

level 4)

•

A matrix P such that A = P ⋅D⋅P (stack level 3)

The diagonalized quadratic form (stack level 2)

The list of variables (stack level 1)

For example:

'X^2+Y^2-Z^2+4*X*Y-16*X*Z' `

['X','Y','Z'] ` G USS

returns

4: [1 –0.333 20.333]

3: [[1 2 –8][0 –3 16][0 0 1]]

2: ’61/3*Z^2+ -1/3*(16*Z+-3*Y)^2+(-8*z+2*Y+X)^2‘

1: [‘X’ ‘Y’ ‘Z’]

Linear Applications

The LINEAR APPLICATIONS menu is available through the „Ø.

Page 11-54

Information on the functions listed in this menu is presented below by using the

calculator’s own help facility. The figures show the help facility entry and the

attached examples.

Function IMAGE

Function ISOM

Page 11-55

Function KER

Function MKISOM

Page 11-56

Chapter 12

Graphics

In this chapter we introduce some of the graphics capabilities of the calculator.

We will present graphics of functions in Cartesian coordinates and polar

coordinates, parametric plots, graphics of conics, bar plots, scatterplots, and a

variety of three-dimensional graphs.

Graphs options in the calculator

To access the list of graphic formats available in the calculator, use the keystroke

sequence „ô(D) Please notice that if you are using the RPN mode

these two keys must be pressed simultaneously to activate any of the graph

functions. After activating the 2D/3D function, the calculator will produce the

PLOT SETUP window, which includes the TYPE field as illustrated below.

Right in front of the TYPE field you will, most likely, see the option Function

highlighted. This is the default type of graph for the calculator. To see the list

of available graph types, press the soft menu key labeled @CHOOS. This will

produce a drop down menu with the following options (use the up- and down-

arrow keys to see all the options):

Page 12-1

These graph options are described briefly next.

Function: for equations of the form y = f(x) in plane Cartesian coordinates

Polar: for equations of the from r = f(θ) in polar coordinates in the plane

Parametric: for plotting equations of the form x = x(t), y = y(t) in the plane

Diff Eq: for plotting the numerical solution of a linear differential equation

Conic: for plotting conic equations (circles, ellipses, hyperbolas, parabolas)

Truth: for plotting inequalities in the plane

Histogram: for plotting frequency histograms (statistical applications)

Bar: for plotting simple bar charts

Scatter: for plotting scatter plots of discrete data sets (statistical applications)

Slopefield: for plotting traces of the slopes of a function f(x,y) = 0.

Fast3D: for plotting curved surfaces in space

Wireframe: for plotting curved surfaces in space showing wireframe grids

Ps-Contour: for plotting contour plots of surfaces

Y- Slice: for plotting a slicing view of a function f(x,y).

Gridmap: for plotting real and imaginary part traces of a complex function

Pr-Surface: for parametric surfaces given by x = x(u,v), y = y(u,v), z = z(u,v).

Plotting an expression of the form y = f(x)

In this section we present an example of a plot of a function of the form y = f(x).

In order to proceed with the plot, first, purge the variable x, if it is defined in the

current directory (x will be the independent variable in the calculator's PLOT

feature, therefore, you don't want to have it pre-defined). Create a sub-

directory called 'TPLOT' (for test plot), or other meaningful name, to perform the

following exercise.

As an example, let's plot the function,

x²

f (x) =

exp(−

)

2π

ꢀ First, enter the PLOT SETUP environment by pressing, „ô. Make

sure that the option Function is selected as the TYPE, and that ‘X’ is

selected as the independent variable (INDEP). Press L@@@OK@@@ to

return to normal calculator display. The PLOT SET UP window should

look similar to this:

Page 12-2

ꢀNote: You will notice that a new variable, called PPAR, shows up in your

soft menu key labels. This stands for Plot PARameters. To see its contents,

press ‚@PPAR. A detailed explanation of the contents of PPAR is

provided later in this Chapter. Press ƒ to drop this line from the stack.

ꢀ

Enter the PLOT environment by pressing „ñ(press them

simultaneously if in RPN mode). Press @ADD to get you into the equation

writer. You will be prompted to fill the right-hand side of an equation

Y1(x) = ꢀ. Type the function to be plotted so that the Equation Writer

shows the following:

ꢀ Press ` to return to the PLOT - FUNCTION window. The expression

‘Y1(X) = EXP(-X^2/2)/√(2*π)’ will be highlighted. Press L@@@OK@@@ to

return to normal calculator display.

Note: Two new variables show up in your soft menu key labels, namely EQ

and Y1. To see the contents of EQ, use ‚@@@EQ@@. The content of EQ is

simply the function name ‘Y1(X)’. The variable EQ is used by the calculator

to store the equation, or equations, to plot.

To see the contents of Y1 press ‚@@@Y1@@. You will get the function Y1(X)

defined as the program:

Page 12-3

<< →X ‘EXP(-X^2/2)/√(2*π)‘ >>.

Press ƒ, twice, to drop the contents of the stack.

ꢀ Enter the PLOT WINDOW environment by entering „ò (press

them simultaneously if in RPN mode). Use a range of –4 to 4 for H-

VIEW, then press @AUTO to generate the V-VIEW automatically. The

PLOT WINDOW screen looks as follows:

ꢀ Plot the graph: @ERASE @DRAW (wait till the calculator finishes the graphs)

ꢀ To see labels:@EDIT L @LABEL @MENU

ꢀ To recover the first graphics menu: LL@PICT

ꢀ To trace the curve: @TRACE @@X,Y@@ . Then use the right- and left-arrow keys

(š™) to move about the curve. The coordinates of the points you

trace will be shown at the bottom of the screen. Check that for x =

1.05 , y = 0.231. Also, check that for x = -1.48 , y = 0.134. Here is

picture of the graph in tracing mode:

ꢀ To recover the menu, and return to the PLOT WINDOW environment,

press L@CANCL@.

Page 12-4

Some useful PLOT operations for FUNCTION plots

In order to discuss these PLOT options, we'll modify the function to force it to

have some real roots (Since the current curve is totally contained above the x

axis, it has no real roots.) Press ‚@@@Y1@@ to list the contents of the function Y1

on the stack: << →X ‘EXP(-X^2/2)/ √(2*π)‘ >>. To edit this expression use:

Launches the line editor

‚˜

Moves cursor to the end of the line

Modifies the expression

Returns to calculator display

ššš-0.1

Next, store the modified expression into variable y by using „@@@Y1@@ if in RPN

mode, or „îK@@@Y1@@ in ALG mode.

x²

The function to be plotted is now,

f (x) =

exp(− ) − 0.1

2π

Enter the PLOT WINDOW environment by entering „ò(press them

simultaneously if in RPN mode.) Keep the range of –4 to 4 for H-VIEW, press

˜@AUTO to generate the V-VIEW. To plot the graph, press @ERASE @DRAW

ꢀ Once the graph is plotted, press @@FCN! to access the function menu.

With this menu you can obtain additional information about the plot

such as intersects with the x-axis, roots, slopes of the tangent line, area

under the curve, etc.

ꢀ For example, to find the root on the left side of the curve, move the

cursor near that point, and press @ROOT. You will get the result: ROOT: -

1.6635…. Press L to recover the menu. Here is the result of

ROOT in the current plot:

ꢀ

If you move the cursor towards the right-hand side of the curve, by

pressing the right-arrow key (™), and press @ROOT, the result now is

Page 12-5

ROOT: 1.6635... The calculator indicated, before showing the root,

that it was found through SIGN REVERSAL. Press L to recover the

menu.

ꢀ

Pressing @ISECT will give you the intersection of the curve with the x-axis,

which is essentially the root. Place the cursor exactly at the root and

press @ISECT. You will get the same message as before, namely SIGN

REVERSAL, before getting the result I-SECT: 1.6635…. The @ISECT

function is intended to determine the intersection of any two curves

closest to the location of the cursor. In this case, where only one curve,

namely, Y1(X), is involved, the intersection sought is that of f(x) with the

x-axis, however, you must place the cursor right at the root to produce

the same result. Press L to recover the menu.

ꢀ

Place the cursor on the curve at any point and press @SLOPE to get the

value of the slope at that point. For example, at the negative root,

SLOPE: 0.16670…. Press L to recover the menu.

To determine the highest point in the curve, place the cursor near the

vertex and press @EXTR The result is EXTRM: 0.. Press L to recover

the menu.

Other buttons available in the first menu are @AREA to calculate the area

under the curve, and @SHADE to shade an area under the curve. Press

L to see more options. The second menu includes one button called

@VIEW that flashes for a few seconds the equation plotted. Press @VIEW.

Alternatively, you can press the button @NEXQ (NEXt eQuation) to see the

name of the function Y1(x). Press L to recover the menu.

ꢀ

The button

gives the value of f(x) corresponding to the cursor

position. Place the cursor anywhere in the curve and press

. The

value will be shown in the lower left corner of the display. Press L

to recover the menu.

Place the cursor in any given point of the trajectory and press TANL to

obtain the equation of the tangent line to the curve at that point. The

equation will be displayed on the lower left corner of the display. Press

L to recover the menu.

If you press

the calculator will plot the derivative function, f'(x) =

df/dx, as well as the original function, f(x). Notice that the two curves

intercept at two points. Move the cursor near the left intercept point

and press @@FCN! @ISECT, to get I-SECT: (-0.6834…,0.21585). Press L

to recover the menu.

ꢀ

To leave the FCN environment, press @PICT (or LPICT).

Press @CANCL to return to the PLOT WINDOW environment. Then, press

L @@@OK@@@ to return to normal calculator display.

Page 12-6

Note: the stack will show all the graph operations performed, properly

identified.

ꢀ Enter the PLOT environment by pressing, simultaneously if in RPN mode,

„ñ. Notice that the highlighted field in the PLOT environment

now contains the derivative of Y1(X). Press L@@@OK@@@ to return to return

to normal calculator display.

ꢀ Press ‚@@EQ@@ to check the contents of EQ. You will notice that it

contains a list instead of a single expression. The list has as elements

an expression for the derivative of Y1(X) and Y1(X) itself. Originally,

EQ contained only Y1(x). After we pressed

in the @FCN@

environment, the calculator automatically added the derivative of Y1(x)

to the list of equations in EQ.

Saving a graph for future use

If you want to save your graph to a variable, get into the PICTURE environment

by pressing š. Then, press @EDIT LL@PICTꢁ. This captures the current

picture into a graphics object. To return to the stack, press @PICT @CANCL.

In level 1 of the stack you will see a graphics object described as Graphic

131 × 64. This can be stored into a variable name, say, PIC1.

To display your figure again, recall the contents of variable PIC1 to the stack.

The stack will show the line: Graphic 131 × 64. To see the graph, enter the

PICTURE environment, by pressing š.

Clear the current picture, @EDIT L@ERASE.

Move the cursor to the upper left corner of the display, by using the š and

— keys.

To display the figure currently in level 1 of the stack press L REPL .

Page 12-7

To return to normal calculator function, press @PICT @CANCL.

Note: To save printing space, we will not include more graphs produced by

following the instructions in this Chapter. The user is invited to produce those

graphics on his or her own.

Graphics of transcendental functions

In this section we use some of the graphics features of the calculator to show the

typical behavior of the natural log, exponential, trigonometric and hyperbolic

functions. You will not see more graphs in this chapter, instead the user should

see them in the calculator.

Graph of ln(X)

Press, simultaneously if in RPN mode, the left-shift key „ and the ô (D)

key to produce the PLOT SETUP window. The field labeled Typewill be

highlighted. If the option Function is not already selected press the soft key

labeled @CHOOS, use the up and down keys to select Function, and press @@@OK@@@

to complete the selection. Check that the field labeled Indep: contains the

variable ‘X’. If that is not so, press the down arrow key twice until the Indep

field is highlighted, press the soft key labeled @EDIT and modify the value of the

independent variable to read ‘X’. Press @@@OK@@@ when done. Press L@@@OK@@@ to

return to normal calculator display.

Next, we’ll resize the plot window. First, press, simultaneously if in RPN mode,

the left-shift key „ and the ñ (A) key to produce the PLOT-FUNCTION

window. If there is any equation highlighted in this window, press @@DEL@@ as

needed to clear the window completely. When the PLOT-FUNCTION window

is empty you will get a prompt message that reads: No Equ., Press ADD.

Press the soft key labeled @@ADD@! . This will trigger the equation writer with the

expression Y1(X)=^ꢀ. Type LN(X). Press ` to return to the PLOT-FUNCTION

window. Press L@@@OK@@@ to return to normal calculator display.

The next step is to press, simultaneously if in RPN mode, the left-shift key „

and the ò(B) key to produce the PLOT WINDOW - FUNCTION window.

Most likely, the display will show the horizontal (H-View) and vertical (V-View)

ranges as: H-View: -6.5

6.5, V-View: -3.9

4.0

These are the default values for the x- and y-range, respectively, of the current

graphics display window. Next, change the H-View values to read: H-View: -1

Page 12-8

10 by using 1\@@@OK@@ 10@@@OK@@@. Next, press the soft key labeled @AUTO

to let the calculator determine the corresponding vertical range. After a couple

of seconds this range will be shown in the PLOT WINDOW-FUNCTION

window. At this point we are ready to produce the graph of ln(X). Press @ERASE

@DRAW to plot the natural logarithm function.

To add labels to the graph press @EDIT L@LABEL. Press @MENU to remove the

menu labels, and get a full view of the graph. Press L to recover the current

graphic menu. Press L@PICT to recover the original graphical menu.

To determine the coordinates of points on the curve press @TRACE (the cursor

moves on top of the curve at a point located near the center of the horizontal

range). Next, press (X,Y) to see the coordinates of the current cursor location.

These coordinates will be shown at the bottom of the screen. Use the right- and

left-arrow keys to move the cursor along the curve. As you move the cursor

along the curve the coordinates of the curve are displayed at the bottom of the

screen. Check that when Y:1.00E0, X:2.72E0. This is the point (e,1), since

ln(e) = 1. Press L to recover the graphics menu.

Next, we will find the intersection of the curve with the x-axis by pressing @FCN

@ROOT. The calculator returns the value Root: 1, confirming that ln(1) = 0. Press

LL@PICT @CANCL to return to the PLOT WINDOW – FUNCTION. Press `

to return to normal calculator display. You will notice that the root found in the

graphics environment was copied to the calculator stack.

Note: When you press J , your variables list will show new variables

called @@@X@@ and @@Y1@@ .Press ‚@@Y1@@ to see the contents of this variable. You

will get the program << → X ‘LN(X)’ >> , which you will recognize as the

program that may result from defining the function ‘Y1(X) = LN(X)’ by using

„à. This is basically what happens when you @@ADD@! a function in the

PLOT – FUNCTION window (the window that results from pressing ꢀñ,

simultaneously if in RPN mode), i.e., the function gets defined and added to

your variable list.

Page 12-9

Next, press ‚@@@X@@@ to see the contents of this variable. A value of 10.275 is

placed in the stack. This value is determined by our selection for the horizontal

display range. We selected a range between -1 and 10 for X. To produce

the graph, the calculator generates values between the range limits using a

constant increment, and storing the values generated, one at a time, in the

variable @@@X@@@ as the graph is drawn. For the horizontal range ( –1,10), the

increment used seems to be 0.275. When the value of X becomes larger than

the maximum value in the range (in this case, when X = 10.275), the drawing

of the graph stops. The last value of X for the graphic under consideration is

kept in variable X. Delete X and Y1 before continuing.

Graph of the exponential function

First, load the function exp(X), by pressing, simultaneously if in RPN mode, the

left-shift key „ and the ñ (V) key to access the PLOT-FUNCTION

window. Press @@DEL@@ to remove the function LN(X), if you didn’t delete Y1 as

suggested in the previous note. Press @@ADD@! and type „¸~x` to

enter EXP(X) and return to the PLOT-FUNCTION window. Press L@@@OK@@@ to

return to normal calculator display.

Next, press, simultaneously if in RPN mode, the left-shift key „ and the

ò (B) key to produce the PLOT WINDOW - FUNCTION window. Change

the H-View values to read:

H-View: -8

by using 8\@@@OK@@ @2@@@OK@@@. Next, press @AUTO. After the vertical range is

calculated, press @ERASE @DRAW to plot the exponential function.

To add labels to the graph press @EDIT L@LABEL. Press @MENU to remove the

menu labels, and get a full view of the graph. Press LL@PICT! @CANCL to

return to the PLOT WINDOW – FUNCTION. Press ` to return to normal

calculator display.

Page 12-10

The PPAR variable

Press J to recover your variables menu, if needed. In your variables menu

you should have a variable labeled PPAR . Press ‚@PPAR to get the contents

of this variable in the stack. Press the down-arrow key, , to launch the stack

editor, and use the up- and down-arrow keys to view the full contents of PPAR.

The screen will show the following values:

PPAR stands for Plot PARameters, and its contents include two ordered pairs of

real numbers, (-8.,-1.10797263281) and (2.,7.38905609893),

which represent the coordinates of the lower left corner and the upper right

corner of the plot, respectively. Next, PPAR lists the name of the independent

variable, X, followed by a number that specifies the increment of the

independent variable in the generation of the plot. The value shown here is the

default value, zero (0.), which specifies increments in X corresponding to 1 pixel

in the graphics display. The next element in PPAR is a list containing first the

coordinates of the point of intersection of the plot axes, i.e., (0.,0.), followed by

a list that specifies the tick mark annotation on the x- and y-axes, respectively {#

10d # 10d}. Next, PPAR lists the type of plot that is to be generated, i.e.,

FUNCTION, and, finally, the y-axis label, i.e., Y.

The variable PPAR, if non-existent, is generated every time you create a plot.

The contents of the function will change depending on the type of plot and on

the options that you select in the PLOT window (the window generated by the

simultaneous activation of the „ and ò(B) keys.

Inverse functions and their graphs

Let y = f(x), if we can find a function y = g(x), such that, g(f(x)) = x, then we say

-1

that g(x) is the inverse function of f(x). Typically, the notation g(x) = f (x) is

used to denote an inverse function. Using this notation we can write: if y = f(x),

-1

then x = f (y). Also, f(f (x)) = x, and f (f(x)) = x.

Page 12-11

As indicated earlier, the ln(x) and exp(x) functions are inverse of each other,

i.e., ln(exp(x)) = x, and exp(ln(x)) = x. This can be verified in the calculator by

typing and evaluating the following expressions in the Equation Writer:

LN(EXP(X)) and EXP(LN(X)). They should both evaluate to X.

-1

When a function f(x) and its inverse f (x) are plotted simultaneously in the

same set of axes, their graphs are reflections of each other about the line y = x.

Let’s check this fact with the calculator for the functions LN(X) and EXP(X) by

following this procedure:

Press, simultaneously if in RPN mode, „ñ. The function Y1(X) = EXP(X)

should be available in the PLOT - FUNCTION window from the previous

exercise. Press @@ADD@! , and type the function Y2(X) = LN(X). Also, load

the function Y3(X) = X. Press L@@@OK@@@ to return to normal calculator display.

Press, simultaneously if in RPN mode, „ò, and change the H-View range

to read:

H-View: -8

Press @AUTO to generate the vertical range. Press @ERASE @DRAW to produce the

graph of y = ln(x), y = exp(x), and y =x, simultaneously if in RPN mode.

You will notice that only the graph of y = exp(x) is clearly visible. Something

went wrong with the @AUTO selection of the vertical range. What happens is that,

when you press @AUTO in the PLOT FUNCTION – WINDOW screen, the

calculator produces the vertical range corresponding to the first function in the

list of functions to be plotted. Which, in this case, happens to be Y1(X) =

EXP(X). We will have to enter the vertical range ourselves in order to display

the other two functions in the same plot.

Press @CANCL to return to the PLOT FUNCTION – WINDOW screen. Modify the

vertical and horizontal ranges to read: H-View: -8

8, V-View: -4

By selecting these ranges we ensure that the scale of the graph is kept 1 vertical

to 1 horizontal. Press @ERASE @DRAW and you will get the plots of the natural

logarithm, exponential, and y = x functions. It will be evident from the graph

that LN(X) and EXP(X) are reflections of each other about the line y = X. Press

@CANCL to return to the PLOT WINDOW – FUNCTION. Press ` to return to

normal calculator display.

Page 12-12

Summary of FUNCTION plot operation

In this section we present information regarding the PLOT SETUP, PLOT-

FUNCTION, and PLOT WINDOW screens accessible through the left-shift key

combined with the soft-menu keys A through D. Based on the graphing

examples presented above, the procedure to follow to produce a FUNCTION

plot (i.e., one that plots one or more functions of the form Y = F(X)), is the

following:

„ô, simultaneously if in RPN mode: Access to the PLOT SETUP window.

If needed, change TYPEto FUNCTION, and enter the name of the independent

variable.

Settings:

ꢀ A check on _Simult means that if you have two or more plots in the same

graph, they will be plotted simultaneously when producing the graph.

ꢀ A check on _Connect means that the curve will be a continuous curve

rather than a set of individual points.

ꢀ A check on _Pixels means that the marks indicated by H-Tick and V-

Tickwill be separated by that many pixels.

ꢀ The default value for both by H-Tickand V-Tickis 10.

Soft key menu options:

ꢀ Use @EDIT to edit functions of values in the selected field.

ꢀ Use @CHOOS to select the type of plot to use when the Type:field is

highlighted. For the current exercises, we want this field set to FUNCTION.

Note: the soft menu keys @EDIT and @CHOOS are not available at the same

time. One or the other will be selected depending on which input field is

highlighted.

ꢀ

Press the AXES soft menu key to select or deselect the plotting of axes in

the graph. If the option ‘plot axes’ is selected, a square dot will appear in

the key label: @AXESꢀ . Absence of the square dot indicates that axes will

not be plotted in the graph.

ꢀ Use @ERASE to erase any graph currently existing in the graphics display

window.

ꢀ Use @DRAW to produce the graph according to the current contents of PPAR

for the equations listed in the PLOT-FUNCTION window.

ꢀ Press L to access the second set of soft menu keys in this screen.

ꢀ Use @RESET to reset any selected field to its default value.

Page 12-13

ꢀ Use @CANCL to cancel any changes to the PLOT SETUP window and return to

normal calculator display.

ꢀ Press @@@OK@@@ to save changes to the options in the PLOT SETUP window and

return to normal calculator display.

„ñ, simultaneously if in RPN mode: Access to the PLOT window (in this

case it will be called PLOT –FUNCTION window).

Soft menu key options:

ꢀ Use @EDIT to edit the highlighted equation.

ꢀ Use @@ADD@! to add new equations to the plot.

Note: @@ADD@! or @EDIT will trigger the equation writer EQW that you can

use to write new equations or edit old equations.

ꢀ Use @@DEL@@ to remove the highlighted equation.

ꢀ Use @CHOOS to add an equation that is already defined in your variables

menu, but not listed in the PLOT – FUNCTION window.

ꢀ Use @ERASE to erase any graph currently existing in the graphics display

window.

ꢀ Use @DRAW to produce the graph according to the current contents of PPAR

for the equations listed in the PLOT-FUNCTION window.

ꢀ Press L to activate the second menu list.

ꢀ Use

and

to move the selected equation one location up or

down, respectively.

ꢀ Use @CLEAR if you want to clear all the equations currently active in the PLOT

– FUNCTION window. The calculator will verify whether or not you want

to clear all the functions before erasing all of them. Select YES, and press

@@@OK@@@ to proceed with clearing all functions. Select NO, and press @@@OK@@@ to

de-activate the option CLEAR.

ꢀ Press @@@OK@@@ when done to return to normal calculator display.

„ò, simultaneously if in RPN mode: Access to the PLOT WINDOW

screen.

Settings:

ꢀ Enter lower and upper limits for horizontal view (H-View) and vertical view

(V-View) ranges in the plot window. Or,

Page 12-14

ꢀ Enter lower and upper limits for horizontal view (H-View), and press @AUTO,

while the cursor is in one of the V-View fields, to generate the vertical view

(V-View) range automatically. Or,

ꢀ Enter lower and upper limits for vertical view (V-View), and press @AUTO,

while the cursor is in one of the H-View fields, to generate the horizontal

view (H-View) range automatically.

ꢀ The calculator will use the horizontal view (H-View) range to generate data

values for the graph, unless you change the options Indep Low, (Indep) High,

and (Indep) Step. These values determine, respectively, the minimum,

maximum, and increment values of the independent variable to be used in

the plot. If the option Default is listed in the fields Indep Low, (Indep) High,

and (Indep) Step, the calculator will use the minimum and maximum values

determined by H-View.

ꢀ A check on _Pixels means that the values of the independent variable

increments (Step:) are given in pixels rather than in plot coordinates.

Soft menu key options:

ꢀ Use @EDIT to edit any entry in the window.

ꢀ Use @AUTO as explained in Settings, above.

ꢀ Use @ERASE to erase any graph currently existing in the graphics display

window.

ꢀ Use @DRAW to produce the graph according to the current contents of PPAR

for the equations listed in the PLOT-FUNCTION window.

ꢀ Press L to activate the second menu list.

ꢀ Use @RESET to reset the field selected (i.e., where the cursor is positioned) to

its default value.

ꢀ Use @CALC to access calculator stack to perform calculations that may be

necessary to obtain a value for one of the options in this window. When

the calculator stack is made available to you, you will also have the soft

menu key options @CANCL and @@@OK@@@ .

ꢀ Use @CANCL in case you want to cancel the current calculation and return to

the PLOT WINDOW screen. Or,

ꢀ Use @@@OK@@@ to accept the results of your calculation and return to the PLOT

WINDOW screen.

ꢀ Use @TYPES to get information on the type of objects that can be used in the

selected option field.

ꢀ Use @CANCL to cancel any changes to the PLOT WINDOW screen and return

to normal calculator display.

ꢀ Press @@@OK@@@ to accept changes to the PLOT WINDOW screen and return to

normal calculator display.

Page 12-15

„ó, simultaneously if in RPN mode: Plots the graph based on the settings

stored in variable PPAR and the current functions defined in the PLOT –

FUNCTION screen. If a graph, different from the one you are plotting, already

exists in the graphic display screen, the new plot will be superimposed on the

existing plot. This may not be the result you desire, therefore, I recommend to

use the @ERASE @DRAW soft menu keys available in the PLOT SETUP, PLOT-

FUNCTION or PLOT WINDOW screens.

Plots of trigonometric and hyperbolic functions

The procedures used above to plot LN(X) and EXP(X), separately or

simultaneously, can be used to plot any function of the form y = f(x). It is left as

an exercise to the reader to produce the plots of trigonometric and hyperbolic

functions and their inverses. The table below suggests the values to use for the

vertical and horizontal ranges in each case. You can include the function Y=X

when plotting simultaneously a function and its inverse to verify their ‘reflection’

about the line Y = X.

H-View range

V-View range

Function

SIN(X)

ASIN(X)

SIN & ASIN

COS(X)

ACOS(X)

COS & ACOS

TAN(X)

ATAN(X)

TAN & ATAN

SINH(X)

ASINH(X)

SINH & ASINH

COSH(X)

Minimum Maximum Minimum Maximum

-3.15

-1.2

-3.2

-3.15

-1.2

-3.2

-3.15

-10

-2

3.15

1.2

3.2

3.15

1.2

3.2

3.15

-2

AUTO

-1.6

1.6

AUTO

-1.6

1.6

-10

-1.8

-2

1.8

-2

-5

-2

-1

-5

AUTO

-5

AUTO

-1

ACOSH(X)

COS & ACOS

Page 12-16

TANH(X)

ATANH(X)

TAN & ATAN

-5

-1.2

-5

1.2

AUTO

-2.5

2.5

Generating a table of values for a function

The combinations „õ(E) and „ö(F), pressed simultaneously if

in RPN mode, let’s the user produce a table of values of functions. For example,

we will produce a table of the function Y(X) = X/(X+10), in the range -5 < X <

5 following these instructions:

ꢀ We will generate values of the function f(x), defined above, for values of x

from –5 to 5, in increments of 0.5. First, we need to ensure that the graph

type is set to ^FUNCTIONin the PLOT SETUP screen („ô, press them

simultaneously, if in RPN mode). The field in front of the Type option will be

highlighted. If this field is not already set to ^FUNCTION, press the soft key

@CHOOS and select the ^FUNCTIONoption, then press @@@OK@@@.

ꢀ Next, press ˜ to highlight the field in front of the option EQ, type the

function expression ‘X/(X+10)’ and press @@@OK@@@.

ꢀ To accept the changes made to the PLOT SETUP screen press L @@@OK@@@.

You will be returned to normal calculator display.

ꢀ The next step is to access the Table Set-up screen by using the keystroke

combination „õ (i.e., soft key E) – simultaneously if in RPN mode.

This will produce a screen where you can select the starting value (Start)

and the increment (Step). Enter the following: 5\@@@OK@@@ 0.5

@@@OK@@@ 0.5@@@OK@@@ (i.e., Zoom factor = 0.5). Toggle the @ꢀ@CHK soft

menu key until a check mark appears in front of the option Small Font if you

so desire. Then press @@@OK@@@. This will return you to normal calculator

display.

The TPAR variable

After finishing the table set up, your calculator will create a variable called

TPAR (Table PARameters) that store information relevant to the table that is to be

generated. To see the contents of this variable, press ‚@TPAR.

ꢀ To see the table, press „ö(i.e., soft menu key F) – simultaneously if

in RPN mode. This will produce a table of values of x = -5, -4.5, …, and

Page 12-17

the corresponding values of f(x), listed as Y1 by default. You can use the

up and down arrow keys to move about in the table. You will notice that

we did not have to indicate an ending value for the independent variable

x. Thus, the table continues beyond the maximum value for x suggested

early, namely x = 5.

Some options available while the table is visible are @ZOOM, @@BIG@, and @DEFN:

ꢀ The @DEFN, when selected, shows the definition of the independent variable.

ꢀ The @@BIG@ key simply changes the font in the table from small to big, and

vice versa. Try it.

ꢀ The @ZOOM key, when pressed, produces a menu with the options: In, Out,

Decimal, Integer, and Trig. Try the following exercises:

ꢀ With the option In highlighted, press @@@OK@@@. The table is expanded so

that the x-increment is now 0.25 rather than 0.5. Simply, what the

calculator does is to multiply the original increment, 0.5, by the zoom

factor, 0.5, to produce the new increment of 0.25. Thus, the zoom in

option is useful when you want more resolution for the values of x in

your table.

ꢀ To increase the resolution by an additional factor of 0.5 press @ZOOM,

select In once more, and press @@@OK@@@. The x-increment is now 0.0125.

ꢀ To recover the previous x-increment, press @ZOOM —@@@OK@@@ to select the

option Un-zoom. The x-increment is increased to 0.25.

ꢀ To recover the original x-increment of 0.5 you can do an un-zoom

again, or use the option zoom out by pressing @ZOOM @@@OK@@@.

ꢀ The option Decimal in @ZOOM produces x-increments of 0.10.

ꢀ The option Integer in @ZOOM produces x-increments of 1.

ꢀ The option Trig in produces increments related to fractions of π, thus

being useful when plotting trigonometric functions.

ꢀ To return to normal calculator display press `.

Plots in polar coordinates

First of all, you may want to delete the variables used in previous examples

(e.g., X, EQ, Y1, PPAR) using function PURGE (I @PURGE). By doing this, all

parameters related to graphics will be cleared. Press J to check that the

variables were indeed purged.

Page 12-18

We will try to plot the function f(θ) = 2(1-sin(θ)), as follows:

ꢀ First, make sure that your calculator’s angle measure is set to radians.

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPEto Polar, by pressing @CHOOS ˜ @@@OK@@@.

ꢀ Press ˜ and type:

³2* „ Ü1-S~‚t @@@OK@@@.

ꢀ The cursor is now in the Indepfield. Press ³~‚t @@@OK@@@ to

change the independent variable to θ.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

window (in this case it will be called PLOT –POLAR window).

ꢀ Change the H-VIEW range to –8 to 8, by using

8\@@@OK@@@8@@@OK@@@, and the V-VIEW range to -6 to 2 by using

6\@@@OK@@@2@@@OK@@@.

Note: the H-VIEW and V-VIEW determine the scales of the display

window only, and their ranges are not related to the range of values of

the independent variable in this case.

ꢀ Change the Indep Low value to 0, and the High value to 6.28 (≈ 2π), by

using: 0@@@OK@@@ 6.28@@@OK@@@.

ꢀ Press @ERASE @DRAW to plot the function in polar coordinates. The result is a

curve shaped like a hearth. This curve is known as a cardiod (cardios,

Greek for heart).

ꢀ Press @EDIT L @LABEL @MENU to see the graph with labels. Press L to

recover the menu. Press L@PICT to recover the original graphics menu.

ꢀ Press @TRACE @x,y@ to trace the curve. The data shown at the bottom of the

display is the angle θ and the radius r, although the latter is labeled Y

(default name of dependent variable).

Page 12-19

ꢀ Press L@CANCL to return to the PLOT WINDOW screen. Press L@@@OK@@@ to

return to normal calculator display.

In this exercise we entered the equation to be plotted directly in the PLOT SETUP

window. We can also enter equations for plotting using the PLOT window, i.e.,

simultaneously if in RPN mode, pressing „ñ. For example, when you

press „ñ after finishing the previous exercise, you will get the equation

‘2*(1-SIN(θ))’ highlighted. Let’s say, we want to plot also the function ‘2*(1-

COS(θ))’ along with the previous equation.

ꢀ Press @@ADD@! , and type 2*„Ü1- T~‚t`, to

enter the new equation.

ꢀ Press @ERASE @DRAW to see the two equations plotted in the same figure. The

result is two intersecting cardioids. Press @CANCL $ to return to normal

calculator display.

Plotting conic curves

The most general form of a conic curve in the x-y plane is:

Ax +By +Cxy+Dx+Ey+F = 0. We also recognize as conic equations those

given in the canonical form for the following figures:

ꢀ

circle:

(x-x ) +(y-y ) = r

ellipse:

(x-x ) /a + (y-y ) /b = 1

ꢀ

parabola:

hyperbola:

(y-b) = K(x-a) or (x-a) = K(y-b)

(x-x ) /a + (y-y ) /b = 1 or xy = K,

where x , y , a, b, and K are constant.

The name conic curves follows because these figures (circles, ellipses, parabolas

or hyperbolas) result from the intersection of a plane with a cone. For example,

a circle is the intersection of a cone with a plane perpendicular to the cone's

main axis.

Page 12-20

The calculator has the ability of plotting one or more conic curves by selecting

Conic as the function TYPE in the PLOT environment. Make sure to delete the

variables PPAR and EQ before continuing. For example, let's store the list of

equations

{ ‘(X-1)^2+(Y-2)^2=3’ , ‘X^2/4+Y^2/3=1’ }

into the variable EQ.

These equations we recognize as those of a circle centered at (1,2) with radius

√3, and of an ellipse centered at (0,0) with semi-axis lengths a = 2 and b = √3.

ꢀ Enter the PLOT environment, by pressing „ô, simultaneously if in RPN

mode, and select Conicas the TYPE. The list of equations will be listed in

the EQ field.

ꢀ Make sure that the independent variable (Indep) is set to ‘X’ and the

dependent variable (Depnd) to ‘Y’.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Enter the PLOT WINDOW environment, by pressing „ò,

simultaneously if in RPN mode.

ꢀ Change the range for H-VIEW to -3 to 3, by using 3\@@@OK@@@3@@@OK@@@.

Also, change the V-VIEW range to -1.5 to

1.5\@@@OK@@@ 2@@@OK@@@.

by using

ꢀ Change the Indep Low: and High: fields to Default by using L @RESET

while each of those fields is highlighted. Select the option Reset value

after pressing @RESET. Press @@@OK@@@ to complete the resetting of values. Press

L to return to the main menu.

ꢀ Plot the graph: @ERASE @DRAW.

Page 12-21

Note: The H-Viewand V-Viewranges were selected to show the intersection

of the two curves. There is no general rule to select those ranges, except based

on what we know about the curves. For example, for the equations shown

above, we know that the circle will extend from -3+1 = -2 to 3+1 = 4 in x, and

from -3+2=-1 to 3+2=5 in y. In addition, the ellipse, which is centered at the

origin (0,0), will extend from -2 to 2 in x, and from -√3 to √3 in y.

Notice that for the circle and the ellipse the region corresponding to the left and

right extremes of the curves are not plotted. This is the case with all circles or

ellipses plotted using Conicas the TYPE.

ꢀ To see labels: @EDIT L@LABEL @MENU

ꢀ To recover the menu: LL@PICT

ꢀ To estimate the coordinates of the point of intersection, press the @(X,Y)@

menu key and move the cursor as close as possible to those points using the

arrow keys. The coordinates of the cursor are shown in the display. For

example, the left point of intersection is close to (-0.692, 1.67), while the

right intersection is near (1.89,0.5).

ꢀ To recover the menu and return to the PLOT environment, press L@CANCL.

ꢀ To return to normal calculator display, press L@@@OK@@@.

Parametric plots

Parametric plots in the plane are those plots whose coordinates are generated

through the system of equations x = x(t) and y = y(t), where t is known as the

parameter. An example of such graph is the trajectory of a projectile, x(t) = x

+ v ⋅COS θ ⋅t, y(t) = y + v ⋅sin θ ⋅t – ½⋅g⋅t . To plot equations like these,

Page 12-22

which involve constant values x , y , v , and θ , we need to store the values of

those parameters in variables. To develop this example, create a sub-directory

called ‘PROJM’ for PROJectile Motion, and within that sub-directory store the

following variables: X0 = 0, Y0 = 10, V0 = 10 , θ0 = 30, and g = 9.806.

Make sure that the calculator’s angle measure is set to DEG. Next, define the

functions (use „à):

X(t) = X0 + V0*COS(θ0)*t

Y(t) = Y0 + V0*SIN(θ0)*t – 0.5*g*t^2

which will add the variables @@@Y@@@ and @@@X@@@ to the soft menu key labels.

To produce the graph itself, follow these steps:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPEto Parametric, by pressing @CHOOS ˜˜@@@OK@@@.

ꢀ Press ˜ and type ‘X(t) + i*Y(t)’ @@@OK@@@ to define the parametric plot as that

of a complex variable. (The real and imaginary parts of the complex

variable correspond to the x- and y-coordinates of the curve.)

ꢀ The cursor is now in the Indep field. Press ³~„t @@@OK@@@ to

change the independent variable to t.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT window

(in this case it will be called PLOT –PARAMETRIC window). Instead of

modifying the horizontal and vertical views first, as done for other types of

plot, we will set the lower and upper values of the independent variable

first as follows:

ꢀ Select the Indep Low field by pressing ˜˜. Change this value to

0@@@OK@@@. Then, change the value of High to 2@@@OK@@@. Enter 0.

1@@@OK@@@ for the Stepvalue (i.e., step = 0.1).

Page 12-23

Note: Through these settings we are indicating that the parameter t will

take values of t = 0, 0.1, 0.2, …, etc., until reaching the value of 2.0.

ꢀ Press @AUTO. This will generate automatic values of the H-View and V-View

ranges based on the values of the independent variable t and the

definitions of X(t) and Y(t) used. The result will be:

ꢀ Press @ERASE @DRAW to draw the parametric plot.

ꢀ Press @EDIT L @LABEL @MENU to see the graph with labels. The window

parameters are such that you only see half of the labels in the x-axis.

ꢀ Press L to recover the menu. Press L@PICT to recover the original

graphics menu.

ꢀ Press TRACE @(X,Y)@ to determine coordinates of any point on the graph.

Use ™ and š to move the cursor about the curve. At the bottom of the

screen you will see the value of the parameter t and coordinates of the

cursor as (X,Y).

ꢀ Press L@CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

A review of your soft menu key labels shows that you now have the following

variables: t, EQ, PPAR, Y, X, g, θ0, V0, Y0, X0. Variables t, EQ, and PPAR are

generated by the calculator to store the current values of the parameter, t, of the

equation to be plotted EQ (which contains ‘X(t) + I∗Y(t)’), and the plot

Page 12-24

parameters. The other variables contain the values of constants used in the

definitions of X(t) and Y(t).

You can store different values in the variables and produce new parametric

plots of the projectile equations used in this example. If you want to erase the

current picture contents before producing a new plot, you need to access either

the PLOT, PLOT WINDOW, or PLOT SETUP screens, by pressing, „ñ ,

„ò, or „ô (the two keys must be pressed simultaneously if in RPN

mode). Then, press @ERASE @DRAW. Press @CANCL to return to the PLOT, PLOT

WINDOW, or PLOT SETUP screen. Press $, or L@@@OK@@@, to return to

normal calculator display.

Generating a table for parametric equations

In an earlier example we generated a table of values (X,Y) for an expression of

the form Y=f(X), i.e., a Function type of graph. In this section, we present the

procedure for generating a table corresponding to a parametric plot. For this

purpose, we’ll take advantage of the parametric equations defined in the

example above.

ꢀ First, let’s access the TABLE SETUP window by pressing „õ,

simultaneously if in RPN mode. For the independent variable change

the Starting value to 0.0, and the Step value to 0.1. Press @@@OK@@@.

ꢀ Generate the table by pressing, simultaneously if in RPN mode,

„ö. The resulting table has three columns representing the

parameter t, and the coordinates of the corresponding points. For this

table the coordinates are labeled X1 and Y1.

ꢀ Use the arrow keys, š™—˜, to move about the table.

ꢀ Press $ to return to normal calculator display.

This procedure for creating a table corresponding to the current type of plot can

be applied to other plot types.

Page 12-25

Plotting the solution to simple differential equations

The plot of a simple differential equation can be obtained by selecting Diff

Eqin the TYPEfield of the PLOT SETUP environment as follows: suppose that we

want to plot x(t) from the differential equation dx/dt = exp(-t ), with initial

conditions: x = 0 at t = 0. The calculator allows for the plotting of the solution

of differential equations of the form Y'(T) = F(T,Y). For our case, we let Yꢁx

and Tꢁt, therefore, F(T,Y)ꢁf(t,x) = exp(-t ).

Before plotting the solution, x(t), for t = 0 to 5, delete the variables EQ and

PPAR.

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPEto Diff Eq.

ꢀ Press ˜ and type ³„ ¸-~ „tQ2@@@OK@@@.

ꢀ The cursor is now in the H-Var field. It should show H-Var:0 and also V-

Var:1. This is the code used by the calculator to identify the variables to be

plotted. H-Var:0means the independent variable (to be selected later) will

be plotted in the horizontal axis. Also, V-Var:1 means the dependent

variable (default name ‘Y’) will be plotted in the vertical axis.

ꢀ Press ˜ . The cursor is now in the Indepfield. Press ³~

„t@@@OK@@@ to change the independent variable to t.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT window

(in this case it will be called PLOT WINDOW – DIFF EQ).

ꢀ Change the H-VIEW and V-VIEW parameters to read: H-VIEW: -15,V-VIEW:

-11.5

ꢀ Change the Initvalue to 0, and the Final value to 5 by using: 0@@@OK@@@

5@@@OK@@@.

ꢀ The values Step and Tol represent the step in the independent variable and

the tolerance for convergence to be used by the numerical solution. Let’s

leave those values with their default settings (if the word default is not

shown in the Step: field, use L @RESET to reset that value to its default

value. Press L to return to the main menu.) Press ˜ .

ꢀ The Init-Soln value represents the initial value of the solution to start the

numerical result. For the present case, we have for initial conditions x(0) =

0, thus, we need to change this value to 0.0, by using 0@@@OK@@@.

ꢀ Press @ERASE @DRAW to plot the solution to the differential equation.

ꢀ Press @EDIT L@LABEL @MENU to see the graph with labels.

Page 12-26

ꢀ Press L to recover the menu. Press L@PICT to recover the original

graphics menu.

ꢀ When we observed the graph being plotted, you'll notice that the graph is

not very smooth. That is because the plotter is using a time step that is too

large. To refine the graph and make it smoother, use a step of 0.1. Try the

following keystrokes: @CANCL ˜˜˜.1@@@OK@@@ @ERASE @DRAW. The plot

will take longer to be completed, but the shape is definitely smoother than

before.

ꢀ Press @EDIT L @LABEL @MENU, to see axes labels and range. Notice that the

labels for the axes are shown as 0 (horizontal) and 1 (vertical). These are

the definitions for the axes as given in the PLOT WINDOW screen (see

above), i.e., H-VAR (t): 0, and V-VAR(x): 1.

ꢀ Press LL@PICT to recover menu and return to PICT environment.

ꢀ Press (X,Y) to determine coordinates of any point on the graph. Use ™

and š to move the cursor in the plot area. At the bottom of the screen

you will see the coordinates of the cursor as (X,Y). The calculator uses X

and Y as the default names for the horizontal and vertical axes,

respectively.

ꢀ Press L@CANCL to return to the PLOT WINDOW environment. Then, press

$ to return to normal calculator display.

More details on using graphical solutions of differential equations are presented

in Chapter 16.

Page 12-27

Truth plots

Truth plots are used to produce two-dimensional plots of regions that satisfy a

certain mathematical condition that can be either true or false. For example,

suppose that you want to plot the region for X^2/36 + Y^2/9 < 1,

proceed as follows:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPE to Truth.

ꢀ Press ˜ and type {‘(X^2/36+Y^2/9 < 1)','(X^2/16+Y^2/9 > 1)’}

@@@OK@@@ to define the conditions to be plotted.

ꢀ The cursor is now in the Indepfield. Leave that as ‘X’ if already set to that

variable, or change it to ‘X’ if needed.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT window

(in this case it will be called PLOT WINDOW – TRUTH window). Let’s keep

the default value for the window’s ranges: H-View: -6.5 6.5, V-View:

-3.9 4.0(To reset them use L@RESET (select Reset all) @@OK@@ L).

Note: if the window’s ranges are not set to default values, the quickest way to

reset them is by using L@RESET@ (select Reset all) @@@OK@@@ L.

ꢀ Press @ERASE @DRAW to draw the truth plot. Because the calculator samples the

entire plotting domain, point by point, it takes a few minutes to produce a

truth plot. The present plot should produce a shaded ellipse of semi-axes 6

and 3 (in x and y, respectively), centered at the origin.

ꢀ Press @EDIT L @LABEL @MENU to see the graph with labels. The window

parameters are such that you only see half of the labels in the x-axis. Press

L to recover the menu. Press L@PICT to recover the original graphics

menu.

ꢀ Press (X,Y) to determine coordinates of any point on the graph. Use the

arrow keys to move the cursor about the region plotted. At the bottom of

the screen you will see the value of the coordinates of the cursor as (X,Y).

ꢀ Press L@CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

You can have more than one condition plotted at the same time if you multiply

the conditions. For example, to plot the graph of the points for which X /36 +

Y /9 < 1, and X /16 + Y /9 > 1, use the following:

Page 12-28

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Press ˜ and type ‘(X^2/36+Y^2/9 < 1)⋅ (X^2/16+Y^2/9 > 1)’@@@OK@@@ to

define the conditions to be plotted.

ꢀ Press @ERASE @DRAW to draw the truth plot. Again, you have to be patient

while the calculator produces the graph. If you want to interrupt the plot,

press $ , once. Then press @CANCEL .

Plotting histograms, bar plots, and scatter plots

Histograms, bar plots and scatter plots are used to plot discrete data stored in

the reserved variable ΣDAT. This variable is used not only for these types of

plots, but also for all kind of statistical applications as will be shown in Chapter

18. As a matter of fact, the use of histogram plots is postponed until we get to

that chapter, for the plotting of a histogram requires to perform a grouping of

data and a frequency analysis before the actual plot. In this section we will

show how to load data in the variable ΣDAT and how to plot bar plots and

scatter plots.

We will use the following data for plotting bar plots and scatter plots:

3.1 2.1 1.1

3.6 3.2 2.2

4.2 4.5 3.3

4.5 5.6 4.4

4.9 3.8 5.5

5.2 2.2 6.6

Bar plots

First, make sure your calculator’s CAS is in Exactmode. Next, enter the data

shown above as a matrix, i.e.,

[[3.1,2.1,1.1],[3.6,3.2,2.2],[4.2,4.5,3.3],

Page 12-29

[4.5,5.6,4.4],[4.9,3.8,5.5],[5.2,2.2,6.6]] `

to store it in ΣDAT, use the function STOΣ (available in the function catalog,

‚N). Press VAR to recover your variables menu. A soft menu key

labeled ΣDAT should be available in the stack. The figure below shows the

storage of this matrix in ALG mode:

To produce the graph:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPEto Bar.

ꢀ A matrix will be shown at the ΣDAT field. This is the matrix we stored

earlier into ΣDAT.

ꢀ Highlight the Col:field. This field lets you choose the column of ΣDAT that

is to be plotted. The default value is 1. Keep it to plot column 1 in ΣDAT.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Change the V-View to read, V-View: 0 5.

ꢀ Press @ERASE @DRAW to draw the bar plot.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

The number of bars to be plotted determines the width of the bar. The H- and V-

VIEW are set to 10, by default. We changed the V-VIEW to better

Page 12-30

accommodate the maximum value in column 1 of ΣDAT. Bar plots are useful

when plotting categorical (i.e., non-numerical) data.

Suppose that you want to plot the data in column 2 of the ΣDAT matrix:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Press ˜˜ to highlight the Col: field and type 2 @@@OK@@@, followed by

L@@@OK@@@.

ꢀ Press „ò, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change V-View to read V-View: 0 6

ꢀ Press @ERASE @DRAW.

ꢀ Press @CANCL to return to the PLOT WINDOW screen, then $ to return to

normal calculator display.

Scatter plots

We will use the same ΣDAT matrix to produce scatter plots. First, we will plot

the values of y vs. x, then those of y vs. z, as follows:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPEto Scatter.

ꢀ Press ˜˜ to highlight the Cols: field. Enter 1@@@OK@@@ 2@@@OK@@@ to

select column 1 as X and column 2 as Y in the Y-vs.-X scatter plot.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Change the plot window ranges to read: H-View: 0 6, V-View: 0 6.

Page 12-31

ꢀ Press @ERASE @DRAW to draw the bar plot. Press @EDIT L @LABEL @MENU to see

the plot unencumbered by the menu and with identifying labels (the cursor

will be in the middle of the plot, however):

ꢀ Press LL@PICT to leave the EDIT environment.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

To plot y vs. z, use:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Press ˜˜ to highlight the Cols: field. Enter 3@@@OK@@@ 2@@@OK@@@ to

select column 3 as X and column 2 as Y in the Y-vs.-X scatter plot.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Change the plot window ranges to read: H-View: 0 7, V-View: 0 7.

ꢀ Press @ERASE @DRAW to draw the bar plot. Press @EDIT L@LABEL @MENU to see

the plot unencumbered by the menu and with identifying labels.

ꢀ Press LL@PICT to leave the EDIT environment.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

Page 12-32

Slope fields

Slope fields are used to visualize the solutions to a differential equation of the

form y’ = f(x,y). Basically, what is presented in the plot are segments tangential

to the solution curves, since y’ = dy/dx, evaluated at any point (x,y), represents

the slope of the tangent line at point (x,y).

For example, to visualize the solution to the differential equation y’ = f(x,y) =

x+y, use the following:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPEto Slopefield.

ꢀ Press ˜ and type ‘X+Y’ @@@OK@@@.

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd:

variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Change the plot window ranges to read: X-Left:-5, X-Right:5, Y-Near:-5, Y-Far:

ꢀ Press @ERASE @DRAW to draw the slope field plot. Press @EDIT L @LABEL

@MENU to see the plot unencumbered by the menu and with identifying labels.

ꢀ Press LL@PICT to leave the EDIT environment.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

If you could reproduce the slope field plot in paper, you can trace by hand lines

that are tangent to the line segments shown in the plot. This lines constitute lines

Page 12-33

of y(x,y) = constant, for the solution of y’ = f(x,y). Thus, slope fields are useful

tools for visualizing particularly difficult equations to solve.

Try also a slope field plot for the function y’ = f(x,y) = - (y/x) , by using:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPEto Slopefield.

ꢀ Press ˜ and type ‘− (Y/X)^2’ @@@OK@@@.

ꢀ Press @ERASE @DRAW to draw the slope field plot. Press @EDIT L @LABEL

@MENU to see the plot unencumbered by the menu and with identifying labels.

ꢀ Press LL@PICT to leave the EDIT environment.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

Fast 3D plots

Fast 3D plots are used to visualize three-dimensional surfaces represented by

equations of the form z = f(x,y). For example, if you want to visualize z = f(x,y)

= x +y , we can use the following:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢀ Change TYPE to Fast3D.

ꢀ Press ˜ and type ‘X^2+Y^2’ @@@OK@@@.

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Keep the default plot window ranges to read: X-Left:-1, X-Right:1, Y-Near:-1, Y-

Far: 1, Z-Low: -1, Z-High: 1, Step Indep: 10, Depnd: 8

Page 12-34

Note: The Step Indep: and Depnd: values represent the number of

gridlines to be used in the plot. The larger these number, the slower it is

to produce the graph, although, the times utilized for graphic generation

are relatively fast. For the time being we’ll keep the default values of 10

and 8 for the Step data.

ꢀ Press @ERASE @DRAW to draw the three-dimensional surface. The result is a

wireframe picture of the surface with the reference coordinate system shown

at the lower left corner of the screen. By using the arrow keys (š™—

˜) you can change the orientation of the surface. The orientation of the

reference coordinate system will change accordingly. Try changing the

surface orientation on your own. The following figures show a couple of

views of the graph:

ꢀ When done, press @EXIT.

ꢀ Press @CANCL to return to the PLOT WINDOW environment.

ꢀ Change the Step data to read: Step Indep: 20 Depnd: 16

ꢀ Press @ERASE @DRAW to see the surface plot. Sample views:

ꢀ When done, press @EXIT.

ꢀ Press @CANCL to return to PLOT WINDOW.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

Try also a Fast 3D plot for the surface z = f(x,y) = sin (x +y )

Page 12-35

ꢀ Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP

window.

ꢀ Press ˜ and type ‘SIN(X^2+Y^2)’ @@@OK@@@.

ꢀ Press @ERASE @DRAW to draw the plot.

ꢀ When done, press @EXIT.

ꢀ Press @CANCL to return to PLOT WINDOW.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

Wireframe plots

Wireframe plots are plots of three-dimensional surfaces described by z = f(x,y).

Unlike Fast 3D plots, wireframe plots are static plots. The user can choose the

viewpoint for the plot, i.e., the point from which the surface is seen. For

example, to produce a wireframe plot for the surface z = x + 2y –3, use the

following:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPEto Wireframe.

ꢀ Press ˜ and type ‘X+2*Y-3’ @@@OK@@@.

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd:

variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Keep the default plot window ranges to read: X-Left:-1, X-Right:1, Y-Near:-1, Y-

Far: 1, Z-Low: -1, Z-High: 1, XE:0,YE:-3, ZE:0, Step Indep: 10 Depnd: 8

The coordinates XE, YE, ZE, stand for “eye coordinates,” i.e., the coordinates

from which an observer sees the plot. The values shown are the default values.

The Step Indep: and Depnd: values represent the number of gridlines to be used

in the plot. The larger these number, the slower it is to produce the graph. For

the time being we’ll keep the default values of 10 and 8 for the Step

ꢀ Press @ERASE @DRAW to draw the three-dimensional surface. The result is a

wireframe picture of the surface.

Page 12-36

ꢀ Press @EDIT L @LABEL @MENU to see the graph with labels and ranges. This

particular version of the graph is limited to the lower part of the display.

We can change the viewpoint to see a different version of the graph.

ꢀ Press LL@PICT @CANCL to return to the PLOT WINDOW environment.

ꢀ Change the eye coordinate data to read : XE:0

YE:-3

ZE:3

ꢀ Press @ERASE @DRAW to see the surface plot.

ꢀ Press @EDIT L @LABEL @MENU to see the graph with labels and ranges.

This version of the graph occupies more area in the display than the

previous one. We can change the viewpoint, once more, to see another

version of the graph.

ꢀ Press LL@PICT @CANCL to return to the PLOT WINDOW environment.

ꢀ Change the eye coordinate data to read : XE:3

YE:3

ZE:3

ꢀ Press @ERASE @DRAW to see the surface plot. This time the bulk of the plot is

located towards the right –hand side of the display.

ꢀ Press @CANCL to return to the PLOT WINDOW environment.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

Page 12-37

Try also a Wireframe plot for the surface z = f(x,y) = x +y

ꢀ Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP

window.

ꢀ Press ˜ and type ‘X^2+Y^2’ @@@OK@@@.

ꢀ Press @ERASE @DRAW to draw the slope field plot. Press @EDIT L@MENU @LABEL

to see the plot unencumbered by the menu and with identifying labels.

ꢀ Press LL@PICT to leave the EDIT environment.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

Ps-Contour plots

Ps-Contour plots are contour plots of three-dimensional surfaces described by z

= f(x,y). The contours produced are projections of level surfaces z = constant

on the x-y plane. For example, to produce a Ps-Contour plot for the surface z =

x +y , use the following:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPE to Ps-Contour.

ꢀ Press ˜ and type ‘X^2+Y^2’ @@@OK@@@.

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd:

variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Change the default plot window ranges to read: X-Left:-2, X-Right:2, Y-Near:-1

Y-Far: 1, Step Indep: 10, Depnd: 8

ꢀ Press @ERASE @DRAW to draw the contour plot. This operation will take some

time, so, be patient. The result is a contour plot of the surface. Notice that

the contour are not necessarily continuous, however, they do provide a

good picture of the level surfaces of the function.

Page 12-38

ꢀ Press @EDITꢁL@LABEL @MENU to see the graph with labels and ranges.

ꢀ Press LL@PICT@CANCL to return to the PLOT WINDOW environment.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

Try also a Ps-Contour plot for the surface z = f(x,y) = sin x cos y.

ꢀ Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP

window.

ꢀ Press ˜ and type ‘SIN(X)*COS(Y)’ @@@OK@@@.

ꢀ Press @ERASE @DRAW to draw the slope field plot. Press @EDIT L@LABEL @MENU to

see the plot unencumbered by the menu and with identifying labels.

ꢀ Press LL@PICT to leave the EDIT environment.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

Y-Slice plots

Y-Slice plots are animated plots of z-vs.-y for different values of x from the

function z = f(x,y). For example, to produce a Y-Slice plot for the surface z =

x -xy , use the following:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPE to Y-Slice.

ꢀ Press ˜ and type ‘X^3+X*Y^3’ @@@OK@@@.

Page 12-39

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Change the default plot window ranges to read: X-Left:-1, X-Right:1, Y-Near:-

1, Y-Far: 1, Z-Low:-1, Z-High:1, Step Indep: 10 Depnd: 8

ꢀ Press @ERASE @DRAW to draw the three-dimensional surface. You will see the

calculator produce a series of curves on the screen, that will immediately

disappear. When the calculator finishes producing all the y-slice curves,

then it will automatically go into animating the different curves. One of the

curves is shown below.

ꢀ Press $ to stop the animation. Press @CANCL to return to the PLOT

WINDOW environment.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

Try also a Ps-Contour plot for the surface z = f(x,y) = (x+y) sin y.

ꢀ Press „ô, simultaneously if in RPN mode, to access the PLOT SETUP

window.

ꢀ Press ˜ and type ‘(X+Y)*SIN(Y)’ @@@OK@@@.

ꢀ Press @ERASE @DRAW to produce the Y-Slice animation.

ꢀ Press $ to stop the animation.

ꢀ Press @CANCL to return to the PLOT WINDOW environment. Then, press

$ , or L@@@OK@@@, to return to normal calculator display.

Gridmap plots

Gridmap plots produce a grid of orthogonal curves describing a function of a

complex variable of the form w =f(z) = f(x+iy), where z = x+iy is a complex

variable. The functions plotted correspond to the real and imaginary part of w

= Φ(x,y) + iΨ(x,y), i.e., they represent curves Φ(x,y) =constant, and Ψ(x,y) =

constant. For example, to produce a Gridmap plot for the function w = sin(z),

use the following:

Page 12-40

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPE to Gridmap.

ꢀ Press ˜ and type ‘SIN(X+i*Y)’ @@@OK@@@.

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd: variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Keep the default plot window ranges to read: X-Left:-1, X-Right:1, Y-Near:-1 Y-

Far: 1, XXLeft:-1 XXRight:1, YYNear:-1, yyFar: 1, Step Indep: 10 Depnd: 8

ꢀ Press @ERASE @DRAW to draw the gridmap plot. The result is a grid of functions

corresponding to the real and imaginary parts of the complex function.

ꢀ Press @EDIT L@LABEL @MENU to see the graph with labels and ranges.

ꢀ Press LL@PICT @CANCL to return to the PLOT WINDOW environment.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

Other functions of a complex variable worth trying for Gridmap plots are:

(1) SIN((X,Y))

(3) EXP((X,Y))

(5) TAN((X,Y))

(7) (X,Y)^3

i.e., F(z) = sin(z) (2)(X,Y)^2

i.e., F(z) = z

i.e., F(z) = e

(4) SINH((X,Y))

i.e., F(z) = sinh(z)

-1

i.e., F(z) = tan(z) (6) ATAN((X,Y))

i.e., F(z) = tan (z)

i.e., F(z) = z

(8) 1/(X,Y)

i.e., F(z) = 1/z

1/2

(9) √ (X,Y)

Pr-Surface plots

Pr-Surface (parametric surface) plots are used to plot a three-dimensional

surface whose coordinates (x,y,z) are described by x = x(X,Y), y = y(X,Y),

z=z(X,Y), where X and Y are independent parameters.

Page 12-41

Note: The equations x = x(X,Y), y = y(X,Y), z=z(X,Y) represent a parametric

description of a surface. X and Y are the independent parameters. Most

textbooks will use (u,v) as the parameters, rather than (X,Y). Thus, the

parametric description of a surface is given as x = x(u,v), y = y(u,v), z=z(u,v).

For example, to produce a Pr-Surface plot for the surface x = x(X,Y) = X sin Y, y

= y(X,Y) = x cos Y, z=z(X,Y)=X, use the following:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPE to Pr-Surface.

ꢀ Press ˜ and type ‘{X*SIN(Y), X*COS(Y), X}’ @@@OK@@@.

ꢀ Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd:variables.

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

WINDOW screen.

ꢀ Keep the default plot window ranges to read: X-Left:-1, X-Right:1, Y-Near:-1, Y-

Far: 1, Z-Low: -1, Z-High:1, XE: 0, YE:-3, zE:0, Step Indep: 10, Depnd: 8

ꢀ Press @ERASE @DRAW to draw the three-dimensional surface.

ꢀ Press @EDITꢁL @LABEL @MENU to see the graph with labels and ranges.

ꢀ Press LL@PICT @CANCL to return to the PLOT WINDOW environment.

ꢀ Press $ , or L@@@OK@@@, to return to normal calculator display.

The VPAR variable

The VPAR (Volume Parameter) variable contains information regarding the

“volume” used to produce a three dimensional graph. Therefore, you will see it

produced whenever you create a three dimensional plot such as Fast3D,

Wireframe, or Pr-Surface.

Page 12-42

Interactive drawing

Whenever we produce a two-dimensional graph, we find in the graphics screen

a soft menu key labeled @EDIT. Pressing @EDIT produces a menu that include the

following options (press L to see additional functions):

Through the examples above, you have the opportunity to try out functions

LABEL, MENU, PICTꢁ, and REPL. Many of the remaining functions, such as

DOT+, DOT-, LINE, BOX, CIRCL, MARK, DEL, etc., can be used to draw points,

lines, circles, etc. on the graphics screen, as described below. To see how to

use these functions we will try the following exercise:

First, we get the graphics screen corresponding to the following instructions:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPEto Function, if needed

ꢀ Change EQ to ‘X’

ꢀ Make sure that Indep: is set to ‘X’ also

ꢀ Press L@@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously if in RPN mode, to access the PLOT

window (in this case it will be called PLOT –POLAR window).

ꢀ Change the H-VIEW range to –10 to 10, by using 10\@@@OK@@@

10@@@OK@@@, and the V-VIEW range to -5 to 5 by using

5\@@@OK@@@ 5@@@OK@@@.

ꢀ Press @ERASE @DRAW to plot the function.

ꢀ Press @EDIT L @LABEL to add labels to the graph. Press LL (or

„«) to recover the original EDIT menu.

Page 12-43

Next, we illustrate the use of the different drawing functions on the resulting

graphics screen. They require use of the cursor and the arrow keys (š™—

˜) to move the cursor about the graphics screen.

DOT+ and DOT-

When DOT+ is selected, pixels will be activated wherever the cursor moves

leaving behind a trace of the cursor position. When DOT- is selected, the

opposite effect occurs, i.e., as you move the cursor, pixels will be deleted.

For example, use the ™— keys to move the cursor somewhere in the middle

of the first quadrant of the x-y plane, then press @DOT+@@. The label will be

selected (DOT+ꢀ@). Press and hold the ™ key to see a horizontal line being

traced. Now, press @DOT-@, to select this option ( @DOT-ꢂ@ ). Press and hold the

š key to see the line you just traced being erased. Press @DOT-, when done,

to deselect this option.

MARK

This command allows the user to set a mark point which can be used for a

number of purposes, such as:

ꢀ Start of line with the LINE or TLINE command

ꢀ Corner for a BOX command

ꢀ Center for a CIRCLE command

Using the MARK command by itself simply leaves an x in the location of the

mark (Press L@MARK to see it in action).

LINE

This command is used to draw a line between two points in the graph. To see

it in action, position the cursor somewhere in the first quadrant, and press

„«@LINE. A MARK is placed over the cursor indicating the origin of the

line. Use the ™ key to move the cursor to the right of the current position, say

about 1 cm to the right, and press @LINE. A line is draw between the first and

the last points.

Notice that the cursor at the end of this line is still active indicating that the

calculator is ready to plot a line starting at that point. Press ˜ to move the

cursor downwards, say about another cm, and press @LINE again. Now you

Page 12-44

should have a straight angle traced by a horizontal and a vertical segments.

The cursor is still active. To deactivate it, without moving it at all, press @LINE.

The cursor returns to its normal shape (a cross) and the LINE function is no

longer active.

TLINE

(Toggle LINE) Move the cursor to the second quadrant to see this function in

action. Press @TLINE. A MARK is placed at the start of the toggle line. Move

the cursor with the arrow keys away from this point, and press @TLINE. A line is

drawn from the current cursor position to the reference point selected earlier.

Pixels that are on in the line path will be turned off, and vice versa. To remove

the most recent line traced, press @TLINE again. To deactivate TLINE, move the

cursor to the original point where TLINE was activated, and press @LINE @LINE.

BOX

This command is used to draw a box in the graph. Move the cursor to a clear

area of the graph, and press @BOX@. This highlights the cursor. Move the cursor

with the arrow keys to a point away, and in a diagonal direction, from the

current cursor position. Press @BOX@ again. A rectangle is drawn whose

diagonal joins the initial and ending cursor positions. The initial position of the

box is still marked with and x. Moving the cursor to another position and

pressing @BOX@ will generate a new box containing the initial point. To deselect

BOX, move the cursor to the original point where BOX was activated, then press

@LINE @LINE.

CIRCL

This command produces a circle. Mark the center of the circle with a MARK

command, then move the cursor to a point that will be part of the periphery of

the circle, and press @CIRCL. To deactivate CIRCL, return the cursor to the

MARK position and press @LINE.

Try this command by moving the cursor to a clear part of the graph, press @MARK.

Move the cursor to another point, then press @CIRCL. A circle centered at the

MARK, and passing through the last point will be drawn.

LABEL

Pressing @LABEL places the labels in the x- and y-axes of the current plot. This

feature has been used extensively through this chapter.

Page 12-45

DEL

This command is used to remove parts of the graph between two MARK

positions. Move the cursor to a point in the graph, and press @MARK. Move the

cursor to a different point, press @MARK again. Then, press @@DEL@. The section of

the graph boxed between the two marks will be deleted.

ERASE

The function ERASE clears the entire graphics window. This command is

available in the PLOT menu, as well as in the plotting windows accessible

through the soft menu keys.

Pressing @MENU will remove the soft key menu labels to show the graphic

unencumbered by those labels. To recover the labels, press L.

SUB

Use this command to extract a subset of a graphics object. The extracted object

is automatically placed in the stack. Select the subset you want to extract by

placing a MARK at a point in the graph, moving the cursor to the diagonal

corner of the rectangle enclosing the graphics subset, and press @@SUB@. This

feature can be used to move parts of a graphics object around the graph.

REPL

This command places the contents of a graphic object currently in stack level 1

at the cursor location in the graphics window. The upper left corner of the

graphic object being inserted in the graph will be placed at the cursor position.

Thus, if you want a graph from the stack to completely fill the graphic window,

make sure that the cursor is placed at the upper left corner of the display.

PICTꢀ

This command places a copy of the graph currently in the graphics window on

to the stack as a graphic object. The graphic object placed in the stack can be

saved into a variable name for storage or other type of manipulation.

Page 12-46

X,Yꢀ

This command copies the coordinates of the current cursor position, in user

coordinates, in the stack.

Zooming in and out in the graphics display

Whenever you produce a two-dimensional FUNCTION graphic interactively,

the first soft-menu key, labeled @ZOOM, lets you access functions that can be used

to zoom in and out in the current graphics display. The ZOOM menu includes

the following functions (press Lto move to the next menu):

We present each of these functions following. You just need to produce a

graph as indicated in Chapter 12, or with one of the programs listed earlier in

this Chapter.

ZFACT, ZIN, ZOUT, and ZLAST

Pressing @ZFACT produces an input screen that allows you to change the current

X- and Y-Factors. The X- and Y-Factors relate the horizontal and vertical user-

defined unit ranges to their corresponding pixel ranges. Change the H-Factor

to read 8., and press @@@OK@@@, then change the V-Factor to read 2., and press

@@@OK@@. Check off the option ꢀRecenter on cursor, and press @@@OK@@.

Back in the graphics display, press @@ZIN@ . The graphic is re-drawn with the new

vertical and horizontal scale factors, centered at the position where the cursor

was located, while maintaining the original PICT size (i.e., the original number

of pixels in both directions). Using the arrow keys, scroll horizontally or

vertically as far as you can of the zoomed-in graph.

To zoom out, subjected to the H- and V-Factors set with ZFACT, press @ZOOM

@ZOUT. The resulting graph will provide more detail than the zoomed-in graph.

Page 12-47

You can always return to the very last zoom window by using @ZLAST.

BOXZ

Zooming in and out of a given graph can be performed by using the soft-menu

key BOXZ. With BOXZ you select the rectangular sector (the “box”) that you

want to zoom in into. Move the cursor to one of the corners of the box (using

the arrow keys), and press @ZOOM @BOXZ. Using the arrow keys once more, move

the cursor to the opposite corner of the desired zoom box. The cursor will trace

the zoom box in the screen. When desired zoom box is selected, press @ZOOM.

The calculator will zoom in the contents of the zoom box that you selected to fill

the entire screen.

If you now press @ZOUT, the calculator will zoom out of the current box using the

H- and V-Factors, which may not recover the graph view from which you started

the zoom box operation.

ZDFLT, ZAUTO

Pressing @ZDFLT re-draws the current plot using the default x- and y-ranges, i.e., -

6.5 to 6.5 in x, and –3.1 to 3.1 in y. The command @ZAUTO, on the other hand,

creates a zoom window using the current independent variable (x) range, but

adjusting the dependent variable (y) range to fit the curve (as when you use the

function @AUTO in the PLOT WINDOW input form („ò, simultaneously in

RPN mode).

HZIN, HZOUT, VZIN and VZOUT

These functions zoom in and out the graphics screen in the horizontal or vertical

direction according to the current H- and V-Factors.

CNTR

Zooms in with the center of the zoom window in the current cursor location. The

zooming factors used are the current H- and V-Factors.

ZDECI

Zooms the graph so as to round off the limits of the x-interval to a decimal

value.

ZINTG

Zooms the graph so that the pixel units become user-define units. For example,

the minimum PICT window has 131 pixels. When you use ZINTG, with the

Page 12-48

cursor at the center of the screen, the window gets zoomed so that the x-axis

extends from –64.5 to 65.5.

ZSQR

Zooms the graph so that the plotting scale is maintained at 1:1 by adjusting the

x scale, keeping the y scale fixed, if the window is wider than taller. This forces

a proportional zooming.

ZTRIG

Zooms the graph so that the x scale incorporates a range from about –3π to

+3π, the preferred range for trigonometric functions.

Note: None of these functions are programmable. They are only useful in an

interactive way. Do not confuse the command @ZFACT in the ZOOM menu with

the function ZFACTOR, which is used for gas dynamic and chemistry

applications (see Chapter 3).

The SYMBOLIC menu and graphs

The SYMBOLIC menu is activated by pressing the P key (fourth key from the

left in fourth row from the top of the keyboard). This menu provides a list of

menus related to the Computer Algebraic System or CAS, these are:

All but one of these menus are available directly in the keyboard by pressing

the appropriate keystroke combination as follows. The Chapter of the user

manual where the menus are described is also listed:

ALGEBRA..

ARITHMETIC..

CALCULUS..

‚× (the 4 key)

„Þ (the 1 key)

„Ö (the 4 key)

Ch. 5

Ch. 13

Page 12-49

SOLVER..

TRIGONOMETRIC..

EXP&LN..

„Î (the 7 key)

‚Ñ (the 8 key)

„Ð (the 8 key)

Ch. 6

Ch. 5

The SYMB/GRAPH menu

The GRAPH sub-menu within the SYMB menu includes the following functions:

DEFINE: same as the keystroke sequence „à (the 2 key)

GROBADD: pastes two GROBs first over the second (See Chapter 22)

PLOT(function): plots a function, similar to „ô

PLOTADD(function): adds this function to the list of functions to plot, similar to

„ô

Plot setup..: same as „ô

SIGNTAB(function): sign table of given function showing intervals of positive

and negative variation, zero points and infinite asymptotes

TABVAL: table of values for a function

TABVAR: variation table of a function

Examples of some of these functions are provided next.

PLOT(X^2-1) is similar to „ô with EQ: X^2 -1. Using @ERASE @DRAW

produces the plot:

PLOTADD(X^2-X) is similar to „ô but adding this function to EQ: X^2 -1.

Using @ERASE @DRAW produces the plot:

Page 12-50

TABVAL(X^2-1,{1, 3}) produces a list of {min max} values of the function in the

interval {1,3}, while SIGNTAB(X^2-1) shows the sign of the function in the

interval (-∞,+), with f(x) > 0 in (-∞,-1), f(x) <0, in (-1,1), and f(x) > 0 in (1,+ ∞).

TABVAR(LN(X)/X) produces the following table of variation:

A detailed interpretation of the table of variation is easier to follow in RPN

mode:

The output is in a graphical format, showing the original function, F(X), the

derivative F’(X) right after derivation and after simplification, and finally a table

of variation. The table consists of two rows, labeled in the right-hand side.

Thus, the top row represents values of X and the second row represents values

Page 12-51

of F. The question marks indicates uncertainty or non-definition. For example,

for X<0, LN(X) is not defined, thus the X lines shows a question mark in that

interval. Right at zero (0+0) F is infinite, for X = e, F = 1/e. F increases before

reaching this value, as indicated by the upward arrow, and decreases after this

value (X=e) becoming slightly larger than zero (+:0) as X goes to infinity. A

plot of the graph is shown below to illustrate these observations:

Function DRAW3DMATRIX

This function takes as argument a n×m matrix, Z, = [ z ], and minimum and

maximum values for the plot. You want to select the values of v

and v

min

max

that they contain the values listed in Z. The general call to the function is,

therefore, DRAW3DMATRIX(Z,v ,v ). To illustrate the use of this function

min max

we first generate a 6×5 matrix using RANM({6,5}), and then call function

DRAW3DMATRIX, as shown below:

The plot is in the style of a FAST3DPLOT. Different views of the plot are shown

below:

Page 12-52

Chapter 13

Calculus Applications

In this Chapter we discuss applications of the calculator’s functions to

operations related to Calculus, e.g., limits, derivatives, integrals, power series,

etc.

The CALC (Calculus) menu

Many of the functions presented in this Chapter are contained in the

calculator’s CALC menu, available through the keystroke sequence „Ö

(associated with the 4 key). The CALC menu shows the following entries:

The first four options in this menu are actually sub-menus that apply to (1)

derivatives and integrals, (2) limits and power series, (3) differential equations,

and (4) graphics. The functions in entries (1) and (2) will be presented in this

Chapter. Differential equations, the subject of item (3), are presented in

Chapter 16. Graphic functions, the subject of item (4), were presented at the

end of Chapter 12. Finally, entries 5. DERVX and 6.INTVX are the functions to

obtain a derivative and a indefinite integral for a function of the default CAS

variable (typically, ‘X’). Functions DERVX and INTVX are discussed in detail

later.

Limits and derivatives

Differential calculus deals with derivatives, or rates of change, of functions and

their applications in mathematical analysis. The derivative of a function is

defined as a limit of the difference of a function as the increment in the

independent variable tends to zero. Limits are used also to check the continuity

of functions.

Page 13-1

Function lim

The calculator provides function lim to calculate limits of functions. This function

uses as input an expression representing a function and the value where the

limit is to be calculated. Function lim is available through the command

catalog (‚N~„l) or through option 2. LIMITS & SERIES… of the

CALC menu (see above).

Note: The functions available in the LIMITS & SERIES menu are shown next:

Function DIVPC is used to divide two polynomials producing a series

expansion. Functions DIVPC, SERIES, TAYLOR0, and TAYLOR are used in series

expansions of functions and discussed in more detail in this Chapter.

Function lim is entered in ALG mode as lim(f(x),x=a) to calculate the

lim f (x)

limit

. In RPN mode, enter the function first, then the expression

x→a

‘x=a’, and finally function lim. Examples in ALG mode are shown next,

including some limits to infinity. The keystrokes for the first example are as

follows (using Algebraic mode, and system flag 117 set to CHOOSE boxes):

„Ö2 @@OK@@ 2 @@OK@@ x+1‚í x‚Å 1`

The infinity symbol is associated with the 0 key, i.e.., „è.

Page 13-2

To calculate one-sided limits, add +0 or -0 to the value to the variable. A “+0”

means limit from the right, while a “-0” means limit from the left. For example,

the limit of

following keystrokes (ALG mode):

as x approaches 1 from the left can be determined with the

x −1

‚N~„l˜$OK$ R!ÜX-

1™@íX@Å1+0`

The result is as follows:

Derivatives

The derivative of a function f(x) at x = a is defined as the limit

f (x + h) − f (x)

= f '(x) = lim

h−>0

Some examples of derivatives using this limit are shown in the following screen

shots:

Functions DERIV and DERVX

The function DERIV is used to take derivatives in terms of any independent

variable, while the function DERVX takes derivatives with respect to the CAS

default variable VX (typically ‘X’). While function DERVX is available directly in

the CALC menu, both functions are available in the DERIV.&INTEG sub-menu

within the CALCL menu ( „Ö).

Function DERIV requires a function, say f(t), and an independent variable, say, t,

while function DERVX requires only a function of VX. Examples are shown next

Page 13-3

in ALG mode. Recall that in RPN mode the arguments must be entered before

the function is applied.

The DERIV&INTEG menu

The functions available in this sub-menu are listed below:

Out of these functions DERIV and DERVX are used for derivatives. The other

functions include functions related to anti-derivatives and integrals (IBP, INTVX,

PREVAL, RISCH, SIGMA, and SIGMAVX), to Fourier series (FOURIER),and to

vector analysis (CURL, DIV, HESS, LAPL). Next we discuss functions DERIV and

DERVX, the remaining functions are presented either later in this Chapter or in

subsequent Chapters.

Calculating derivatives with ∂

The symbol is available as ‚¿ (the T key). This symbol can be used to

enter a derivative in the stack or in the Equation Writer (see Chapter 2). If you

use the symbol to write a derivative into the stack, follow it immediately with the

independent variable, then by a pair of parentheses enclosing the function to

Page 13-4

be differentiated. Thus, to calculate the derivative d(sin(r),r), use, in ALG mode:

‚¿~„r„ÜS~„r`

In RPN mode, this expression must be enclosed in quotes before entering it into

the stack. The result in ALG mode is:

In the Equation Writer, when you press ‚¿, the calculator provides the

following expression:

The insert cursor (ꢀ) will be located right at the denominator awaiting for the

user to enter an independent variable, say, s: ~„s. Then, press the right-

arrow key (™) to move to the placeholder between parentheses:

Next, enter the function to be differentiated, say, s*ln(s):

Page 13-5

To evaluate the derivative in the Equation Writer, press the up-arrow key —,

four times, to select the entire expression, then, press @EVAL. The derivative will

be evaluated in the Equation Writer as:

Note: The symbol ∂ is used formally in mathematics to indicate a partial

derivative, i.e., the derivative of a function with more than one variable.

However, the calculator does not distinguish between ordinary and partial

derivatives, utilizing the same symbol for both. The user must keep this

distinction in mind when translating results from the calculator to paper.

The chain rule

The chain rule for derivatives applies to derivatives of composite functions. A

general expression for the chain-rule is d{f[g(x)]}/dx = (df/dg)⋅ (dg/dx). Using

the calculator, this formula results in:

The terms d1 in front of g(x) and f(g(x)) in the expression above are

abbreviations the calculator uses to indicate a first derivative when the

independent variable, in this case x, is clearly defined. Thus, the latter result is

interpreted as in the formula for the chain rule shown above. Here is another

example of a chain rule application:

Page 13-6

Derivatives of equations

You can use the calculator to calculate derivatives of equations, i.e., expressions

in which derivatives will exist in both sides of the equal sign. Some examples

are shown below:

Notice that in the expressions where the derivative sign (∂) or function DERIV

was used, the equal sign is preserved in the equation, but not in the cases

where function DERVX was used. In these cases, the equation was re-written

with all its terms moved to the left-hand side of the equal sign. Also, the equal

sign was removed, but it is understood that the resulting expression is equal to

zero.

Implicit derivatives

Implicit derivatives are possible in expressions such as:

Application of derivatives

Derivatives can be used for analyzing the graphs of functions and for

optimizing functions of one variable (i.e., finding maxima and minima). Some

applications of derivatives are shown next.

Page 13-7

Analyzing graphics of functions

In Chapter 11 we presented some functions that are available in the graphics

screen for analyzing graphics of functions of the form y = f(x). These functions

include (X,Y) and TRACE for determining points on the graph, as well as

functions in the ZOOM and FCN menu. The functions in the ZOOM menu

allow the user to zoom in into a graph to analyze it in more detail. These

functions are described in detail in Chapter 12. Within the functions of the

FCN menu, we can use the functions SLOPE, EXTR, F’, and TANL to determine

the slope of a tangent to the graph, the extrema (minima and maxima) of the

function, to plot the derivative, and to find the equation of the tangent line.

Try the following example for the function y = tan(x).

ꢀ Press „ô, simultaneously in RPN mode, to access to the PLOT

SETUP window.

ꢀ Change TYPEto FUNCTION, if needed, by using [@CHOOS].

ꢀ Press ˜ and type in the equation ‘TAN(X)’.

ꢀ Make sure the independent variable is set to ‘X’.

ꢀ Press L @@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously, to access the PLOT window

ꢀ Change H-VIEW range to –2 to 2, and V-VIEW range to –5 to 5.

ꢀ Press @ERASE @DRAW to plot the function in polar coordinates.

The resulting plot looks as follows:

ꢀ Notice that there are vertical lines that represent asymptotes. These are

not part of the graph, but show points where TAN(X) goes to ∞ at

certain values of X.

ꢀ Press @TRACE @(X,Y)@, and move the cursor to the point X: 1.08E0, Y:

1.86E0. Next, press L@@FCN@ @SLOPE. The result is Slope:

4.45010547846.

ꢀ Press LL@TANL. This operation produces the equation of the

tangent line, and plots its graph in the same figure. The result is shown

in the figure below:

Page 13-8

ꢀ Press L @PICT @CANCL $ to return to normal calculator display.

Notice that the slope and tangent line that you requested are listed in

the stack.

Function DOMAIN

Function DOMAIN, available through the command catalog (‚N),

provides the domain of definition of a function as a list of numbers and

specifications. For example,

indicates that between –∞ and 0, the function LN(X) is not defined (?), while

from 0 to +∞, the function is defined (+). On the other hand,

indicates that the function is not defined between –∞ and -1, nor between 1

and +∞. The domain of this function is, therefore, -1<X<1.

Function TABVAL

This function is accessed through the command catalog or through the GRAPH

sub-menu in the CALC menu. Function TABVAL takes as arguments a function of

the CAS variable, f(X), and a list of two numbers representing a domain of

interest for the function f(X). Function TABVAL returns the input values plus the

range of the function corresponding to the domain used as input. For example,

Page 13-9

This result indicates that the range of the function

f (X ) =

X ²+1

⎧

⎨

⎩

⎫

⎬

⎭

corresponding to the domain D = { -1,5 } is R =

Function SIGNTAB

Function SIGNTAB, available through the command catalog (‚N),

provides information on the sign of a function through its domain. For example,

for the TAN(X) function,

SIGNTAB indicates that TAN(X) is negative between –π/2 and 0, and positive

between 0 and π /2. For this case, SIGNTAB does not provide information (?)

in the intervals between –∞ and -π /2, nor between +π /2 and ∞. Thus,

SIGNTAB, for this case, provides information only on the main domain of

TAN(X), namely -π /2 < X < +π /2.

A second example of function SIGNTAB is shown below:

For this case, the function is negative for X<-1 and positive for X> -1.

Function TABVAR

This function is accessed through the command catalog or through the GRAPH

sub-menu in the CALC menu. It uses as input the function f(VX), where VX is the

default CAS variable. The function returns the following, in RPN mode:

Page 13-10

ꢀ Level 3: the function f(VX)

ꢀ Two lists, the first one indicates the variation of the function (i.e., where

it increases or decreases) in terms of the independent variable VX, the

second one indicates the variation of the function in terms of the

dependent variable.

ꢀ A graphic object showing how the variation table was computed.

Example: Analyze the function Y = X -4X -11X+30, using the function TABVAR.

Use the following keystrokes, in RPN mode:

'X^3-4*X^2-11*X+30' `‚N ~t(select TABVAR) @@OK@@

This is what the calculator shows in stack level 1:

This is a graphic object. To be able to the result in its entirety, press ˜. The

variation table of the function is shown as follows:

Press $ to recover normal calculator display. Press ƒ to drop this last

result from the stack.

Two lists, corresponding to the top and bottom rows of the graphics matrix

shown earlier, now occupy level 1. These lists may be useful for programming

purposes. Press ƒ to drop this last result from the stack.

Page 13-11

The interpretation of the variation table shown above is as follows: the function

F(X) increases for X in the interval (-∞, -1), reaching a maximum equal to 36 at

X = -1. Then, F(X) decreases until X = 11/3, reaching a minimum of –400/27.

After that F(X) increases until reaching +∞. Also, at X = ∞, F(X) = ∞.

Using derivatives to calculate extreme points

“Extreme points,” or extrema, is the general designation for maximum and

minimum values of a function in a given interval. Since the derivative of a

function at a given point represents the slope of a line tangent to the curve at

that point, then values of x for which f’(x) =0 represent points where the graph

of the function reaches a maximum or minimum. Furthermore, the value of the

second derivative of the function, f”(x), at those points determines whether the

point is a relative or local maximum [f”(x)<0] or minimum [f”(x)>0]. These

ideas are illustrated in the figure below.

In this figure we limit ourselves to determining extreme points of the function y =

f(x) in the x-interval [a,b]. Within this interval we find two points, x = x and x

= x , where f’(x)=0. The point x = x , where f”(x)>0, represents a local

minimum, while the point x = x , where f”(x)<0, represents a local maximum.

From the graph of y = f(x) it follows that the absolute maximum in the interval

[a,b] occurs at x = a, while the absolute minimum occurs at x = b.

For example, to determine where the critical points of function 'X^3-4*X^2-

11*X+30' occur, we can use the following entries in ALG mode:

Page 13-12

We find two critical points, one at x = 11/3 and one at x = -1. To evaluate the

second derivative at each point use:

The last screen shows that f”(11/3) = 14, thus, x = 11/3 is a relative minimum.

For x = -1, we have the following:

This result indicates that f”(-1) = -14, thus, x = -1 is a relative maximum.

Evaluate the function at those points to verify that indeed f(-1) > f(11/3).

Higher order derivatives

Higher order derivatives can be calculated by applying a derivative function

several times, e.g.,

Page 13-13

Anti-derivatives and integrals

An anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx. For

example, since d(x ) /dx = 3x , an anti-derivative of f(x) = 3x is F(x) = x + C,

where C is a constant. One way to represent an anti-derivative is as a

indefinite integral, i.e.,

C = constant.

, if and only if, f(x) = dF/dx, and

f (x)dx = F(x) + C

∫

Functions INT, INTVX, RISCH, SIGMA and SIGMAVX

The calculator provides functions INT, INTVX, RISCH, SIGMA and SIGMAVX to

calculate anti-derivatives of functions. Functions INT, RISCH, and SIGMA work

with functions of any variable, while functions INTVX, and SIGMAVX utilize

functions of the CAS variable VX (typically, ‘x’). Functions INT and RISCH

require, therefore, not only the expression for the function being integrated, but

also the independent variable name. Function INT, requires also a value of x

where the anti-derivative will be evaluated. Functions INTVX and SIGMAVX

require only the expression of the function to integrate in terms of VX. Some

examples are shown next in ALG mode:

Please notice that functions SIGMAVX and SIGMA are designed for integrands

that involve some sort of integer function like the factorial (!) function shown

Page 13-14

above. Their result is the so-called discrete derivative, i.e., one defined for

integer numbers only.

Definite integrals

In a definite integral of a function, the resulting anti-derivative is evaluated at the

upper and lower limit of an interval (a,b) and the evaluated values subtracted.

^bf (x)dx = F(b) − F(a),

Symbolically,

where f(x) = dF/dx.

∫

The PREVAL(f(x),a,b) function of the CAS can simplify such calculation by

returning f(b)-f(a) with x being the CAS variable VX.

To calculate definite integrals the calculator also provides the integral symbol as

the keystroke combination ‚Á (associated with the U key). The simplest

way to build an integral is by using the Equation Writer (see Chapter 2 for an

example). Within the Equation Writer, the symbol ‚Á produces the

integral sign and provides placeholders for the integration limits (a,b), for the

function, f(x), and for the variable of integration (x). The following screen shots

show how to build a particular integral. The insert cursor is first located in the

lower limit of integration, enter a value and press the right-arrow key (™) to

move to the upper limit of integration. Enter a value in that location and press

™ again to move to the integrand location. Type the integrand expression,

and press once more to move to the differential place holder, type the variable

of integration in that location and the integral is ready to be calculated.

At this point, you can press ` to return the integral to the stack, which will

show the following entry (ALG mode shown):

Page 13-15

This is the general format for the definite integral when typed directly into the

stack, i.e., ∫ (lower limit, upper limit, integrand, variable of integration)

Pressing ` at this point will evaluate the integral in the stack:

The integral can be evaluated also in the Equation Writer by selecting the entire

expression an using the soft menu key @EVAL.

Step-by-step evaluation of derivatives and integrals

With the Step/Step option in the CAS MODES windows selected (see Chapter

1), the evaluation of derivatives and integrals will be shown step by step. For

example, here is the evaluation of a derivative in the Equation Writer:

[[[[[

Notice the application of the chain rule in the first step, leaving the derivative of

the function under the integral explicitly in the numerator. In the second step, the

resulting fraction is rationalized (eliminating the square root from the

denominator), and simplified. The final version is shown in the third step. Each

step is shown by pressing the @EVAL menu key, until reaching the point where

further application of function EVAL produce no more changes in the

expression.

Page 13-16

The following example shows the evaluation of a definite integral in the

Equation Writer, step-by-step:

[[[[[

Notice that the step-by-step process provides information on the intermediate

steps followed by the CAS to solve this integral. First, CAS identifies a square

root integral, next, a rational fraction, and a second rational expression, to

come up with the final result. Notice that these steps make a lot of sense to the

calculator, although not enough information is provided to the user on the

individual steps.

Integrating an equation

Integrating an equation is straightforward, the calculator simply integrates both

sides of the equation simultaneously, e.g.,

Page 13-17

Techniques of integration

Several techniques of integration can be implemented in the calculators, as

shown in the following examples.

Substitution or change of variables

Suppose we want to calculate the integral

. If we use step-by-

step calculation in the Equation Writer, this is the sequence of variable

substitutions:

This second step shows the proper substitution to use, u = x -1.

The last four steps show the progression of the solution: a square root, followed

by a fraction, a second fraction, and the final result. This result can be

simplified by using function @SIMP, to read:

Page 13-18

Integration by parts and differentials

A differential of a function y = f(x), is defined as dy = f’(x) dx, where f’(x) is the

derivative of f(x). Differentials are used to represent small increments in the

variables. The differential of a product of two functions, y = u(x)v(x), is given by

dy = u(x)dv(x) +du(x)v(x), or, simply, d(uv) = udv - vdu. Thus, the integral of udv

= d(uv) - vdu, is written as

. Since by the definition of

udv = d(uv) − vdu

∫

a differential, ∫dy = y, we write the previous expression as

udv = uv − vdu

∫

This formulation, known as integration by parts, can be used to find an integral

if dv is easily integrable. For example, the integral ∫xe dx can be solved by

integration by parts if we use u = x, dv = e dx, since, v = e . With du = dx, the

integral becomes ∫xe dx = ∫udv = uv - ∫vdu = xe - ∫e dx = xe - e .

The calculator provides function IBP, under the CALC/DERIV&INTG menu, that

takes as arguments the original function to integrate, namely, u(X)*v’(X), and

the function v(X), and returns u(X)*v(X) and -v(X)*u’(X). In other words, function

IBP returns the two terms of the right-hand side in the integration by parts

equation. For the example used above, we can write in ALG mode:

Thus, we can use function IBP to provide the components of an integration by

parts. The next step will have to be carried out separately.

It is important to mention that the integral can be calculated directly by using,

for example,

Page 13-19

Integration by partial fractions

Function PARTFRAC, presented in Chapter 5, provides the decomposition of a

fraction into partial fractions. This technique is useful to reduce a complicated

fraction into a sum of simple fractions that can then be integrated term by term.

For example, to integrate

X ⁵+ 5

X ⁴+ 2X ³+ X

∫

we can decompose the fraction into its partial component fractions, as follows:

The direct integration produces the same result, with some switching of the terms

(Rigorous mode set in the CAS – see Chapter 2):

Improper integrals

These are integrals with infinite limits of integration. Typically, an improper

integral is dealt with by first calculating the integral as a limit to infinity, e.g.,

^∞dx

x²

= lim

x²

∫

ε→∞

Page 13-20

Using the calculator, we proceed as follows:

Alternatively, you can evaluate the integral to infinity from the start, e.g.,

Integration with units

An integral can be calculated with units incorporated into the limits of

integration, as in the example shown below that uses ALG mode, with the CAS

set to Approx mode. The left-hand side figure shows the integral typed in the

line editor before pressing `. The right-hand figure shows the result after

pressing `.

If you enter the integral with the CAS set to Exact mode, you will be asked to

change to Approx mode, however, the limits of the integral will be shown in a

different format as shown here:

These limits represent 1×1_mm and 0×1_mm, which is the same as 1_mm and

0_mm, as before. Just be aware of the different formats in the output.

Page 13-21

Some notes in the use of units in the limits of integrations:

1 – The units of the lower limit of integration will be the ones used in the final

result, as illustrated in the two examples below:

2 - Upper limit units must be consistent with lower limit units. Otherwise, the

calculator simply returns the unevaluated integral. For example,

3 – The integrand may have units too. For example:

4 – If both the limits of integration and the integrand have units, the resulting

units are combined according to the rules of integration. For example,

Infinite series

∞

h(n)(x − a)ⁿ

An infinite series has the form

. The infinite series typically

∑

n=0,1

starts with indices n = 0 or n = 1. Each term in the series has a coefficient h(n)

that depends on the index n.

Page 13-22

Taylor and Maclaurin’s series

A function f(x) can be expanded into an infinite series around a point x=x by

using a Taylor’s series, namely,

f ⁽ⁿ⁾(x_o)

∞

f (x) =

⋅(x − x_o)ⁿ

∑

n=0

(n)

(0)

where f (x) represents the n-th derivative of f(x) with respect to x, f (x) = f(x).

If the value x is zero, the series is referred to as a Maclaurin’s series, i.e.,

f ⁽ⁿ⁾(0)

∞

f (x) =

⋅ xⁿ

∑

n=0

Taylor polynomial and reminder

In practice, we cannot evaluate all terms in an infinite series, instead, we

approximate the series by a polynomial of order k, P (x), and estimate the order

of a residual, R (x), such that

f ⁽ⁿ⁾(x_o)

∞

f (x) =

⋅(x − x )ⁿ+

⋅(x − x_o)ⁿ

∑

n=0

n=k+1

f (x) = P (x) + R_k(x).

i.e.,

The polynomial P (x) is referred to as Taylor’s polynomial. The order of the

residual is estimated in terms of a small quantity h = x-x , i.e., evaluating the

polynomial at a value of x very close to x . The residual if given by

f ^(k+1)(ξ)

⋅ h^k+1

R_k(x) =

Page 13-23

where ξ is a number near x = x . Since ξ is typically unknown, instead of an

estimate of the residual, we provide an estimate of the order of the residual in

n+1

k+1

reference to h, i.e., we say that R (x) has an error of order h , or R ≈ O(h ).

k+1

If h is a small number, say, h<<1, then h

will be typically very small, i.e.,

k+1

<<h << …<< h << 1. Thus, for x close to x , the larger the number of

elements in the Taylor polynomial, the smaller the order of the residual.

Functions TAYLR, TAYLR0, and SERIES

Functions TAYLR, TAYLR0, and SERIES are used to generate Taylor polynomials,

as well as Taylor series with residuals. These functions are available in the

CALC/LIMITS&SERIES menu described earlier in this Chapter.

Function TAYLOR0 performs a Maclaurin series expansion, i.e., about X = 0, of

an expression in the default independent variable, VX (typically ‘X’). The

expansion uses a 4-th order relative power, i.e., the difference between the

highest and lowest power in the expansion is 4. For example,

Function TAYLR produces a Taylor series expansion of a function of any variable

x about a point x = a for the order k specified by the user. Thus, the function has

the format TAYLR(f(x-a),x,k). For example,

Function SERIES produces a Taylor polynomial using as arguments the function

f(x) to be expanded, a variable name alone (for Maclaurin’s series) or an

expression of the form ‘variable = value’ indicating the point of expansion of a

Taylor series, and the order of the series to be produced. Function SERIES

returns two output items a list with four items, and an expression for h = x - a, if

the second argument in the function call is ‘x=a’, i.e., an expression for the

Page 13-24

increment h. The list returned as the first output object includes the following

items:

lim f (x)

1 - Bi-directional limit of the function at point of expansion, i.e.,

x→a

2 - An equivalent value of the function near x = a

3 - Expression for the Taylor polynomial

4 - Order of the residual or remainder

Because of the relatively large amount of output, this function is easier to handle

in RPN mode. For example:

Drop the contents of stack level 1 by pressing ƒ, and then enter μ, to

decompose the list. The results are as follows:

In the right-hand side figure above, we are using the line editor to see the series

expansion in detail.

Page 13-25

Chapter 14

Multi-variate Calculus Applications

Multi-variate calculus refers to functions of two or more variables. In this

Chapter we discuss the basic concepts of multi-variate calculus including partial

derivatives and multiple integrals.

Multi-variate functions

A function of two or more variables can be defined in the calculator by using

the DEFINE function („à). To illustrate the concept of partial derivative,

we will define a couple of multi-variate functions, f(x,y) = x cos(y), and g(x,y,z)

2 1/2

= (x +y ) sin(z), as follows:

We can evaluate the functions as we would evaluate any other calculator

function, e.g.,

Graphics of two-dimensional functions are possible using Fast3D, Wireframe,

Ps-Contour, Y-Slice, Gridmap, and Pr-Surface plots as described in Chapter 12.

Partial derivatives

Consider the function of two variables z = f(x,y), the partial derivative of the

function with respect to x is defined by the limit

Page 14-1

∂f

∂x

f (x + h, y) − f (x, y)

= lim

h→0

Similarly,

∂f

∂y

f (x, y + k) − f (x, y)

= lim

k→0

We will use the multi-variate functions defined earlier to calculate partial

derivatives using these definitions. Here are the derivatives of f(x,y) with respect

to x and y, respectively:

Notice that the definition of partial derivative with respect to x, for example,

requires that we keep y fixed while taking the limit as hꢀ0. This suggest a way

to quickly calculate partial derivatives of multi-variate functions: use the rules of

ordinary derivatives with respect to the variable of interest, while considering all

other variables as constant. Thus, for example,

∂

∂x

∂

∂y

(

xcos(y)

)

= cos(y),

(

xcos(y)

)

= −xsin(y)

which are the same results as found with the limits calculated earlier.

Consider another example,

∂

∂x

(

yx²+ y²

)

= 2yx + 0 = 2xy

In this calculation we treat y as a constant and take derivatives of the expression

with respect to x.

Similarly, you can use the derivative functions in the calculator, e.g., DERVX,

DERIV, ∂ (described in detail in Chapter 13) to calculate partial derivatives.

Recall that function DERVX uses the CAS default variable VX (typically, ‘X’),

Page 14-2

therefore, with DERVX you can only calculate derivatives with respect to X.

Some examples of first-order partial derivatives are shown next:

[[[[[

Higher-order derivatives

The following second-order derivatives can be defined

⎛

⎞

∂ f

∂ ∂f ∂ f

∂ ∂f

⎛

⎞

⎜

⎟

⎜

⎟

∂x²

∂y²

⎟

∂x ∂x

∂y ∂y

⎝

⎠

⎝

⎠

⎛

⎞

∂ f

∂ ∂f

∂ f

∂ ∂f

⎛

⎞

⎜

⎟

⎜

⎟

⎜

∂y∂x ∂y ∂x ∂x∂y ∂x ∂y

⎝

⎠

⎝

⎠

The last two expressions represent cross-derivatives, the partial derivatives signs

in the denominator shows the order of derivation. In the left-hand side, the

derivation is taking first with respect to x and then with respect to y, and in the

right-hand side, the opposite is true. It is important to indicate that, if a

function is continuous and differentiable, then

∂ f

∂y∂x ∂x∂y

Page 14-3

Third-, fourth-, and higher order derivatives are defined in a similar manner.

To calculate higher order derivatives in the calculator, simply repeat the

derivative function as many times as needed. Some examples are shown

below:

The chain rule for partial derivatives

Consider the function z = f(x,y), such that x = x(t), y = y(t). The function z

actually represents a composite function of t if we write it as z = f[x(t),y(t)]. The

chain rule for the derivative dz/dt for this case is written as

∂z ∂z ∂x ∂z ∂y

⋅

∂v ∂x ∂v ∂y ∂v

To see the expression that the calculator produces for this version of the chain

rule use:

The result is given by d1y(t)⋅d2z(x(t),y(t))+d1x(t)⋅d1z(x(y),y(t)). The term d1y(t)

is to be interpreted as “the derivative of y(t) with respect to the 1 independent

variable, i.e., t”, or d1y(t) = dy/dt. Similarly, d1x(t) = dx/dt. On the other

hand, d1z(x(t),y(t)) means “the first derivative of z(x,y) with respect to the first

independent variable, i.e., x”, or d1z(x(t),y(t)) = ∂z/∂x. Similarly, d2z(x(t),y(t))

= ∂z/∂y. Thus, the expression above is to be interpreted as:

dz/dt = (dy/dt)⋅(∂z/∂y) + (dx/dt)⋅(∂z/∂x).

Page 14-4

Total differential of a function z = z(x,y)

From the last equation, if we multiply by dt, we get the total differential of the

function z = z(x,y), i.e., dz = (∂z/∂x)⋅dx + (∂z/∂y)⋅dy.

A different version of the chain rule applies to the case in which z = f(x,y), x =

x(u,v), y = y(u,v), so that z = f[x(u,v), y(u,v)]. The following formulas represent

the chain rule for this situation:

∂z ∂z ∂x ∂z ∂y

∂u ∂x ∂u ∂y ∂u

∂z ∂z ∂x ∂z ∂y

∂v ∂x ∂v ∂y ∂v

⋅

Determining extrema in functions of two variables

In order for the function z = f(x,y) to have an extreme point (extrema) at (x ,y ),

its derivatives ∂f/∂x and ∂f/∂y must vanish at that point. These are necessary

conditions. The sufficient conditions for the function to have an extreme at point

(x ,y ) are ∂f/∂x = 0, ∂f/∂y = 0, and Δ = (∂ f/∂x )⋅ (∂ f/∂y )-[∂ f/∂x∂y] > 0.

The point (x ,y ) is a relative maximum if ∂ f/∂x < 0, or a relative minimum if

∂ f/∂x > 0. The value Δ is referred to as the discriminant.

If Δ = (∂ f/∂x )⋅ (∂ f/∂y )-[∂ f/∂x∂y] < 0, we have a condition known as a

saddle point, where the function would attain a maximum in x if we were to

hold y constant, while, at the same time, attaining a minimum if we were to

hold x constant, or vice versa.

Example 1 – Determine the extreme points (if any) of the function f(X,Y) = X -3X-

Y +5. First, we define the function f(X,Y), and its derivatives fX(X,Y) = ∂f/∂X,

fY(X,Y) = ∂f/∂Y. Then, we solve the equations fX(X,Y) = 0 and fY(X,Y) = 0,

simultaneously:

Page 14-5

We find critical points at (X,Y) = (1,0), and (X,Y) = (-1,0). To calculate the

discriminant, we proceed to calculate the second derivatives, fXX(X,Y) = ∂ f/

∂X , fXY(X,Y) = ∂ f/∂X/∂Y, and fYY(X,Y) = ∂ f/∂Y .

The last result indicates that the discriminant is Δ = -12X, thus, for (X,Y) = (1,0),

Δ <0 (saddle point), and for (X,Y) = (-1,0), Δ>0 and ∂ f/∂X <0 (relative

maximum). The figure below, produced in the calculator, and edited in the

computer, illustrates the existence of these two points:

Using function HESS to analyze extrema

Function HESS can be used to analyze extrema of a function of two variables as

shown next. Function HESS, in general, takes as input a function of n

independent variables φ(x , x , …,x ), and a vector of the functions [‘x ’

‘x ’…’x ’]. Function HESS returns the Hessian matrix of the function φ, defined

as the matrix H = [h ] = [∂ φ/∂x ∂x ], the gradient of the function with respect to

the n-variables, grad f = [ ∂φ/∂x , ∂φ/∂x , … ∂φ/∂x ], and the list of

variables [‘x ’ ‘x ’…’x ’].

Page 14-6

Applications of function HESS are easier to visualize in the RPN mode.

Consider as an example the function φ(X,Y,Z) = X + XY + XZ, we’ll apply

function HESS to function φ in the following example. The screen shots show the

RPN stack before and after applying function HESS.

When applied to a function of two variables, the gradient in level 2, when

made equal to zero, represents the equations for critical points, i.e., ∂φ/∂x = 0,

while the matrix in level 3 represent second derivatives. Thus, the results from

the HESS function can be used to analyze extrema in functions of two variables.

For example, for the function f(X,Y) = X -3X-Y +5, proceed as follows in RPN

mode:

‘X^3-3*X-Y^2+5’ ` [‘X’,’Y’] `

Enter function and variables

Apply function HESS

Find critical points

HESS

SOLVE

Decompose vector

‘s1’ K ‘s2’ K

Store critical points

The variables s1 and s2, at this point, contain the vectors [‘X=-1’,’Y=0] and

[‘X=1’,’Y=0], respect

‘H’ K

J @@@H@@@ @@s1@@ SUBST ‚ï

Store Hessian matrix

Substitute s1 into H

The resulting matrix A has a elements a = ∂ φ/∂X = -6., a = ∂ φ/∂X =

-2., and a = a = ∂ φ/∂X∂Y = 0. The discriminant, for this critical point

s1(-1,0) is Δ = (∂ f/∂x )⋅ (∂ f/∂y )-[∂ f/∂x∂y] = (-6.)(-2.) = 12.0 > 0. Since

∂ φ/∂X <0, point s1 represents a relative maximum.

Next, we substitute the second point, s2, into H:

J @@@H@@@ @@s2@@ SUBST ‚ï

Substitute s2 into H

Page 14-7

The resulting matrix has elements a = ∂ φ/∂X = 6., a = ∂ φ/∂X = -2.,

and a = a = ∂ φ/∂X∂Y = 0. The discriminant, for this critical point s2(1,0)

is Δ = (∂ f/∂x )⋅ (∂ f/∂y )-[∂ f/∂x∂y] = (6.)(-2.) = -12.0 < 0, indicating a

saddle point.

Multiple integrals

^bf (x)dx

A physical interpretation of an ordinary integral,

, is the area under

∫

the curve y = f(x) and abscissas x = a and x = b. The generalization to three

dimensions of an ordinary integral is a double integral of a function f(x,y) over

a region R on the x-y plane representing the volume of the solid body contained

under the surface f(x,y) above the region R. The region R can be described as

R = {a<x<b, f(x)<y<g(x)} or as R = {c<y<d, r(y)<x<s(y)}. Thus, the double

integral can be written as

s( y)

^g(x)φ(x, y)dydx =

φ(x, y)dydx

∫∫^{φ(x, y)dA =}∫_a∫_{f (x)}

∫ ∫

r( y)

Calculating a double integral in the calculator is straightforward. A double

integral can be built in the Equation Writer (see example in Chapter 2). An

example follows. This double integral is calculated directly in the Equation

Writer by selecting the entire expression and using function @EVAL. The result is

3/2. Step-by-step output is possible by setting the Step/Step option in the CAS

MODES screen.

Page 14-8

Jacobian of coordinate transformation

Consider the coordinate transformation x = x(u,v), y = y(u,v). The Jacobian of

this transformation is defined as

∂x ∂x

⎛

⎜

⎝

⎞

⎟

⎠

∂u ∂v

| J |= det(J) = det

∂y ∂y

∂u ∂v

When calculating an integral using such transformation, the expression to use

^is∫∫^{φ(x, y)dydx =}∫∫^{φ[x(u,v), y(u,v)] | J | dudv}

, where R’ is the region R

expressed in (u,v) coordinates.

Double integral in polar coordinates

To transform from polar to Cartesian coordinates we use x(r,θ) = r cos θ, and

y(r, θ) = r sin θ. Thus, the Jacobian of the transformation is

∂x ∂x

cos(θ) − r ⋅sin(θ )

sin(θ) r ⋅cos(θ)

∂r ∂θ

| J |=

= r

∂y ∂y

∂r ∂θ

With this result, integrals in polar coordinates are written as

Page 14-9

^{g(θ )}φ(r,θ)rdrdθ

∫∫^{φ(r,θ)dA =}∫_α∫_{f (θ )}

where the region R’ in polar coordinates is R’ = {α < θ < β, f(θ) < r < g(θ)}.

Double integrals in polar coordinates can be entered in the calculator, making

sure that the Jacobian |J| = r is included in the integrand. The following is an

example of a double integral calculated in polar coordinates, shown step-by-

step:

Page 14-10

Chapter 15

Vector Analysis Applications

In this Chapter we present a number of functions from the CALC menu that

apply to the analysis of scalar and vector fields. The CALC menu was

presented in detail in Chapter 13. In particular, in the DERIV&INTEG menu we

identified a number of functions that have applications in vector analysis,

namely, CURL, DIV, HESS, LAPL. For the exercises in this Chapter, change your

angle measure to radians.

Definitions

A function defined in a region of space such as φ(x,y,z) is known as a scalar

field, examples are temperature, density, and voltage near a charge. If the

function is defined by a vector, i.e., F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, it is

referred to as a vector field.

The following operator, referred to as the ‘del’ or ‘nabla’ operator, is a vector-

based operator that can be applied to a scalar or vector function:

∂

∇

[ ]= i ⋅ [ ]+ j ⋅ [ ]+ k ⋅ [ ]

∂x ∂y ∂z

When this operator is applied to a scalar function we can obtain the gradient

of the function, and when applied to a vector function we can obtain the

divergence and the curl of that function. A combination of gradient and

divergence produces another operator, called the Laplacian of a scalar

function. These operations are presented next.

Gradient and directional derivative

The gradient of a scalar function φ(x,y,z) is a vector function defined by

∂φ

∂x

∂φ

∂y

∂φ

∂z

gradφ = ∇φ = i ⋅

+ j ⋅

+ k ⋅

The dot product of the gradient of a function with a given unit vector represents

the rate of change of the function along that particular vector. This rate of

change is called the directional derivative of the function, D φ(x,y,z) = u•∇φ.

Page 15-1

At any particular point, the maximum rate of change of the function occurs in

the direction of the gradient, i.e., along a unit vector u = ∇φ/|∇φ|.

The value of that directional derivative is equal to the magnitude of the gradient

at any point D φ(x,y,z) = ∇φ •∇φ/|∇φ| = |∇φ|

max

The equation φ(x,y,z) = 0 represents a surface in space. It turns out that the

gradient of the function at any point on this surface is normal to the surface.

Thus, the equation of a plane tangent to the curve at that point can be found by

using a technique presented in Chapter 9.

The simplest way to obtain the gradient is by using function DERIV, available in

the CALC menu, e.g.,

A program to calculate the gradient

The following program, which you can store into variable GRADIENT, uses

function DERIV to calculate the gradient of a scalar function of X,Y,Z.

Calculations for other base variables will not work. If you work frequently in the

(X,Y,Z) system, however, this function will facilitate calculations:

<< X Y Z 3 ꢀARRY DERIV >>

Type the program while in RPN mode. After switching to ALG mode, you can

call the function GRADIENT as in the following example:

Using function HESS to obtain the gradient

The function HESS can be used to obtain the gradient of a function as shown

next. As indicated in Chapter 14, function HESS takes as input a function of n

independent variables φ(x , x , …,x ), and a vector of the functions [‘x ’

‘x ’…’x ’]. Function HESS returns the Hessian matrix of the function φ, defined

Page 15-2

as the matrix H = [h ] = [∂φ/∂x ∂x ], the gradient of the function with respect to

the n-variables, grad f = [ ∂φ/∂x , ∂φ/∂x , … ∂φ/∂x ], and the list of

variables [‘x ’ ‘x ’…’x ’]. Consider as an example the function φ(X,Y,Z) = X +

XY + XZ, we’ll apply function HESS to this scalar field in the following example

in RPN mode:

Thus, the gradient is [2X+Y+Z, X, X]. Alternatively, one can use function DERIV

as follows: DERIV(X^2+X*Y+X*Z,[X,Y,Z]), to obtain the same result.

Potential of a gradient

Given the vector field, F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, if there exists a

function φ(x,y,z), such that f = ∂φ/∂x, g = ∂φ/∂y, and h = ∂φ/∂z, then φ(x,y,z) is

referred to as the potential function for the vector field F. It follows that F = grad

φ = ∇φ.

The calculator provides function POTENTIAL, available through the command

catalog (‚N), to calculate the potential function of a vector field, if it

exists. For example, if F(x,y,z) = xi + yj + zk, applying function POTENTIAL we

find:

Since function SQ(x) represents x , this results indicates that the potential

function for the vector field F(x,y,z) = xi + yj + zk, is φ(x,y,z) = (x +y +z )/2.

Notice that the conditions for the existence of φ(x,y,z), namely, f = ∂φ/∂x, g =

∂φ/∂y, and h = ∂φ/∂z, are equivalent to the conditions: ∂f/∂y = ∂g/∂x, ∂f/∂z =

∂h/∂x, and ∂g/∂z = ∂h/∂y. These conditions provide a quick way to determine

if the vector field has an associated potential function. If one of the conditions

∂f/∂y = ∂g/∂x, ∂f/∂z = ∂h/∂x, ∂g/∂z = ∂h/∂y, fails, a potential function

φ(x,y,z) does not exist. In such case, function POTENTIAL returns an error

message. For example, the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, does

Page 15-3

not have a potential function associated with it, since, ∂f/∂z ≠ ∂h/∂x. The

calculator response in this case is shown below:

Divergence

The divergence of a vector function, F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,

is defined by taking a “dot-product” of the del operator with the function, i.e.,

∂f ∂g ∂h

divF = ∇ • F =

∂x ∂y ∂z

Function DIV can be used to calculate the divergence of a vector field. For

example, for F(X,Y,Z) = [XY,X +Y +Z ,YZ], the divergence is calculated, in ALG

mode, as follows:

Laplacian

The divergence of the gradient of a scalar function produces an operator called

the Laplacian operator. Thus, the Laplacian of a scalar function φ(x,y,z) is given

∂ φ ∂ φ ∂ φ

∇ φ = ∇ • ∇φ =

∂x²∂x²∂x²

The partial differential equation ∇ φ = 0 is known as Laplace’s equation.

Function LAPL can be used to calculate the Laplacian of a scalar function. For

example, to calculate the Laplacian of the function φ(X,Y,Z) = (X +Y )cos(Z),

use:

Page 15-4

Curl

The curl of a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, is defined by a

“cross-product” of the del operator with the vector field, i.e.,

∂

∂x

∂

∂y

∂

∂z

curlF = ∇× F =

[ ]

f (x, y, z) g(x, y, z) h(x, y, z)

⎛

⎞

⎛

⎞

∂h ∂g

∂y ∂z

∂f ∂h

−

∂h ∂g

−

⎛

⎞

⎟

⎠

⎜

⎟

⎜

⎟

= i

−

+ j

+ k

⎜

∂z ∂x

∂y ∂z

⎝

⎠

⎝

⎠

The curl of vector field can be calculated with function CURL. For example, for

the function F(X,Y,Z) = [XY,X +Y +Z ,YZ], the curl is calculated as follows:

Irrotational fields and potential function

In an earlier section in this chapter we introduced function POTENTIAL to

calculate the potential function φ(x,y,z) for a vector field, F(x,y,z) = f(x,y,z)i+

g(x,y,z)j+ h(x,y,z)k, such that F = grad φ = ∇φ. We also indicated that the

conditions for the existence of φ, were: ∂f/∂y = ∂g/∂x, ∂f/∂z = ∂h/∂x, and ∂g/

∂z = ∂h/∂y. These conditions are equivalent to the vector expression

curl F = ∇×F = 0.

A vector field F(x,y,z), with zero curl, is known as an irrotational field. Thus, we

conclude that a potential function φ(x,y,z) always exists for an irrotational field

F(x,y,z).

Page 15-5

As an example, in an earlier example we attempted to find a potential function

for the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, and got an error message

back from function POTENTIAL. To verify that this is a rotational field (i.e., ∇×F

≠ 0), we use function CURL on this field:

On the other hand, the vector field F(x,y,z) = xi + yj + zk, is indeed irrotational

as shown below:

Vector potential

Given a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, if there exists a

vector function Φ(x,y,z) = φ(x,y,z)i+ψ(x,y,z)j+η(x,y,z)k, such that F = curl Φ = ∇

× Φ, then function Φ(x,y,z) is referred to as the vector potential of F(x,y,z).

The calculator provides function VPOTENTIAL, available through the command

catalog (‚N), to calculate the vector potential, Φ(x,y,z), given the vector

field, F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k. For example, given the vector

field, F(x,y,z) = -(yi+zj+xk), function VPOTENTIAL produces

i.e., Φ(x,y,z) = -x /2j + (-y /2+zx)k.

It should be indicated that there is more than one possible vector potential

functions Φ for a given vector field F. For example, the following screen shot

shows that the curl of the vector function Φ = [X +Y +Z ,XYZ,X+Y+Z] is the

vector F = ∇× Φ = [1-XY,2Z-1,ZY-2Y]. Application of function VPOTENTIAL

Page 15-6

produces the vector potential function Φ = [0, ZYX-2YX, Y-(2ZX-X)], which is

different from Φ . The last command in the screen shot shows that indeed F =

∇× Φ . Thus, a vector potential function is not uniquely determined.

The components of the given vector field, F(x,y,z) = f(x,y,z)i+g(x,y,z)j

+h(x,y,z)k, and those of the vector potential function, Φ(x,y,z) =

φ(x,y,z)i+ψ(x,y,z)j+η(x,y,z)k, are related by f = ∂η/∂y - ∂ψ/∂x, g = ∂φ/∂z - ∂η/

∂x, and h = ∂ψ/∂x - ∂φ/∂y.

A condition for function Φ(x,y,z) to exists is that div F = ∇•F = 0, i.e., ∂f/∂x +

∂g/∂y + ∂f/∂z = 0. Thus, if this condition is not satisfied, the vector potential

function Φ(x,y,z) does not exist. For example, given F = [X+Y,X-Y,Z^2], function

VPOTENTIAL returns an error message, since function F does not satisfy the

condition ∇•F = 0:

The condition ∇•F ≠ 0 is verified in the following screen shot:

Page 15-7

Chapter 16

Differential Equations

In this Chapter we present examples of solving ordinary differential equations

(ODE) using calculator functions. A differential equation is an equation

involving derivatives of the independent variable. In most cases, we seek the

dependent function that satisfies the differential equation.

Basic operations with differential equations

In this section we present some uses of the calculator for entering, checking and

visualizing the solution of ODEs.

Entering differential equations

The key to using differential equations in the calculator is typing in the

derivatives in the equation. The easiest way to enter a differential equation is to

type it in the equation writer. For example, to type the following ODE:

(x-1)⋅(dy(x)/dx) + 2⋅x⋅y(x) = e sin x, use:

‚O „ Ü~ „x -1 ™™™*‚¿ ~„x

™~„y„Ü~„x™™ Q2 ™™+2*

~„ x * ~„ y „Ü~„x ™™™™

‚= „¸ ~„ x ™*S~„x `

The derivative dy/dx is represented by ∂x(y(x)) or by d1y(x). For solution

or calculation purposes, you need to specify y(x) in the expression, i.e., the

dependent variable must include its independent variable(s) in any derivative in

the equation.

You can also type an equation directly into the stack by using the symbol ∂ in

the derivatives. For example, to type the following ODE involving second-order

derivatives: d u(x)/dx + 3u(x)⋅(du(x)/dx) + u(x) = 1/x, directly into the

screen, use:

³‚∂ ~„x„Ü‚¿~„x„ Ü~ „u

„Ü ~„x™™™+3*~ „u„Ü

~„x™*‚¿~„x„ Ü~„u„ Ü

~„x ™™ +~„u„ Ü ~„x™ Q2

‚ Å 1/ ~„x`

The result is ‘∂x(∂x(u(x)))+3*u(x)*∂x(u(x))+u^2=1/x’. This format

shows up in the screen when the _Textbook option in the display setting

Page 16-1

(H@DISP) is not selected. Press ˜ to see the equation in the Equation

Writer.

An alternative notation for derivatives typed directly in the stack is to use ‘d1’ for

the derivative with respect to the first independent variable, ‘d2’ for the

derivative with respect to the second independent variable, etc. A second-

order derivative, e.g., d x/dt , where x = x(t), would be written as ‘d1d1x(t)’,

while (dx/dt) would be written ‘d1x(t)^2’. Thus, the PDE ∂ y/∂t – g(x,y)⋅

2 2

(∂ y/∂x ) = r(x,y), would be written, using this notation, as ‘d2d2y(x,t)-

g(x,y)*d1d1y(x,t)^2=r(x,y)’.

The notation using ‘d’ and the order of the independent variable is the notation

preferred by the calculator when derivatives are involved in a calculation. For

example, using function DERIV, in ALG mode, as shown next

DERIV(‘x*f(x,t)+g(t,y) = h(x,y,t)’,t), produces the following expression:

‘x*d2f(x,t)+d1g(t,y)=d3h(x,y,t)’. Translated to paper, this

expression represents the partial differential equation x⋅(∂f/∂t) + ∂g/∂t = ∂h/∂t.

Because the order of the variable t is different in f(x,t), g(t,y), and h(x,y,t),

derivatives with respect to t have different indices, i.e., d2f(x,t), d1g(t,y), and

d3h(x,y,t). All of them, however, represent derivatives with respect to the same

variable.

Expressions for derivatives using the order-of-variable index notation do not

translate into derivative notation in the equation writer, as you can check by

pressing ˜ while the last result is in stack level 1. However, the calculator

understands both notations and operates accordingly regarding of the notation

used.

Checking solutions in the calculator

To check if a function satisfy a certain equation using the calculator, use

function SUBST (see Chapter 5) to replace the solution in the form ‘y = f(x)’ or ‘y

= f(x,t)’, etc., into the differential equation. You may need to simplify the result

by using function EVAL to verify the solution. For example, to check that u = A

sin ω t is a solution of the equation d u/dt + ω ⋅u = 0, use the following:

In ALG mode:

SUBST(‘∂t(∂t(u(t)))+ ω0^2*u(t) = 0’,‘u(t)=A*SIN (ω0*t)’) `

Page 16-2

EVAL(ANS(1)) `

In RPN mode:

‘∂t(∂t(u(t)))+ ω0^2*u(t) = 0’ ` ‘u(t)=A*SIN (ω0*t)’ `

SUBST EVAL

The result is

‘0=0’.

For this example, you could also use: ‘∂t(∂t(u(t))))+ ω0^2*u(t) = 0’ to enter the

differential equation.

Slope field visualization of solutions

Slope field plots, introduced in Chapter 12, are used to visualize the solutions to

a differential equation of the form dy/dx = f(x,y). A slope field plot shows a

number of segments tangential to the solution curves, y = f(x). The slope of the

segments at any point (x,y) is given by dy/dx = f(x,y), evaluated at any point

(x,y), represents the slope of the tangent line at point (x,y).

Example 1 -- Trace the solution to the differential equation y’ = f(x,y) = sin x cos

y, using a slope field plot. To solve this problem, follow the instructions in

Chapter 12 for slopefield plots.

If you could reproduce the slope field plot in paper, you can trace by hand lines

that are tangent to the line segments shown in the plot. This lines constitute lines

of y(x,y) = constant, for the solution of y’ = f(x,y). Thus, slope fields are useful

tools for visualizing particularly difficult equations to solve.

In summary, slope fields are graphical aids to sketch the curves y = g(x) that

correspond to solutions of the differential equation dy/dx = f(x,y).

The CALC/DIFF menu

The DIFFERENTIAL EQNS.. sub-menu within the CALC („Ö) menu provides

functions for the solution of differential equations. The menu is listed below with

system flag 117 set to CHOOSE boxes:

Page 16-3

These functions are briefly described next. They will be described in more detail

in later parts of this Chapter.

DESOLVE: Differential Equation SOLVEr, provides a solution if possible

-1

ILAP: Inverse LAPlace transform, L [F(s)] = f(t)

LAP: LAPlace transform, L[f(t)]=F(s)

LDEC: solves Linear Differential Equations with Constant coefficients, including

systems of differential equations with constant coefficients

Solution to linear and non-linear equations

An equation in which the dependent variable and all its pertinent derivatives

are of the first degree is referred to as a linear differential equation. Otherwise,

the equation is said to be non-linear. Examples of linear differential equations

are: d x/dt + β⋅(dx/dt) + ω ⋅x = A sin ω t, and ∂C/∂t + u⋅(∂C/∂x) = D⋅(∂ C/

∂x ).

An equation whose right-hand side (not involving the function or its derivatives)

is equal to zero is called a homogeneous equation. Otherwise, it is called non-

homogeneous. The solution to the homogeneous equation is known as a

general solution. A particular solution is one that satisfies the non-

homogeneous equation.

Function LDEC

The calculator provides function LDEC (Linear Differential Equation Command)

to find the general solution to a linear ODE of any order with constant

coefficients, whether it is homogeneous or not. This function requires you to

provide two pieces of input:

•

the right-hand side of the ODE

the characteristic equation of the ODE

Page 16-4

Both of these inputs must be given in terms of the default independent variable

for the calculator’s CAS (typically ‘X’). The output from the function is the

general solution of the ODE. The function LDEC is available through in the

CALC/DIFF menu. The examples are shown in the RPN mode, however,

translating them to the ALG mode is straightforward.

Example 1 – To solve the homogeneous ODE: d y/dx -4⋅(d y/dx )-11⋅(dy/

dx)+30⋅y = 0, enter: 0 ` 'X^3-4*X^2-11*X+30'` LDEC μ.

The solution is:

where cC0, cC1, and cC2 are constants of integration. While this result seems

very complicated, it can be simplified if we take

K1 = (10*cC0-(7+cC1-cC2))/40, K2 = -(6*cC0-(cC1+cC2))/24,

and

K3 = (15*cC0+(2*cC1-cC2))/15.

Then, the solution is

–3x

y = K ⋅e

+ K ⋅e + K ⋅e .

The reason why the result provided by LDEC shows such complicated

combination of constants is because, internally, to produce the solution, LDEC

utilizes Laplace transforms (to be presented later in this chapter), which

transform the solution of an ODE into an algebraic solution. The combination

of constants result from factoring out the exponential terms after the Laplace

transform solution is obtained.

Example 2 – Using the function LDEC, solve the non-homogeneous ODE:

d y/dx -4⋅(d y/dx )-11⋅(dy/dx)+30⋅y = x .

Enter:

'X^2' ` 'X^3-4*X^2-11*X+30' ` LDEC μ

Page 16-5

The solution, shown partially here in the Equation Writer, is:

Replacing the combination of constants accompanying the exponential terms

–3x

with simpler values, the expression can be simplified to y = K ⋅e

+ K ⋅e

K ⋅e + (450⋅x +330⋅x+241)/13500.

We recognize the first three terms as the general solution of the homogeneous

equation (see Example 1, above). If y represents the solution to the

–3x

homogeneous equation, i.e., y = K ⋅e

+ K ⋅e + K ⋅e . You can prove

that the remaining terms in the solution shown above, i.e., y =

(450⋅x +330⋅x+241)/13500, constitute a particular solution of the ODE.

Note: This result is general for all non-homogeneous linear ODEs, i.e., given

the solution of the homogeneous equation, y (x), the solution of the

corresponding non-homogeneous equation, y(x), can be written as

y(x) = y (x) + y (x),

where y (x) is a particular solution to the ODE.

To verify that y = (450⋅x +330⋅x+241)/13500, is indeed a particular solution

of the ODE, use the following:

'd1d1d1Y(X)-4*d1d1Y(X)-11*d1Y(X)+30*Y(X) = X^2'`

'Y(X)=(450*X^2+330*X+241)/13500' `

SUBST EV L

Allow the calculator about ten seconds to produce the result: ‘X^2 = X^2’.

Example 3 - Solving a system of linear differential equations with constant

coefficients.

Consider the system of linear differential equations:

x ’(t) + 2x ’(t) = 0,

Page 16-6

2x ’(t) + x ’(t) = 0.

1 2

2 1

⎡

⎤

In algebraic form, this is written as: A⋅x’(t) = 0, where

. The

A =

⎢

⎥

⎣

⎦

system can be solved by using function LDEC with arguments [0,0] and matrix

A, as shown in the following screen using ALG mode:

The solution is given as a vector containing the functions [x (t), x (t)]. Pressing

˜ will trigger the Matrix Writer allowing the user to see the two components

of the vector. To see all the details of each component, press the @EDIT! soft

menu key. Verify that the components are:

Function DESOLVE

The calculator provides function DESOLVE (Differential Equation SOLVEr) to

solve certain types of differential equations. The function requires as input the

differential equation and the unknown function, and returns the solution to the

equation if available. You can also provide a vector containing the differential

equation and the initial conditions, instead of only a differential equation, as

input to DESOLVE. The function DESOLVE is available in the CALC/DIFF

menu. Examples of DESOLVE applications are shown below using RPN mode.

Example 1 – Solve the first-order ODE:

dy/dx + x ⋅y(x) = 5.

In the calculator use:

'd1y(x)+x^2*y(x)=5' ` 'y(x)' ` DESOLVE

The solution provided is

{‘y = (INT(5*EXP(xt^3/3),xt,x)+cC0)*1/EXP(x^3/3)}’ }, i.e.,

Page 16-7

The variable ODETYPE

You will notice in the soft-menu key labels a new variable called @ODETY

(ODETYPE). This variable is produced with the call to the DESOL function and

holds a string showing the type of ODE used as input for DESOLVE. Press @ODETY

to obtain the string “1st order linear”.

Example 2 -- Solve the second-order ODE:

d y/dx + x (dy/dx) = exp(x).

In the calculator use:

‘d1d1y(x)+x*d1y(x) = EXP(x)’ ` ‘y(x)’ ` DESOLVE

The result is an expression having two implicit integrations, namely,

For this particular equation, however, we realize that the left-hand side of the

equation represents d/dx(x dy/dx), thus, the ODE is now written:

d/dx(x dy/dx ) = exp x,

and

x dy/dx = exp x + C.

Next, we can write

dy/dx = (C + exp x)/x = C/x + e /x.

In the calculator, you may try to integrate:

‘d1y(x) = (C + EXP(x))/x’ ` ‘y(x)’ ` DESOLVE

The result is

{ ‘y(x) = INT((EXP(xt)+C)/xt,xt,x)+C0’ }, i.e.,

Page 16-8

e^x+ C

y(x) = ⋅

dx + C₀

∫

Performing the integration by hand, we can only get it as far as:

e^x

y(x) = ⋅ dx + C ⋅ln x + C₀

∫

because the integral of exp(x)/x is not available in closed form.

Example 3 – Solving an equation with initial conditions. Solve

d y/dt + 5y = 2 cos(t/2),

with initial conditions

In the calculator, use:

y(0) = 1.2, y’(0) = -0.5.

[‘d1d1y(t)+5*y(t) = 2*COS(t/2)’ ‘y(0) = 6/5’ ‘d1y(0) = -1/2’] `

‘y(t)’ `

DESOLVE

Notice that the initial conditions were changed to their Exact expressions, ‘y(0)

= 6/5’, rather than ‘y(0)=1.2’, and ‘d1y(0) = -1/2’, rather than, ‘d1y(0) = -0.5’.

Changing to these Exact expressions facilitates the solution.

Note: To obtain fractional expressions for decimal values use function ꢀQ

(See Chapter 5).

The solution is:

Press μμto simplify the result to

‘y(t) = -((19*√5*SIN(√5*t)-(148*COS(√5*t)+80*COS(t/2)))/190)’.

Page 16-9

Press J @ODETY to get the string “Linear w/ cst coeff” for the ODE type in

this case.

Laplace Transforms

The Laplace transform of a function f(t) produces a function F(s) in the image

domain that can be utilized to find the solution of a linear differential equation

involving f(t) through algebraic methods. The steps involved in this application

are three:

1. Use of the Laplace transform converts the linear ODE involving f(t) into an

algebraic equation.

2. The unknown F(s) is solved for in the image domain through algebraic

manipulation.

3. An inverse Laplace transform is used to convert the image function found in

step 2 into the solution to the differential equation f(t).

Definitions

The Laplace transform for function f(t) is the function F(s) defined as

∞

L{ f (t)} = F(s) = f (t)⋅e^−stdt.

∫

The image variable s can be, and it generally is, a complex number.

Many practical applications of Laplace transforms involve an original function

f(t) where t represents time, e.g., control systems in electric or hydraulic circuits.

In most cases one is interested in the system response after time t>0, thus, the

definition of the Laplace transform, given above, involves an integration for

values of t larger than zero.

The inverse Laplace transform maps the function F(s) onto the original function

-1

f(t) in the time domain, i.e., L {F(s)} = f(t).

The convolution integral or convolution product of two functions f(t) and g(t),

where g is shifted in time, is defined as

( f * g )( t ) =

f (u ) ⋅ g (t − u ) ⋅ du .

∫

Page 16-10

Laplace transform and inverses in the calculator

The calculator provides the functions LAP and ILAP to calculate the Laplace

transform and the inverse Laplace transform, respectively, of a function f(VX),

where VX is the CAS default independent variable, which you should set to ‘X’.

Thus, the calculator returns the transform or inverse transform as a function of X.

The functions LAP and ILAP are available under the CALC/DIFF menu. The

examples are worked out in the RPN mode, but translating them to ALG mode is

straightforward. For these examples, set the CAS mode to Real and Exact.

Example 1 – You can get the definition of the Laplace transform use the

following: ‘f(X)’ ` L P in RPN mode, or L P(f(X))in ALG mode.

The calculator returns the result (RPN, left; ALG, right):

Compare these expressions with the one given earlier in the definition of the

Laplace transform, i.e.,

∞

L{ f (t)} = F(s) =

f (t) ⋅ e^−stdt,

∫

and you will notice that the CAS default variable X in the equation writer screen

replaces the variable s in this definition. Therefore, when using the function

LAP you get back a function of X, which is the Laplace transform of f(X).

Example 2 – Determine the Laplace transform of f(t) = e ⋅sin(t). Use:

‘EXP(2*X)*SIN(X)’ ` LAP The calculator returns the result: 1/(SQ(X-2)+1).

Press μ to obtain, 1/(X -4X+5).

When you translate this result in paper you would write

F(s) = L{e^2t⋅sint} =

s²− 4⋅ s + 5

Page 16-11

Example 3 – Determine the inverse Laplace transform of F(s) = sin(s). Use:

‘SIN(X)’ ` ILAP. The calculator takes a few seconds to return the result:

‘ILAP(SIN(X))’, meaning that there is no closed-form expression f(t), such that f(t)

-1

= L {sin(s)}.

Example 4 – Determine the inverse Laplace transform of F(s) = 1/s . Use:

‘1/X^3’ ` ILAP μ. The calculator returns the result: ‘X^2/2’, which is

-1

interpreted as L {1/s } = t /2.

Example 5 – Determine the Laplace transform of the function f(t) = cos (a⋅t+b).

Use: ‘COS(a*X+b)’ ` LAP . The calculator returns the result:

Press μ to obtain –(a sin(b) – X cos(b))/(X +a ). The transform is interpreted

as follows: L {cos(a⋅t+b)} = (s⋅cos b – a⋅sin b)/(s +a ).

Laplace transform theorems

To help you determine the Laplace transform of functions you can use a number

of theorems, some of which are listed below. A few examples of the theorem

applications are also included.

ꢀ Differentiation theorem for the first derivative. Let f be the initial condition

for f(t), i.e., f(0) = f , then

L{df/dt} = s⋅F(s) - f .

Example 1 – The velocity of a moving particle v(t) is defined as v(t) = dr/dt,

where r = r(t) is the position of the particle. Let r = r(0), and R(s) =L{r(t)}, then,

the transform of the velocity can be written as V(s) = L{v(t)}=L{dr/dt}= s⋅R(s)-r .

ꢀ Differentiation theorem for the second derivative. Let f = f(0), and (df/dt)

= df/dt| , then L{d f/dt } = s ⋅F(s) - s⋅f – (df/dt) .

t=0

Page 16-12

Example 2 – As a follow up to Example 1, the acceleration a(t) is defined as

a(t) = d r/dt . If the initial velocity is v = v(0) = dr/dt| , then the Laplace

t=0

transform of the acceleration can be written as:

A(s) = L{a(t)} = L{d r/dt }= s ⋅R(s) - s⋅r – v .

ꢀ Differentiation theorem for the n-th derivative.

(k)

Let f

= d f/dx |

, and f = f(0), then

t = 0

n-1

(n-2)

(n-1)

L{d f/dt } = s ⋅F(s) – s ⋅f −…– s⋅f

– f

ꢀ Linearity theorem. L{af(t)+bg(t)} = a⋅L{f(t)} + b⋅L{g(t)}.

ꢀ Differentiation theorem for the image function. Let F(s) = L{f(t)}, then d F/

ds = L{(-t) ⋅f(t)}.

–at

Example 3 – Let f(t) = e , using the calculator with ‘EXP(-a*X)’ ` LAP, you

get ‘1/(X+a)’, or F(s) = 1/(s+a). The third derivative of this expression can be

calculated by using:

‘X’ ` ‚¿ ‘X’ `‚¿ ‘X’ ` ‚¿ μ

The result is

‘-6/(X^4+4*a*X^3+6*a^2*X^2+4*a^3*X+a^4)’, or

2 2

d F/ds = -6/(s +4⋅a⋅s +6⋅a ⋅s +4⋅a ⋅s+a ).

Now, use ‘(-X)^3*EXP(-a*X)’ ` LAP μ. The result is exactly the same.

ꢀ Integration theorem. Let F(s) = L{f(t)}, then

{

f (u)du

}

= ⋅ F(s).

∫

ꢀ Convolution theorem. Let F(s) = L{f(t)} and G(s) = L{g(t)}, then

Page 16-13

{

}

f (u)g(t − u)du = L{( f * g)(t)} =

∫

L{ f (t)}⋅L{g(t)} = F(s)⋅G(s)

Example 4 – Using the convolution theorem, find the Laplace transform of

(f*g)(t), if f(t) = sin(t), and g(t) = exp(t). To find F(s) = L{f(t)}, and G(s) = L{g(t)},

use: ‘SIN(X)’ ` LAP μ. Result, ‘1/(X^2+1)’, i.e., F(s) = 1/(s +1).

Also, ‘EXP(X)’ ` LAP. Result, ‘1/(X-1)’, i.e., G(s) = 1/(s-1). Thus, L{(f*g)(t)} =

3 2

F(s)⋅G(s) = 1/(s +1)⋅1/(s-1) = 1/((s-1)(s +1)) = 1/(s -s +s-1).

ꢀ Shift theorem for a shift to the right. Let F(s) = L{f(t)}, then

–as

L{f(t-a)}=e ⋅L{f(t)} = e ⋅F(s).

ꢀ Shift theorem for a shift to the left. Let F(s) = L{f(t)}, and a >0, then

L{ f (t + a)}= e ⋅ F(s) − f (t)⋅e^−st⋅dt .

⎛

⎝

⎞

⎠

⎜

⎟

∫

ꢀ Similarity theorem. Let F(s) = L{f(t)}, and a>0, then L{f(a⋅t)} = (1/a)⋅F(s/a).

–bt

ꢀ Damping theorem. Let F(s) = L{f(t)}, then L{e ⋅f(t)} = F(s+b).

ꢀ Division theorem. Let F(s) = L{f(t)}, then

^∞F(u)du.

f (t)

⎧

⎨

⎩

⎫

⎬

⎭

∫

ꢀ Laplace transform of a periodic function of period T:

L{ f (t)} =

⋅

f (t)⋅e^−st⋅dt.

1− e^−sT

∫

•

Limit theorem for the initial value: Let F(s) = L{f(t)}, then

f₀= lim f (t) = lim[s ⋅ F(s)].

t→0

s→∞

Limit theorem for the final value: Let F(s) = L{f(t)}, then

Page 16-14

f_∞= lim f (t) = lim[s ⋅ F(s)].

t→∞

s→0

Dirac’s delta function and Heaviside’s step function

In the analysis of control systems it is customary to utilize a type of functions that

represent certain physical occurrences such as the sudden activation of a switch

(Heaviside’s step function, H(t)) or a sudden, instantaneous, peak in an input to

the system (Dirac’s delta function, δ(t)). These belong to a class of functions

known as generalized or symbolic functions [e.g., see Friedman, B., 1956,

Principles and Techniques of Applied Mathematics, Dover Publications Inc.,

New York (1990 reprint) ].

The formal definition of Dirac’s delta function, δ(x), is δ(x) = 0, for x ≠0, and

^∞δ (x)dx =1.0.

∫

−∞

∞

Also, if f(x) is a continuous function, then

f (x)δ (x − x₀)dx = f (x₀).

∫

−∞

An interpretation for the integral above, paraphrased from Friedman (1990), is

that the δ-function “picks out” the value of the function f(x) at x = x . Dirac’s

delta function is typically represented by an upward arrow at the point x = x0,

indicating that the function has a non-zero value only at that particular value of

x .

Heaviside’s step function, H(x), is defined as

1, x > 0

0, x < 0

⎧

⎨

⎩

H(x) =

Also, for a continuous function f(x),

∞

f (x)H(x − x₀)dx =

f (x)dx.

∫

−∞

x₀

Dirac’s delta function and Heaviside’s step function are related by dH/dx =

δ(x). The two functions are illustrated in the figure below.

Page 16-15

H(x x₀)

(x x₀)

x₀

You can prove that

from which it follows that

L{H(t)} = 1/s,

L{U ⋅H(t)} = U /s,

-1

where U is a constant. Also, L {1/s}=H(t),

-1

and

L { U /s}= U ⋅H(t).

–as

Also, using the shift theorem for a shift to the right, L{f(t-a)}=e ⋅L{f(t)} =

–as

–ks

⋅F(s), we can write L{H(t-k)}=e ⋅L{H(t)} = e ⋅(1/s) = (1/s)⋅e

Another important result, known as the second shift theorem for a shift to the

-1 –as

right, is that L {e ⋅F(s)}=f(t-a)⋅H(t-a), with F(s) = L{f(t)}.

In the calculator the Heaviside step function H(t) is simply referred to as ‘1’. To

check the transform in the calculator use: 1 ` LAP. The result is ‘1/X’, i.e.,

L{1} = 1/s. Similarly, ‘U0’ ` LAP , produces the result ‘U0/X’, i.e., L{U }

= U /s.

You can obtain Dirac’s delta function in the calculator by using: 1` ILAP

The result is

‘Delta(X)’.

This result is simply symbolic, i.e., you cannot find a numerical value for, say

‘Delta(5)’.

This result can be defined the Laplace transform for Dirac’s delta function,

-1

because from L {1.0}= δ(t), it follows that L{δ(t)} = 1.0

–as

Also, using the shift theorem for a shift to the right, L{f(t-a)}=e ⋅L{f(t)} =

–as

–ks

⋅F(s), we can write L{δ(t-k)}=e ⋅L{δ(t)} = e ⋅1.0 = e

Page 16-16

Applications of Laplace transform in the solution of linear ODEs

At the beginning of the section on Laplace transforms we indicated that you

could use these transforms to convert a linear ODE in the time domain into an

algebraic equation in the image domain. The resulting equation is then solved

for a function F(s) through algebraic methods, and the solution to the ODE is

found by using the inverse Laplace transform on F(s).

The theorems on derivatives of a function, i.e.,

L{df/dt} = s⋅F(s) - f ,

L{d f/dt } = s ⋅F(s) - s⋅f – (df/dt) ,

and, in general,

n-1

(n-2)

(n-1)

L{d f/dt } = s ⋅F(s) – s ⋅f −…– s⋅f

– f

are particularly useful in transforming an ODE into an algebraic equation.

Example 1 – To solve the first order equation,

–t

dh/dt + k⋅h(t) = a⋅e ,

by using Laplace transforms, we can write:

–t

L{dh/dt + k⋅h(t)} = L{a⋅e },

–t

L{dh/dt} + k⋅L{h(t)} = a⋅L{e }.

–t

Note: ‘EXP(-X)’ ` LAP , produces ‘1/(X+1)’, i.e., L{e }=1/(s+1).

With H(s) = L{h(t)}, and L{dh/dt} = s⋅H(s) - h , where h = h(0), the transformed

equation is

Use the calculator to solve for H(s), by writing:

‘X*H-h0+k*H=a/(X+1)’ ` ‘H’ ISOL

s⋅H(s)-h +k⋅H(s) = a/(s+1).

Page 16-17

The result is

‘H=((X+1)*h0+a)/(X^2+(k+1)*X+k)’.

To find the solution to the ODE, h(t), we need to use the inverse Laplace

transform, as follows:

OBJꢀ ƒ ƒ

ILAP μ

Isolates right-hand side of last expression

Obtains the inverse Laplace transform

The result is

simplifying, results in

. Replacing X with t in this expression and

-t

-kt

h(t) = a/(k-1)⋅e +((k-1)⋅h -a)/(k-1)⋅e .

Check what the solution to the ODE would be if you use the function LDEC:

‘a*EXP(-X)’ ` ‘X+k’ ` LDEC μ

The result is:

, i.e.,

h(t) = a/(k-1)⋅e +((k-1)⋅cC -a)/(k-1)⋅e .

-t

-kt

Thus, cC0 in the results from LDEC represents the initial condition h(0).

Note: When using the function LDEC to solve a linear ODE of order n in f(X),

the result will be given in terms of n constants cC0, cC1, cC2, …, cC(n-1),

(n-1)

representing the initial conditions f(0), f’(0), f”(0), …, f

(0).

Example 2 – Use Laplace transforms to solve the second-order linear equation,

d y/dt +2y = sin 3t.

Using Laplace transforms, we can write:

L{d y/dt +2y} = L{sin 3t},

L{d y/dt } + 2⋅L{y(t)} = L{sin 3t}.

Page 16-18

Note: ‘SIN(3*X)’ ` LAP μ produces ‘3/(X^2+9)’, i.e.,

L{sin 3t}=3/(s +9).

With Y(s) = L{y(t)}, and L{d y/dt } = s ⋅Y(s) - s⋅y – y , where y = h(0) and y

= h’(0), the transformed equation is

s ⋅Y(s) – s⋅y – y + 2⋅Y(s) = 3/(s +9).

Use the calculator to solve for Y(s), by writing:

‘X^2*Y-X*y0-y1+2*Y=3/(X^2+9)’ ` ‘Y’ ISOL

The result is

‘Y=((X^2+9)*y1+(y0*X^3+9*y0*X+3))/(X^4+11*X^2+18)’.

To find the solution to the ODE, y(t), we need to use the inverse Laplace

transform, as follows:

OBJꢀ ƒ ƒ

ILAPμ

Isolates right-hand side of last expression

Obtains the inverse Laplace transform

The result is

i.e.,

y(t) = -(1/7) sin 3x + y cos √2x + (√2 (7y +3)/14) sin √2x.

Check what the solution to the ODE would be if you use the function LDEC:

‘SIN(3*X)’ ` ‘X^2+2’ ` LDEC μ

The result is:

i.e., the same as before with cC0 = y0 and cC1 = y1.

Page 16-19

Note: Using the two examples shown here, we can confirm what we indicated

earlier, i.e., that function ILAP uses Laplace transforms and inverse transforms to

solve linear ODEs given the right-hand side of the equation and the

characteristic equation of the corresponding homogeneous ODE.

Example 3 – Consider the equation

d y/dt +y = δ(t-3),

where δ(t) is Dirac’s delta function.

Using Laplace transforms, we can write:

L{d y/dt +y} = L{δ(t-3)},

L{d y/dt } + L{y(t)} = L{δ(t-3)}.

With ‘Delta(X-3)’ ` LAP , the calculator produces EXP(-3*X), i.e., L{δ(t-3)}

–3s

= e . With Y(s) = L{y(t)}, and L{d y/dt } = s ⋅Y(s) - s⋅y – y , where y = h(0)

–3s

and y = h’(0), the transformed equation is s ⋅Y(s) – s⋅y – y + Y(s) = e . Use

the calculator to solve for Y(s), by writing:

‘X^2*Y-X*y0-y1+Y=EXP(-3*X)’ ` ‘Y’ ISOL

The result is

‘Y=(X*y0+(y1+EXP(-(3*X))))/(X^2+1)’.

To find the solution to the ODE, y(t), we need to use the inverse Laplace

transform, as follows:

OBJꢀ ƒ ƒ

ILAP

Isolates right-hand side of last expression

Obtains the inverse Laplace transform

The result is

‘y1*SIN(X)+y0*COS(X)+SIN(X-3)*Heaviside(X-3)’.

Page 16-20

Notes:

[1]. An alternative way to obtain the inverse Laplace transform of the

expression ‘(X*y0+(y1+EXP(-(3*X))))/(X^2+1)’ is by separating the

expression into partial fractions, i.e.,

‘y0*X/(X^2+1) + y1/(X^2+1) + EXP(-3*X)/(X^2+1)’,

and use the linearity theorem of the inverse Laplace transform

-1

L {a⋅F(s)+b⋅G(s)} = a⋅L {F(s)} + b⋅L {G(s)},

to write,

-1

–3s

L {y ⋅s/(s +1)+y /(s +1)) + e /(s +1)) } =

-1

-1 –3s

y ⋅L {s/(s +1)}+ y ⋅L {1/(s +1)}+ L {e /(s +1))},

Then, we use the calculator to obtain the following:

-1

‘X/(X^2+1)’ ` ILAP

Result, ‘COS(X)’, i.e., L {s/(s +1)}= cos t.

-1

‘1/(X^2+1)’ ` ILAP

Result, ‘SIN(X)’, i.e., L {1/(s +1)}= sin t.

‘EXP(-3*X)/(X^2+1)’ ` ILAP Result, SIN(X-3)*Heaviside(X-3)’.

[2]. The very last result, i.e., the inverse Laplace transform of the expression

‘(EXP(-3*X)/(X^2+1))’, can also be calculated by using the second shifting

theorem for a shift to the right

-1 –as

L {e ⋅F(s)}=f(t-a)⋅H(t-a),

if we can find an inverse Laplace transform for 1/(s +1). With the calculator,

-1 –3s

try ‘1/(X^2+1)’ ` ILAP. The result is ‘SIN(X)’. Thus, L {e /(s +1)}} =

sin(t-3)⋅H(t-3),

Check what the solution to the ODE would be if you use the function LDEC:

‘Delta(X-3)’ ` ‘X^2+1’ ` LDEC μ

Page 16-21

The result is:

‘SIN(X-3)*Heaviside(X-3) + cC1*SIN(X) + cC0*COS(X)’.

Please notice that the variable X in this expression actually represents the

variable t in the original ODE. Thus, the translation of the solution in paper may

be written as:

y(t) = Co ⋅cost + C₁⋅sint + sin(t − 3)⋅ H(t − 3)

When comparing this result with the previous result for y(t), we conclude that

cC = y , cC = y .

Defining and using Heaviside’s step function in the calculator

The previous example provided some experience with the use of Dirac’s delta

function as input to a system (i.e., in the right-hand side of the ODE describing

the system). In this example, we want to use Heaviside’s step function, H(t). In

the calculator we can define this function as:

‘H(X) = IFTE(X>0, 1, 0)’ `„à

This definition will create the variable @@@H@@@ in the calculator’s soft menu key.

Example 1 – To see a plot of H(t-2), for example, use a FUNCTION type of

plot (see Chapter 12):

ꢁ Press „ô, simultaneously in RPN mode, to access to the PLOT SETUP

window.

ꢁ Change TYPEto FUNCTION, if needed

ꢁ Change EQ to ‘H(X-2)’.

ꢁ Make sure that Indepis set to ‘X’.

ꢁ Press L @@@OK@@@ to return to normal calculator display.

ꢀ Press „ò, simultaneously, to access the PLOT window.

ꢁ Change the H-VIEW range to 0 to 20, and the V-VIEW range to -2 to 2.

ꢁ

Press @ERASE @DRAW to plot the function .

Page 16-22

Use of the function H(X) with LDEC, LAP, or ILAP, is not allowed in the calculator.

You have to use the main results provided earlier when dealing with the

-1

Heaviside step function, i.e., L{H(t)} = 1/s, L {1/s}=H(t),

–ks

-1 –as

L{H(t-k)}=e ⋅L{H(t)} = e ⋅(1/s) = ⋅(1/s)⋅e and L {e ⋅F(s)}=f(t-a)⋅H(t-a).

Example 2 – The function H(t-t ) when multiplied to a function f(t), i.e., H(t-t )f(t),

has the effect of switching on the function f(t) at t = t . For example, the solution

obtained in Example 3, above, was y(t) = y cos t + y sin t + sin(t-3)⋅H(t-3).

Suppose we use the initial conditions y = 0.5, and y = -0.25. Let’s plot this

function to see what it looks like:

ꢀ Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP

window.

ꢁ Change TYPEto FUNCTION, if needed

ꢁ Change EQ to ‘0.5*COS(X)-0.25*SIN(X)+SIN(X-3)*H(X-3)’.

ꢁ Make sure that Indepis set to ‘X’.

ꢁ H-VIEW: 0 20, V-VIEW: -3 2.

ꢁ Press @ERASE @DRAW to plot the function.

ꢁ Press @EDIT L@LABEL to see the plot.

The resulting graph will look like this:

Notice that the signal starts with a relatively small amplitude, but suddenly, at

t=3, it switches to an oscillatory signal with a larger amplitude. The difference

between the behavior of the signal before and after t = 3 is the “switching on”

of the particular solution y (t) = sin(t-3)⋅H(t-3). The behavior of the signal before

t = 3 represents the contribution of the homogeneous solution, y (t) = y cos t +

y sin t.

The solution of an equation with a driving signal given by a Heaviside step

function is shown below.

Example 3 – Determine the solution to the equation, d y/dt +y = H(t-3),

Page 16-23

where H(t) is Heaviside’s step function. Using Laplace transforms, we can

write: L{d y/dt +y} = L{H(t-3)}, L{d y/dt } + L{y(t)} = L{H(t-3)}. The last term in

–3s

this expression is: L{H(t-3)} = (1/s)⋅e . With Y(s) = L{y(t)}, and L{d y/dt } =

s ⋅Y(s) - s⋅y – y , where y = h(0) and y = h’(0), the transformed equation is

–3s

s ⋅Y(s) – s⋅y – y + Y(s) = (1/s)⋅e . Change CAS mode to Exact, if

necessary. Use the calculator to solve for Y(s), by writing:

‘X^2*Y-X*y0-y1+Y=(1/X)*EXP(-3*X)’ ` ‘Y’ ISOL

The result is

‘Y=(X^2*y0+X*y1+EXP(-3*X))/(X^3+X)’.

To find the solution to the ODE, y(t), we need to use the inverse Laplace

transform, as follows:

OBJꢀ ƒ ƒ

ILAP

Isolates right-hand side of last expression

Obtains the inverse Laplace transform

The result is

‘y1*SIN(X-1)+y0*COS(X-1)-(COS(X-3)-1)*Heaviside(X-3)’.

Thus, we write as the solution: y(t) = y cos t + y sin t + H(t-3)⋅(1+sin(t-3)).

Check what the solution to the ODE would be if you use the function LDEC:

‘H(X-3)’ `[ENTER] ‘X^2+1’ ` LDEC

The result is:

Please notice that the variable X in this expression actually represents the

variable t in the original ODE, and that the variable ttt in this expression is a

dummy variable. Thus, the translation of the solution in paper may be written

as:

∞

y(t) = Co⋅cost + C₁⋅sint + sint ⋅ H(u −3)⋅e^−ut⋅du.

∫

Page 16-24

Example 4 – Plot the solution to Example 3 using the same values of y and y

used in the plot of Example 1, above. We now plot the function

y(t) = 0.5 cos t –0.25 sin t + (1+sin(t-3))⋅H(t-3).

In the range 0 < t < 20, and changing the vertical range to (-1,3), the graph

should look like this:

Again, there is a new component to the motion switched at t=3, namely, the

particular solution y (t) = [1+sin(t-3)]⋅H(t-3), which changes the nature of the

solution for t>3.

The Heaviside step function can be combined with a constant function and with

linear functions to generate square, triangular, and saw tooth finite pulses, as

follows:

ꢀ Square pulse of size U in the interval a < t < b:

f(t) = Uo[H(t-a)-H(t-b)].

ꢀ Triangular pulse with a maximum value Uo, increasing from a < t < b,

decreasing from b < t < c:

f(t) = U ⋅ ((t-a)/(b-a)⋅[H(t-a)-H(t-b)]+(1-(t-b)/(b-c))[H(t-b)-H(t-c)]).

ꢀ Saw tooth pulse increasing to a maximum value Uo for a < t < b, dropping

suddenly down to zero at t = b:

f(t) = U ⋅ (t-a)/(b-a)⋅[H(t-a)-H(t-b)].

ꢀ Saw tooth pulse increasing suddenly to a maximum of Uo at t = a, then

decreasing linearly to zero for a < t < b:

Page 16-25

f(t) = U ⋅[1-(t-a)/(b-1)]⋅[H(t-a)-H(t-b)].

Examples of the plots generated by these functions, for Uo = 1, a = 2, b = 3, c

= 4, horizontal range = (0,5), and vertical range = (-1, 1.5), are shown in the

figures below:

Fourier series

Fourier series are series involving sine and cosine functions typically used to

expand periodic functions. A function f(x) is said to be periodic, of period T, if

f(x+T) = f(t). For example, because sin(x+2π) = sin x, and cos(x+2π) = cos x,

the functions sin and cos are 2π-periodic functions. If two functions f(x) and g(x)

are periodic of period T, then their linear combination h(x) = a⋅f(x) + b⋅g(x), is

also periodic of period T. A T-periodic function f(t) can be expanded into a

series of sine and cosine functions known as a Fourier series given by

∞

2nπ

⎛

⎜

⎝

⎞

⎟

⎠

f (t) = a +

a ⋅cos

t + b_n⋅sin

∑

n=1

where the coefficients a and b are given by

T / 2

2nπ

a₀=

f (t)⋅dt, a_n=

f (t)⋅cos

t ⋅dt,

∫

−T / 2

T / 2

2nπ

b_n=

f (t) ⋅sin

t ⋅ dt.

∫

−T / 2

Page 16-26

The following exercises are in ALG mode, with CAS mode set to Exact. (When

you produce a graph, the CAS mode will be reset to Approx. Make sure to set

it back to Exact after producing the graph.) Suppose, for example, that the

function f(t) = t +t is periodic with period T = 2. To determine the coefficients

a , a , and b for the corresponding Fourier series, we proceed as follows:

First, define function f(t) = t +t :

Next, we’ll use the Equation Writer to calculate the coefficients:

Thus, the first three terms of the function are:

f(t) ≈ 1/3 – (4/π )⋅cos (π⋅t)+(2/π)⋅sin (π⋅t).

A graphical comparison of the original function with the Fourier expansion

using these three terms shows that the fitting is acceptable for t < 1, or

thereabouts. But, then, again, we stipulated that T/2 = 1. Therefore, the fitting

is valid only between –1 < t < 1.

Page 16-27

Function FOURIER

An alternative way to define a Fourier series is by using complex numbers as

follows:

+∞

2inπt

f (t) =

c ⋅exp(

∑

n=−∞

where

2 ⋅i ⋅ n ⋅π

c_n=

f (t) ⋅ exp(

⋅t) ⋅ dt, n = −∞,...,−2,−1,0,1,2,...∞.

∫

Function FOURIER provides the coefficient c of the complex-form of the Fourier

series given the function f(t) and the value of n. The function FOURIER requires

you to store the value of the period (T) of a T-periodic function into the CAS

variable PERIOD before calling the function. The function FOURIER is available

in the DERIV sub-menu within the CALC menu („Ö).

Fourier series for a quadratic function

Determine the coefficients c , c , and c for the function f(t) = t +t, with period

T = 2. (Note: Because the integral used by function FOURIER is calculated in

the interval [0,T], while the one defined earlier was calculated in the interval

[-T/2,T/2], we need to shift the function in the t-axis, by subtracting T/2 from t,

i.e., we will use g(t) = f(t-1) = (t-1) +(t-1).)

Using the calculator in ALG mode, first we define functions f(t) and g(t):

Page 16-28

Next, we move to the CASDIR sub-directory under HOME to change the value

of variable PERIOD, e.g., „ (hold) §`J @CASDI `2 K @PERIOD

Return to the sub-directory where you defined functions f and g, and calculate

the coefficients (Accept change to Complex mode when requested):

Thus,

c = 1/3, c = (π⋅i+2)/π , c = (π⋅i+1)/(2π ).

The Fourier series with three elements will be written as

g(t) ≈ Re[(1/3) + (π⋅i+2)/π ⋅exp(i⋅π⋅t)+ (π⋅i+1)/(2π )⋅exp(2⋅i⋅π⋅t)].

A plot of the shifted function g(t) and the Fourier series fitting follows:

Page 16-29

The fitting is somewhat acceptable for 0<t<2, although not as good as in the

previous example.

A general expression for c

The function FOURIER can provide a general expression for the coefficient c of

the complex Fourier series expansion. For example, using the same function g(t)

as before, the general term c is given by (figures show normal font and small

font displays):

The general expression turns out to be, after simplifying the previous result,

(nπ + 2i) ⋅e^2inπ+ 2i²n²π + 3nπ − 2i

c_n=

2n³π ⋅e^2inπ

We can simplify this expression even further by using Euler’s formula for

2inπ

complex numbers, namely, e

= cos(2nπ) + i⋅sin(2nπ) = 1 + i⋅0 = 1, since

cos(2nπ) = 1, and sin(2nπ) = 0, for n integer.

Using the calculator you can simplify the expression in the equation writer

2inπ

(‚O) by replacing e

simplification:

= 1. The figure shows the expression after

Page 16-30

The result is

c = (i⋅n⋅π+2)/(n ⋅π ).

Putting together the complex Fourier series

Having determined the general expression for c , we can put together a finite

complex Fourier series by using the summation function (Σ) in the calculator as

follows:

ꢀ First, define a function c(n) representing the general term c in the complex

Fourier series.

ꢀ Next, define the finite complex Fourier series, F(X,k), where X is the

independent variable and k determines the number of terms to be used.

Ideally we would like to write this finite complex Fourier series as

2⋅i ⋅π ⋅n

F(X ,k) =

c(n)⋅exp(

⋅ X )

∑

n=−k

However, because the function c(n) is not defined for n = 0, we will be

better advised to re-write the expression as

F(X ,k,c0) = c0 +

2⋅i ⋅π ⋅ n

[c(n)⋅exp(

⋅ X ) + c(−n)⋅exp(−

⋅ X )],

∑

n=1

Page 16-31

Or, in the calculator entry line as:

DEFINE(‘F(X,k,c0) = c0+Σ(n=1,k,c(n)*EXP(2*i*π*n*X/T)+

c(-n)*EXP(-(2*i*π*n*X/T))’),

where T is the period, T = 2. The following screen shots show the definition of

function F and the storing of T = 2:

The function @@@F@@@ can be used to generate the expression for the complex

Fourier series for a finite value of k. For example, for k = 2, c = 1/3,and using

t as the independent variable, we can evaluate F(t,2,1/3) to get:

This result shows only the first term (c0) and part of the first exponential term in

the series. The decimal display format was changed to Fix with 2 decimals to

be able to show some of the coefficients in the expansion and in the exponent.

As expected, the coefficients are complex numbers.

The function F, thus defined, is fine for obtaining values of the finite Fourier

series. For example, a single value of the series, e.g., F(0.5,2,1/3), can be

obtained by using (CAS modes set to Exact, step-by-step, and Complex):

Page 16-32

Accept change to Approx mode if requested. The result is the value

–0.40467…. The actual value of the function g(0.5) is g(0.5) = -0.25. The

following calculations show how well the Fourier series approximates this value

as the number of components in the series, given by k, increases:

F (0.5, 1, 1/3) = (-0.303286439037,0.)

F (0.5, 2, 1/3) = (-0.404607622676,0.)

F (0.5, 3, 1/3) = (-0.192401031886,0.)

F (0.5, 4, 1/3) = (-0.167070735979,0.)

F (0.5, 5, 1/3) = (-0.294394690453,0.)

F (0.5, 6, 1/3) = (-0.305652599743,0.)

To compare the results from the series with those of the original function, load

these functions into the PLOT – FUNCTION input form („ñ, simultaneously

if using RPN mode):

Change the limits of the Plot Window („ò) as follows:

Press the soft-menu keys @ERASE @DRAW to produce the plot:

Notice that the series, with 5 terms, “hugs” the graph of the function very closely

in the interval 0 to 2 (i.e., through the period T = 2). You can also notice a

Page 16-33

periodicity in the graph of the series. This periodicity is easy to visualize by

expanding the horizontal range of the plot to (-0.5,4):

Fourier series for a triangular wave

Consider the function

x, if 0 < x <1

2 − x, if 1 < x < 2

⎧

⎨

⎩

g(x) =

which we assume to be periodic with period T = 2. This function can be

defined in the calculator, in ALG mode, by the expression

DEFINE(‘g(X) = IFTE(X<1,X,2-X)’)

If you started this example after finishing example 1 you already have a value

of 2 stored in CAS variable PERIOD. If you are not sure, check the value of this

variable, and store a 2 in it if needed. The coefficient c for the Fourier series

is calculated as follows:

The calculator will request a change to Approx mode because of the integration

of the function IFTE() included in the integrand. Accepting, the change to

Approx produces c = 0.5. If we now want to obtain a generic expression for

the coefficient c use:

Page 16-34

The calculator returns an integral that cannot be evaluated numerically because

it depends on the parameter n. The coefficient can still be calculated by

typing its definition in the calculator, i.e.,

i ⋅ 2⋅ n ⋅ π⋅ X

⎛

⎞

⎟

⎠

⋅

X ⋅ EXP −

⋅dX +

⎜

∫

⎝

i ⋅ 2⋅ n ⋅ π⋅ X

⎛

⎞

⎟

⎠

⋅

(2 − X)⋅ EXP −

⋅dX

⎜

∫

⎝

where T = 2 is the period. The value of T can be stored using:

Typing the first integral above in the Equation Writer, selecting the entire

expression, and using @EVAL@, will produce the following:

inπ

Recall the e

= cos(nπ) + i⋅sin(nπ) = (-1) . Performing this substitution in the

result above we have:

Page 16-35

Press `` to copy this result to the screen. Then, reactivate the Equation

Writer to calculate the second integral defining the coefficient c , namely,

inπ

2inπ

Once again, replacing e

= (-1) , and using e

= 1, we get:

Press `` to copy this second result to the screen. Now, add ANS(1) and

ANS(2) to get the full expression for c :

Pressing ˜ will place this result in the Equation Writer, where we can simplify

(@SIMP@) it to read:

inπ

Once again, replacing e

= (-1) , results in

Page 16-36

This result is used to define the function c(n) as follows:

DEFINE(‘c(n) = - (((-1)^n-1)/(n^2*π^2*(-1)^n)’)

i.e.,

Next, we define function F(X,k,c0) to calculate the Fourier series (if you

completed example 1, you already have this function stored):

DEFINE(‘F(X,k,c0) = c0+Σ(n=1,k,c(n)*EXP(2*i*π*n*X/T)+

c(-n)*EXP(-(2*i*π*n*X/T))’),

To compare the original function and the Fourier series we can produce the

simultaneous plot of both functions. The details are similar to those of example

1, except that here we use a horizontal range of 0 to 2 and a vertical range

from 0 to 1, and adjust the equations to plot as shown here:

The resulting graph is shown below for k = 5 (the number of elements in the

series is 2k+1, i.e., 11, in this case):

Page 16-37

From the plot it is very difficult to distinguish the original function from the

Fourier series approximation. Using k = 2, or 5 terms in the series, shows not so

good a fitting:

The Fourier series can be used to generate a periodic triangular wave (or saw

tooth wave) by changing the horizontal axis range, for example, from –2 to 4.

The graph shown below uses k = 5:

Fourier series for a square wave

A square wave can be generated by using the function

0, if 0 < x <1

g(x) = 1, if 1 < x < 3

⎧

⎪

⎨

⎪

0, if 3 < x < 4

⎩

Page 16-38

In this case, the period T, is 4. Make sure to change the value of variable @@@T@@@

to 4 (use: 4 K @@@T@@ `). Function g(X) can be defined in the calculator

by using

DEFINE(‘g(X) = IFTE((X>1) AND (X<3),1,0)’)

The function plotted as follows (horizontal range: 0 to 4, vertical range:0 to 1.2

Using a procedure similar to that of the triangular shape in example 2, above,

you can find that

⎛

⎜

⎞

⎠

c₀=

⋅

1⋅ dX = 0.5

⎟

∫

⎝

and

inπ/2

3inπ/2

We can simplify this expression by using e

= i and e

= (-i) to get:

Page 16-39

The simplification of the right-hand side of c(n), above, is easier done on paper

(i.e., by hand). Then, retype the expression for c(n) as shown in the figure to the

left above, to define function c(n). The Fourier series is calculated with

F(X,k,c0), as in examples 1 and 2 above, with c0 = 0.5. For example, for k =

5, i.e., with 11 components, the approximation is shown below:

A better approximation is obtained by using k = 10, i.e.,

For k = 20, the fitting is even better, but it takes longer to produce the graph:

Fourier series applications in differential equations

Suppose we want to use the periodic square wave defined in the previous

example as the excitation of an undamped spring-mass system whose

homogeneous equation is: d y/dX + 0.25y = 0.

We can generate the excitation force by obtaining an approximation with k

=10 out of the Fourier series by using SW(X) = F(X,10,0.5):

Page 16-40

We can use this result as the first input to the function LDEC when used to obtain

a solution to the system d y/dX + 0.25y = SW(X), where SW(X) stands for

Square Wave function of X. The second input item will be the characteristic

equation corresponding to the homogeneous ODE shown above, i.e.,

‘X^2+0.25’ .

With these two inputs, function LDEC produces the following result (decimal

format changed to Fix with 3 decimals).

Pressing ˜ allows you to see the entire expression in the Equation writer.

Exploring the equation in the Equation Writer reveals the existence of two

constants of integration, cC0 and cC1. These values would be calculated using

initial conditions. Suppose that we use the values cC0 = 0.5 and cC1 = -0.5,

we can replace those values in the solution above by using function SUBST (see

Chapter 5). For this case, use SUBST(ANS(1),cC0=0.5) `, followed by

SUBST(ANS(1),cC1=-0.5) `. Back into normal calculator display we can

use:

The latter result can be defined as a function, FW(X), as follows (cutting and

pasting the last result into the command):

We can now plot the real part of this function. Change the decimal mode to

Standard, and use the following:

Page 16-41

The solution is shown below:

Fourier Transforms

Before presenting the concept of Fourier transforms, we’ll discuss the general

definition of an integral transform. In general, an integral transform is a

transformation that relates a function f(t) to a new function F(s) by an integration

F(s) = ^bκ (s,t) ⋅ f (t)⋅ dt.

of the form

The function κ(s,t) is known as the

∫

kernel of the transformation.

The use of an integral transform allows us to resolve a function into a given

spectrum of components. To understand the concept of a spectrum, consider

the Fourier series

∞

f (t) = a +

(

a ⋅cosω x + b ⋅sinω x

)

∑

n=1

representing a periodic function with a period T. This Fourier series can be re-

∞

f (x) = a + A ⋅cos(ϖ x +φ ),

written as

where

∑

n=1

⎛

⎞

b_n

a_n

A_n= a_n²+ b_n², φ_n= tan⁻¹

⎜

⎟

⎝

⎠

for n =1,2, …

Page 16-42

The amplitudes A will be referred to as the spectrum of the function and will be

a measure of the magnitude of the component of f(x) with frequency f = n/T.

The basic or fundamental frequency in the Fourier series is f = 1/T, thus, all

other frequencies are multiples of this basic frequency, i.e., f = n⋅f . Also, we

can define an angular frequency, ω = 2nπ/T = 2π⋅f = 2π⋅ n⋅f = n⋅ω , where

ω is the basic or fundamental angular frequency of the Fourier series.

Using the angular frequency notation, the Fourier series expansion is written as

∞

f (x) = a + A ⋅cos(ω x +φ ).

∑

n=1

∞

= a +

(

a ⋅cosω x + b ⋅sinω x

)

∑

n=1

A plot of the values A vs. ω is the typical representation of a discrete

spectrum for a function. The discrete spectrum will show that the function has

components at angular frequencies ω which are integer multiples of the

fundamental angular frequency ω .

Suppose that we are faced with the need to expand a non-periodic function into

sine and cosine components. A non-periodic function can be thought of as

having an infinitely large period. Thus, for a very large value of T, the

fundamental angular frequency, ω = 2π/T, becomes a very small quantity, say

Δω. Also, the angular frequencies corresponding to ω = n⋅ω = n⋅Δω, (n = 1,

2, …, ∞), now take values closer and closer to each other, suggesting the need

for a continuous spectrum of values.

The non-periodic function can be written, therefore, as

f (x) = ^∞[C(ω) ⋅cos(ω ⋅ x) + S(ω) ⋅sin(ω ⋅ x)]dω,

∫

where

∞

C(ω) =

⋅

f (x) ⋅cos(ω ⋅ x)⋅ dx,

∫

−∞

2π

Page 16-43

and

∞

S(ω) =

⋅

f (x) ⋅sin(ω ⋅ x)⋅ dx.

∫

−∞

2π

The continuous spectrum is given by

A(ω) = [C(ω)]²+[S(ω)]²

The functions C(ω), S(ω), and A(ω) are continuous functions of a variable ω,

which becomes the transform variable for the Fourier transforms defined below.

Example 1 – Determine the coefficients C(ω), S(ω), and the continuous spectrum

A(ω), for the function f(x) = exp(-x), for x > 0, and f(x) = 0, x < 0.

In the calculator, set up and evaluate the following integrals to calculate C(ω)

and S(ω), respectively. CAS modes are set to Exact and Real.

Their results are, respectively:

The continuous spectrum, A(ω) is calculated as:

Page 16-44

Define this expression as a function by using function DEFINE („à). Then,

plot the continuous spectrum, in the range 0 < ω < 10, as:

Definition of Fourier transforms

Different types of Fourier transforms can be defined. The following are the

definitions of the sine, cosine, and full Fourier transforms and their inverses used

in this Chapter.

Fourier sine transform

∞

Fs{ f (t)} = F(ω) =

⋅

f (t)⋅sin(ω ⋅t)⋅ dt

∫

Inverse sine transform

F_s⁻¹{F(ω)} = f (t) = ^∞F(ω)⋅sin(ω ⋅t)⋅ dt

∫

Fourier cosine transform

∞

Fc{ f (t)} = F(ω) =

⋅

f (t)⋅cos(ω ⋅t)⋅ dt

∫

Inverse cosine transform

F_c⁻¹{F(ω)} = f (t) = ^∞F(ω)⋅cos(ω ⋅t)⋅ dt

∫

Fourier transform (proper)

∞

F{ f (t)}= F(ω) =

⋅

f (t)⋅e^−iωt⋅dt

∫

−∞

2π

Inverse Fourier transform (proper)

∞

F⁻¹{F(ω)} = f (t) =

_−∞F(ω)⋅e^−iωt⋅dt

∫

2π

Example 1 – Determine the Fourier transform of the function f(t) = exp(- t), for t

>0, and f(t) = 0, for t<0.

Page 16-45

The continuous spectrum, F(ω), is calculated with the integral:

^∞e^{−(1+iω )t}dt = lim

^εe^{−(1+iω )t}dt

∫

ε→∞

2π

1− exp(−(1+ iω)t)

⎡

⎢

⎤

= lim

⋅

⎥

ε →∞

1+ iω

2π

⎣

⎦

This result can be rationalized by multiplying numerator and denominator by

the conjugate of the denominator, namely, 1-iω. The result is now:

1−iω

⎛

⎞ ⎛

⎞

⎟

⎠

F(ω) =

⋅

⎟ ⎜

⎜

2π 1+ iω 2π 1+ iω

⎝

⎠ ⎝

1+ω

⎛

⎞

⎟

⎠

−i ⋅

⎜

2π 1+ω

⎝

which is a complex function.

The absolute value of the real and imaginary parts of the function can be

plotted as shown below

Notes:

The magnitude, or absolute value, of the Fourier transform, |F(ω)|, is the

frequency spectrum of the original function f(t). For the example shown above,

2 1/2

|F(ω)| = 1/[2π(1+ω )] . The plot of |F(ω)| vs. ω was shown earlier.

Some functions, such as constant values, sin x, exp(x), x , etc., do not have

Fourier transform. Functions that go to zero sufficiently fast as x goes to infinity

do have Fourier transforms.

Page 16-46

Properties of the Fourier transform

Linearity: If a and b are constants, and f and g functions, then F{a⋅f + b⋅g} = a

F{f }+ b F{g}.

Transformation of partial derivatives. Let u = u(x,t). If the Fourier transform

transforms the variable x, then

F{∂u/∂x} = iω F{u}, F{∂ u/∂x } = -ω F{u},

F{∂u/∂t} = ∂F{u}/∂t, F{∂ u/∂t } = ∂ F{u}/∂t

Convolution: For Fourier transform applications, the operation of convolution is

defined as

( f * g)(x) =

⋅

f (x −ξ)⋅ g(ξ) ⋅ dξ.

∫

2π

The following property holds for convolution:

F{f*g} = F{f}⋅F{g}.

Fast Fourier Transform (FFT)

The Fast Fourier Transform is a computer algorithm by which one can calculate

very efficiently a discrete Fourier transform (DFT). This algorithm has

applications in the analysis of different types of time-dependent signals, from

turbulence measurements to communication signals.

The discrete Fourier transform of a sequence of data values {x }, j = 0, 1, 2, …,

n-1, is a new finite sequence {X }, defined as

n−1

X _k=

x ⋅exp(−i ⋅ 2πkj / n),

k = 0,1,2,...,n −1

∑

j=0

The direct calculation of the sequence X involves n products, which would

involve enormous amounts of computer (or calculator) time particularly for large

values of n. The Fast Fourier Transform reduces the number of operations to the

order of n⋅log n. For example, for n = 100, the FFT requires about 664

operations, while the direct calculation would require 10,000 operations. Thus,

Page 16-47

the number of operations using the FFT is reduced by a factor of 10000/664 ≈

15.

The FFT operates on the sequence {x } by partitioning it into a number of shorter

sequences. The DFT’s of the shorter sequences are calculated and later

combined together in a highly efficient manner. For details on the algorithm

refer, for example, to Chapter 12 in Newland, D.E., 1993, “An Introduction to

Random Vibrations, Spectral & Wavelet Analysis – Third Edition,” Longman

Scientific and Technical, New York.

The only requirement for the application of the FFT is that the number n be a

power of 2, i.e., select your data so that it contains 2, 4, 8, 16, 32, 62, etc.,

points.

Examples of FFT applications

FFT applications usually involve data discretized from a time-dependent signal.

The calculator can be fed that data, say from a computer or a data logger, for

processing. Or, you can generate your own data by programming a function

and adding a few random numbers to it.

Example 1 – Define the function f(x) = 2 sin (3x) + 5 cos(5x) + 0.5*RAND,

where RAND is the uniform random number generator provided by the

calculator. Generate 128 data points by using values of x in the interval

(0,12.8). Store those values in an array, and perform a FFT on the array.

First, we define the function f(x) as a RPN program:

<< ꢀ x ‘2*SIN(3*x) + 5*COS(5*x)’ EVAL RAND 5 * + ꢀNUM >>

and store this program in variable @@@@f@@@. Next, type the following program to

generate 2 data values between a and b. The program will take the values of

m, a, and b:

<< ꢀ m a b << ‘2^m’ EVAL ꢀ n << ‘(b-a)/(n+1)’ EVAL ꢀ Dx << 1 n FOR j

‘a+(j-1)*Dx’ EVAL f NEXT n ꢀARRY >> >> >> >>

Store this program under the name GDATA (Generate DATA). Then, run the

program for the values, m = 5, a = 0, b = 100. In RPN mode, use:

5#0#100@GDATA!

Page 16-48

The figure below is a box plot of the data produced. To obtain the graph, first

copy the array just created, then transform it into a column vector by using:

OBJꢀ 1 + ꢀARRY (Functions OBJꢀ and ꢀARRY are available in the

command catalog, ‚N). Store the array into variable ΣDAT by using

function STOΣ (also available through ‚N). Select Bar in the TYPE for

graphs, change the view window to H-VIEW: 0 32, V-VIEW: -10 10, and

BarWidth to 1. Press @CANCL $ to return to normal calculator display.

To perform the FFT on the array in stack level 1 use function FFT available in the

MTH/FFT menu on array ΣDAT: @£DAT FFT. The FFT returns an array of complex

numbers that are the arrays of coefficients X of the DFT. The magnitude of the

coefficients X represents a frequency spectrum of the original data. To obtain

the magnitude of the coefficients you could transform the array into a list, and

then apply function ABS to the list. This is accomplished by using: OBJꢀ μ

ƒ ꢀLIST „Ê

Finally, you can convert the list back to a column vector to be stored in ΣDAT, as

follows:

OBJꢀ 1 ` 2 ꢀLIST ꢀARRY STOΣ

To plot the spectrum, follow the instructions for producing a bar plot given

earlier. The vertical range needs to be changed to –1 to 80. The spectrum of

frequencies is the following:

The spectrum shows two large components for two frequencies (these are the

sinusoidal components, sin (3x) and cos(5x)), and a number of smaller

components for other frequencies.

Page 16-49

Example 2 – To produce the signal given the spectrum, we modify the program

GDATA to include an absolute value, so that it reads:

<< ꢀ m a b << ‘2^m’ EVAL ꢀ n << ‘(b-a)/(n+1)’ EVAL ꢀ Dx << 1 n FOR j

‘a+(j-1)*Dx’ EVAL f ABS NEXT n ꢀARRY >> >> >> >>

Store this version of the program under GSPEC (Generate SPECtrum). Run the

program with m = 6, a = 0, b = 100. In RPN mode, use:

6#0#100@GSPEC!

Press ` when done, to keep an additional copy of the spectrum array.

Convert this row vector into a column vector and store it into ΣDAT. Following

the procedure for generating a bar plot, the spectrum generated for this

example looks as shown below. The horizontal range in this case is 0 to 64,

while the vertical range is –1 to 10:

To reproduce the signal whose spectrum is shown , use function IFFT. Since we

left a copy of the spectrum in the stack (a row vector), all you need to do if find

function IFFT in the MTH/FFT menu or through the command catalog, ‚N.

As an alternative, you could simply type the function name, i.e., type

~~ifft`. The signal is shown as an array (row vector) with

complex numbers. We are interested only in the real part of the elements. To

extract the real part of the complex numbers, use function RE from the CMPLX

menu (see Chapter 4), e.g., type ~~re`. What results is another

row vector. Convert it into a column vector, store it into ΣDAT, and plot a bar

plot to show the signal. The signal for this example is shown below, using a

horizontal range of 0 to 64, and a vertical range of –1 to 1:

Page 16-50

Except for a large peak at t = 0, the signal is mostly noise. A smaller vertical

scale (-0.5 to 0.5) shows the signal as follows:

Solution to specific second-order differential equations

In this section we present and solve specific types of ordinary differential

equations whose solutions are defined in terms of some classical functions, e.g.,

Bessel’s functions, Hermite polynomials, etc. Examples are presented in RPN

mode.

The Cauchy or Euler equation

An equation of the form x ⋅(d y/dx ) + a⋅x⋅ (dy/dx) + b⋅y = 0, where a and b

are real constants, is known as the Cauchy or Euler equation. A solution to the

Cauchy equation can be found by assuming that y(x) = x .

Type the equation as: ‘x^2*d1d1y(x)+a*x*d1y(x)+b*y(x)=0’ `

Then, type and substitute the suggested solution: ‘y(x) = x^n’ ` @SUBST

The result is: ‘x^2*(n*(x^(n-1-1)*(n-1)))+a*x*(n*x^(n-1))+b*x^n =0, which

simplifies to ‘n*(n-1)*x^n+a*n*x^n+b*x^n = 0’. Dividing by x^n, results in

an auxiliary algebraic equation: ‘n*(n-1)+a*n+b = 0’, or.

n²+ (a −1) ⋅ n + b = 0

ꢀ If the equation has two different roots, say n and n , then the general

solution of this equation is y(x) = K ⋅x

+ K ⋅x

ꢀ If b = (1-a) /4, then the equation has a double root n = n = n =

(1-a)/2, and the solution turns out to be y(x) = (K + K ⋅ln x)x .

Legendre’s equation

An equation of the form (1-x )⋅(d y/dx )-2⋅x⋅ (dy/dx)+n⋅ (n+1) ⋅y = 0, where n

is a real number, is known as the Legendre’s differential equation. Any solution

for this equation is known as a Legendre’s function. When n is a nonnegative

integer, the solutions are called Legendre’s polynomials. Legendre’s

polynomial of order n is given by

Page 16-51

(2n − 2m)!

P (x) = (−1)^m⋅

⋅x^n−2m

∑

2ⁿ⋅ m!⋅(n − m)!⋅(n − 2m)!

m=0

(2n)!

2ⁿ⋅(n!)²

(2n − 2)!

⋅ xⁿ−

⋅ xⁿ⁻²+ ... −..

2ⁿ⋅1!⋅(n −1)!(n − 2)!

where M = n/2 or (n-1)/2, whichever is an integer.

Legendre’s polynomials are pre-programmed in the calculator and can be

recalled by using the function LEGENDRE given the order of the polynomial, n.

The function LEGENDRE can be obtained from the command catalog (‚N)

or through the menu ARITHMETIC/POLYNOMIAL menu (see Chapter 5). In

RPN mode, the first six Legendre polynomials are obtained as follows:

0 LEGENDRE, result: 1,

1 LEGENDRE, result: ‘X’,

i.e.,

P (x) = 1.0.

P (x) = x.

2 LEGENDRE, result: ‘(3*X^2-1)/2’,

i.e.,

P (x) = (3x -1)/2.

3 LEGENDRE, result: ‘(5*X^3-3*X)/2’,

i.e.,

P (x) =(5x -3x)/2.

4 LEGENDRE, result: ‘(35*X^4-30*X^2+3)/8’, i.e.,

P (x) =(35x -30x +3)/8.

5 LEGENDRE, result: ‘(63*X^5-70*X^3+15*X)/8’, i.e.,

P (x) =(63x -70x +15x)/8.

The ODE (1-x )⋅(d y/dx )-2⋅x⋅ (dy/dx)+[n⋅ (n+1)-m /(1-x )] ⋅y = 0, has for

2 m/2

solution the function y(x) = P (x)= (1-x )

⋅(d Pn/dx ). This function is

referred to as an associated Legendre function.

Bessel’s equation

2 2

The ordinary differential equation x ⋅(d y/dx ) + x⋅ (dy/dx)+ (x -ν ) ⋅y = 0,

where the parameter ν is a nonnegative real number, is known as Bessel’s

differential equation. Solutions to Bessel’s equation are given in terms of

Bessel functions of the first kind of order ν:

(−1)^m⋅ x^2m

⋅ m!⋅Γ(ν + m +1)

∞

J (x) = x^ν⋅

∑

2m+ν

m=0

Page 16-52

where ν is not an integer, and the function Gamma Γ(α) is defined in Chapter

If ν = n, an integer, the Bessel functions of the first kind for n = integer are

defined by

(−1)^m⋅ x^2m

∞

J (x) = xⁿ⋅

∑

2m+n

⋅ m!⋅(n + m)!

m=0

Regardless of whether we use ν (non-integer) or n (integer) in the calculator, we

can define the Bessel functions of the first kind by using the following finite

series:

Thus, we have control over the function’s order, n, and of the number of

elements in the series, k. Once you have typed this function, you can use

function DEFINE to define function J(x,n,k). This will create the variable @@@J@@@ in

the soft-menu keys. For example, to evaluate J (0.1) using 5 terms in the series,

calculate J(0.1,3,5), i.e., in RPN mode: .1#3#5@@@J@@@ The

result is 2.08203157E-5.

If you want to obtain an expression for J (x) with, say, 5 terms in the series, use

J(x,0,5). The result is

‘1-0.25*x^2+0.015625*x^4-4.3403777E-4*x^6+6.782168E-6*x^8-

6.78168*x^10’.

For non-integer values ν, the solution to the Bessel equation is given by

y(x) = K ⋅J (x)+K ⋅J (x).

2 -ν

For integer values, the functions Jn(x) and J-n(x) are linearly dependent, since

J (x) = (-1) ⋅J (x),

-n

therefore, we cannot use them to obtain a general function to the equation.

Instead, we introduce the Bessel functions of the second kind defined as

Page 16-53

Y (x) = [J (x) cos νπ – J (x)]/sin νπ,

−ν

for non-integer ν, and for n integer, with n > 0, by

(−1)^m−1⋅(h_m+ h_m+n

)

∞

xⁿ

Y_n(x) = ⋅ J_n(x)⋅(ln +γ ) +

⋅

⋅x^2m

∑

2^2m+n⋅m!⋅(m + n)!

m=0

x⁻^{n n−1}(n − m −1)!

−

⋅

⋅x^2m

∑

2^2m−n⋅m!

m=0

where γ is the Euler constant, defined by

γ = lim[1+ + +... + − ln r] ≈ 0.57721566490...,

r→∞

and h represents the harmonic series

h_m= 1+ + + ... +

For the case n = 0, the Bessel function of the second kind is defined as

(−1)^m−1⋅ h_m

∞

⎡

⎢

⎣

⎤

⎥

⎦

Y₀(x) = ⋅ J (x)⋅(ln + γ ) +

⋅x^2m

∑

2^2m⋅(m!)²

m=0

With these definitions, a general solution of Bessel’s equation for all values of ν

is given by y(x) = K ⋅J (x)+K ⋅Y (x).

In some instances, it is necessary to provide complex solutions to Bessel’s

equations by defining the Bessel functions of the third kind of order ν as

(1)

(2)

(x) = J (x)+i⋅Y (x), and H (x) = J (x)−i⋅Y (x),

These functions are also known as the first and second Hankel functions of order

ν.

In some applications you may also have to utilize the so-called modified Bessel

-ν

functions of the first kind of order ν defined as I (x)= i ⋅J (i⋅x), where i is the unit

imaginarynumber. Thesefunctionsaresolutionstothedifferentialequation

x ⋅(d y/dx ) + x⋅ (dy/dx)- (x +ν ) ⋅y = 0.

Page 16-54

The modified Bessel functions of the second kind,

K (x) = (π/2)⋅[I (x)−I (x)]/sin νπ,

-ν

are also solutions of this ODE.

You can implement functions representing Bessel’s functions in the calculator in

a similar manner to that used to define Bessel’s functions of the first kind, but

keeping in mind that the infinite series in the calculator need to be translated

into a finite series.

Chebyshev or Tchebycheff polynomials

-1

2 1/2

The functions T (x) = cos(n⋅cos x), and U (x) = sin[(n+1) cos x]/(1-x )

n = 0, 1, … are called Chebyshev or Tchebycheff polynomials of the first and

second kind, respectively. The polynomials Tn(x) are solutions of the differential

equation (1-x )⋅(d y/dx ) − x⋅ (dy/dx) + n ⋅y = 0.

In the calculator the function TCHEBYCHEFF generates the Chebyshev or

Tchebycheff polynomial of the first kind of order n, given a value of n > 0. If

the integer n is negative (n < 0), the function TCHEBYCHEFF generates a

Tchebycheff polynomial of the second kind of order n whose definition is

U (x) = sin(n⋅arccos(x))/sin(arccos(x)).

You can access the function TCHEBYCHEFF through the command catalog

(‚N).

The first four Chebyshev or Tchebycheff polynomials of the first and second kind

are obtained as follows:

0 TCHEBYCHEFF, result: 1,

-0 TCHEBYCHEFF, result: 1,

1 TCHEBYCHEFF, result: ‘X’,

-1 TCHEBYCHEFF, result: 1,

2 TCHEBYCHEFF, result: ‘2*X^2-1,

-2 TCHEBYCHEFF, result: ‘2*X’,

i.e.,

T (x) = 1.0.

U (x) = 1.0.

T (x) = x.

U (x) =1.0.

T (x) =2x -1.

U (x) =2x.

3 TCHEBYCHEFF, result: ‘4*X^3-3*X’, i.e.,

-3 TCHEBYCHEFF, result: ‘4*X^2-1’, i.e.,

T (x) = 4x -3x.

U (x) = 4x -1.

Page 16-55

Laguerre’s equation

Laguerre’s equation is the second-order, linear ODE of the form x⋅(d y/dx )

+(1−x)⋅ (dy/dx) + n⋅y = 0. Laguerre polynomials, defined as

e^xd ⁿ(xⁿ⋅e^−x)

L₀(x) =1, L_n(x) =

⋅

, n =1,2,...

dxⁿ

are solutions to Laguerre’s equation. Laguerre’s polynomials can also be

calculated with:

(−1)^m

⎛ ⎞

⎜ ⎟

L (x) =

⋅

⋅ x .

∑

⎝ ⎠

m=0

n(n −1)

(−1)ⁿ

=1− n⋅ x +

⋅ x²−...+....+

⋅ xⁿ

The term

⎛ ⎞

m!(n − m)!

⎜ ⎟

= C(n,m)

⎝ ⎠

is the m-th coefficient of the binomial expansion (x+y) . It also represents the

number of combinations of n elements taken m at a time. This function is

available in the calculator as function COMB in the MTH/PROB menu (see also

Chapter 17).

You can define the following function to calculate Laguerre’s polynomials:

When done typing it in the equation writer press use function DEFINE to create

the function L(x,n) into variable @@@L@@@ .

To generate the first four Laguerre polynomials use, L(x,0), L(x,1), L(x,2), L(x,3).

The results are:

L (x) = .

L (x) = 1-x.

Page 16-56

L (x) = 1-2x+ 0.5x

L (x) = 1-3x+1.5x -0.16666…x .

Weber’s equation and Hermite polynomials

Weber’s equation is defined as d y/dx +(n+1/2-x /4)y = 0, for n = 0, 1, 2,

… A particular solution of this equation is given by the function , y(x) =

exp(-x /4)H (x/√2), where the function H (x) is the Hermite polynomial:

d ⁿ

dxⁿ

H₀* =1, H_n*(x) = (−1)ⁿe^x

(e^−x), n = 1,2,..

In the calculator, the function HERMITE, available through the menu

ARITHMETIC/POLYNOMIAL. Function HERMITE takes as argument an integer

number, n, and returns the Hermite polynomial of n-th degree. For example, the

first four Hermite polynomials are obtained by using:

0 HERMITE, result: 1,

i.e., H = 1.

1 HERMITE, result: ’2*X’,

2 HERMITE, result: ’4*X^2-2’,

3 HERMITE, result: ’8*X^3-12*X’,

i.e., H = 2x.

i.e., H = 4x -2.

i.e., H = 8x -12x.

Numerical and graphical solutions to ODEs

Differential equations that cannot be solved analytically can be solved

numerically or graphically as illustrated below.

Numerical solution of first-order ODE

Through the use of the numerical solver (‚Ï), you can access an input

form that lets you solve first-order, linear ordinary differential equations. The

use of this feature is presented using the following example. The method used

in the solution is a fourth-order Runge-Kutta algorithm preprogrammed in the

calculator.

Example 1 -- Suppose we want to solve the differential equation, dv/dt = -1.5

1/2

, with v = 4 at t = 0. We are asked to find v for t = 2.

Page 16-57

First, create the expression defining the derivative and store it into variable EQ.

The figure to the left shows the ALG mode command, while the right-hand side

figure shows the RPN stack before pressing K.

Then, enter the NUMERICAL SOLVER environment and select the differential

equation solver: ‚Ï˜@@@OK@@@ . Enter the following parameters:

To solve, press: @SOLVE (wait) @EDIT@. The result is 0.2499 ≈ 0.25. Press @@@OK@@@.

Solution presented as a table of values

Suppose we wanted to produce a table of values of v, for t = 0.00, 0.25, …,

2.00, we will proceed as follows:

First, prepare a table to write down your results. Write down in your table the

step-by-step results:

0.00

0.25

…

2.00

Next, within the SOLVE environment, change the final value of the independent

variable to 0.25, use :

—.25@@OK@@ ™™ @SOLVE (wait) @EDIT

(Solves for v at t = 0.25, v = 3.285 …. )

@@OK@@ INIT+ — . 5 @@OK@@ ™™@SOLVE (wait) @EDIT

(Changes initial value of t to 0.25, and final value of t to 0.5, solve for v(0.5) =

2.640…)

Page 16-58

@@OK@@ @INIT+—.75@@OK@@ ™™@SOLVE (wait) @EDIT

(Changes initial value of t to 0.5, and final value of t to 0.75, solve for v(0.75)

= 2.066…)

@@OK@@ @INIT+—1@@OK@@ ™ ™ @SOLVE (wait) @EDIT

(Changes initial value of t to 0.75, and final value of t to 1, solve for v(1) =

1.562…)

Repeat for t = 1.25, 1.50, 1.75, 2.00. Press @@OK@@ after viewing the last result in

@EDIT. To return to normal calculator display, press $ or L@@OK@@. The

different solutions will be shown in the stack, with the latest result in level 1.

The final results look as follows (rounded to the third decimal):

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

4.000

3.285

2.640

2.066

1.562

1.129

0.766

0.473

0.250

Graphical solution of first-order ODE

When we can not obtain a closed-form solution for the integral, we can always

plot the integral by selecting Diff Eqin the TYPEfield of the PLOT

environment as follows: suppose that we want to plot the position x(t) for a

velocity function v(t) = exp(-t ), with x = 0 at t = 0. We know there is no

closed-form expression for the integral, however, we know that the definition of

v(t) is dx/dt = exp(-t ).

The calculator allows for the plotting of the solution of differential equations of

the form Y'(T) = F(T,Y). For our case, we let Y = x and T = t, therefore, F(T,Y) =

f(t, x) = exp(-t ). Let's plot the solution, x(t), for t = 0 to 5, by using the following

keystroke sequence:

Page 16-59

ꢀ „ô (simultaneously, if in RPN mode) to enter PLOT environment

ꢀ Highlight the field in front of TYPE, using the —˜keys. Then, press

@CHOOS, and highlight Diff Eq, using the —˜keys. Press @@OK@@.

ꢀ Change field F: to ‘EXP(- t^2)’

ꢀ Make sure that the following parameters are set to: H-VAR: 0, V-VAR:

ꢀ Change the independent variable to t .

ꢀ Accept changes to PLOT SETUP: L@@OK@@

ꢀ „ò (simultaneously, if in RPN mode). To enter PLOT WINDOW

environment

ꢀ Change the horizontal and vertical view window to the following settings:

H-VIEW: -1

5; V-VIEW: -1 ꢀꢀꢀꢀ1.5

ꢀ Also, use the following values for the remaining parameters: Init: 0, Final:

5, Step: Default, Tol: 0.0001, Init-Soln: 0

ꢀ To plot the graph use: @ERASE @DRAW

When you observe the graph being plotted, you'll notice that the graph is not

very smooth. That is because the plotter is using a time step that may be a bit

large for a smooth graph. To refine the graph and make it smoother, use a step

of 0.1. Press @CANCL and change the Step : value to 0.1, then use @ERASE @DRAW

once more to repeat the graph. The plot will take longer to be completed, but

the shape is definitely smoother than before. Try the following: @EDIT L@LABEL

@MENU to see axes labels and range.

Notice that the labels for the axes are shown as 0 (horizontal, for t) and 1

(vertical, for x). These are the definitions for the axes as given in the PLOT

SETUP window („ô) i.e., H-VAR: 0, and V-VAR: 1. To see the graphical

solution in detail use the following:

Page 16-60

LL@PICT To recover menu and return to PICT environment.

@(X,Y)@ To determine coordinates of any point on the graph.

Use the š™ keys to move the cursor around the plot area. At the bottom of

the screen you will see the coordinates of the cursor as (X,Y), i.e., the calculator

uses X and Y as the default names for the horizontal and vertical axes,

respectively. Press L@CANCL to recover the menu and return to the PLOT

WINDOW environment. Finally, press $ to return to normal display.

Numerical solution of second-order ODE

Integration of second-order ODEs can be accomplished by defining the solution

as a vector. As an example, suppose that a spring-mass system is subject to a

damping force proportional to its speed, so that the resulting differential

d ²x

dt²

equation is:

or,

= −18.75⋅ x −1.962⋅

x" = - 18.75 x - 1.962 x',

subject to the initial conditions, v = x' = 6, x = 0, at t = 0. We want to find x,

x' at t = 2.

Re-write the ODE as: w' = Aw, where w = [ x x' ] , and A is the 2 x 2 matrix

shown below.

⎡ ⎤

⎡

⎤ ⎡ ⎤

⋅

⎢ ⎥

⎢

⎥ ⎢ ⎥

−18.75 −1.962 x'

⎣ ⎦

⎣

⎦ ⎣ ⎦

The initial conditions are now written as w = [0 6] , for t = 0. (Note: The

symbol [ ] means the transpose of the vector or matrix).

To solve this problem, first, create and store the matrix A, e.g., in ALG mode:

Then, activate the numerical differential equation solver by using: ‚ Ï

˜ @@@OK@@@ . To solve the differential equation with starting time t = 0 and final

Page 16-61

time t = 2, the input form for the differential equation solver should look as

follows (notice that the Init: value for the Soln: is a vector [0, 6]):

Press @SOLVE (wait) @EDIT to solve for w(t=2). The solution reads [.16716… -

.6271…], i.e., x(2) = 0.16716, and x'(2) = v(2) = -0.6271. Press @CANCL to

return to SOLVE environment.

Solution presented as a table of values

In the previous example we were interested only in finding the values of the

position and velocity at a given time t. If we wanted to produce a table of

values of x and x', for t = 0.00, 0.25, …, 2.00, we will proceed as follows:

First, prepare a table to write down your results:

0.00

6.00

0.00

0.25

…

2.00

…

Next, within the SOLVE environment, change the final value of the independent

variable to 0.25, use:

—.25@@OK@@ ™™ @SOLVE (wait) @EDIT

(Solves for w at t = 0.25, w = [0.968 1.368]. )

@@OK@@ INIT+ — . 5 @@OK@@ ™™@SOLVE (wait) @EDIT

(Changes initial value of t to 0.25, and final value of t to 0.5, solve again for

w(0.5) = [0.748 -2.616])

@@OK@@ @INIT+ —.75@@OK@@™™@SOLVE (wait) @EDIT

(Changes initial value of t to 0.5, and final value of t to 0.75, solve again for

w(0.75) = [0.0147 -2.859])

@@OK@@ @INIT+ —1@@OK@@ ™ ™ @SOLVE (wait) @EDIT

Page 16-62

(Changes initial value of t to 0.75, and final value of t to 1, solve again for w(1)

= [-0.469 -0.607])

Repeat for t = 1.25, 1.50, 1.75, 2.00. Press @@OK@@ after viewing the last result in

@EDIT. To return to normal calculator display, press $ or L@@OK@@. The

different solutions will be shown in the stack, with the latest result in level 1.

The final results look as follows:

0.00

0.25

0.50

0.000 6.000

0.968 1.368

0.748 -2.616

1.25 -0.354 1.281

1.50 0.141 1.362

1.75 0.227 0.268

2.00 0.167 -0.627

0.75 -0.015 -2.859

1.00 -0.469 -0.607

Graphical solution for a second-order ODE

Start by activating the differential equation numerical solver, ‚ Ï ˜

@@@OK@@@ . The SOLVE screen should look like this:

Notice that the initial condition for the solution (Soln: w Init:[0., …) includes the

vector [0, 6]. Press L@@OK@@.

Next, press „ô (simultaneously, if in RPN mode) to enter the PLOT

environment. Highlight the field in front of TYPE, using the —˜keys. Then,

press @CHOOS, and highlight Diff Eq, using the —˜keys. Press @@OK@@.

Modify the rest of the PLOT SETUP screen to look like this:

Page 16-63

Notice that the option V-Var: is set to 1, indicating that the first element in the

vector solution, namely, x’, is to be plotted against the independent variable t.

Accept changes to PLOT SETUP by pressing L@@OK@@.

Press „ò (simultaneously, if in RPN mode) to enter the PLOT WINDOW

environment. Modify this input form to look like this:

To plot the x’ vs. t graph use: @ERASE @DRAW . The plot of x’ vs. t looks like this:

To plot the second curve we need to use the PLOT SETUP input form once, more.

To reach this form from the graph above use: @CANCL L@@OK@@

„ô(simultaneously, if in RPN mode) . Change the value of the V-Var: field

to 2, and press @DRAW (do not press @ERASE or you would loose the graph

produced above). Use: @EDIT L @LABEL @MENU to see axes labels and range.

Notice that the x axis label is the number 0 (indicating the independent

variable), while the y-axis label is the number 2 (indicating the second variable,

i.e., the last variable plotted). The combined graph looks like this:

Page 16-64

Press LL @PICT @CANCL $ to return to normal calculator display.

Numerical solution for stiff first-order ODE

Consider the ODE: dy/dt = -100y+100t+101, subject to the initial condition

y(0) = 1.

Exact solution

This equation can be written as dy/dt + 100 y = 100 t + 101, and solved

using an integrating factor, IF(t) = exp(100t), as follows (RPN mode, with CAS

set to Exact mode):

‘(100*t+101)*EXP(100*t)’ ` ‘t’ ` RISCH

The result is

‘(t+1)*EXP(100*t)’.

Next, we add an integration constant, by using: ‘C’ `+

Then, we divide by FI(x), by using: ‘EXP(100*t)’ `/.

100t

The result is: ‘((t+1)*EXP(100*t)+C)/EXP(100*t)’, i.e., y(t) = 1+ t +C⋅e

. Use of

the initial condition y(0) = 1, results in 1 = 1 + 0 + C⋅e , or C = 0, the

particular solution being y(t) = 1+t.

Numerical solution

If we attempt a direct numerical solution of the original equation dy/dt = -

100y+100t+101, using the calculator’s own numerical solver, we find that the

calculator takes longer to produce a solution that in the previous first-order

example. To check this out, set your differential equation numerical solver (‚

Ï˜@@@OK@@@) to:

Page 16-65

Here we are trying to obtain the value of y(2) given y(0) = 1. With the Soln:

Finalfield highlighted, press @SOLVE. You can check that a solution takes about

6 seconds, while in the previous first-order example the solution was almost

instantaneous. Press $ to cancel the calculation.

This is an example of a stiff ordinary differential equation. A stiff ODE is one

whose general solution contains components that vary at widely different rates

under the same increment in the independent variable. In this particular case,

100t

the general solution, y(t) = 1+ t +C⋅e

, contains the components ‘t’ and

100t

‘C⋅e

’, which vary at very different rates, except for the cases C=0 or C≈0

100t

(e.g., for C = 1, t =0.1, C⋅e

=22026).

The calculator’s ODE numerical solver allows for the solution of stiff ODEs by

selecting the option _Stiffin the SOLVE Y’(T) = F(T,Y)screen. With this

option selected you need to provide the values of ∂f/∂y and ∂f/∂t. For the case

under consideration ∂f/∂y = -100 and ∂f/∂t = 100.

Enter those values in the corresponding fields of the SOLVE Y’(T) = F(T,Y)

screen:

When done, move the cursor to the ^Soln:Final field and press @SOLVE.

This

time, the solution in produced in about 1 second. Press @EDIT to see the

solution: 2.9999999999, i.e., 3.0.

Page 16-66

Note: The option Stiff is also available for graphical solutions of differential

equations.

Numerical solution to ODEs with the SOLVE/DIFF menu

The SOLVE soft menu is activated by using 74 MENU in RPN mode. This menu

is presented in detail in Chapter 6. One of the sub-menus, DIFF, contains

functions for the numerical solution of ordinary differential equations for use in

programming. These functions are described next using RPN mode and system

flag 117 set to SOFT menus. (See note in page 16-75).

The functions provided by the SOLVE/DIFF menu are the following:

Function RKF

This function is used to compute the solution to an initial value problem for a

th th

first-order differential equation using the Runge-Kutta-Fehlbert 4 -5 order

solution scheme. Suppose that the differential equation to be solved is given by

dy/dx = f(x,y), with y = 0 at x = 0, and that you will allow a convergence

criteria ε for the solution. You can also specify an increment in the independent

variable, Δx, to be used by the function. To run this function you will prepare

your stack as follows:

3: {‘x’, ‘y’, ‘f(x,y)’}

{ ε Δx }

final

The value in the first stack level is the value of the independent variable where

you want to find your solution, i.e., you want to find, y = f (x ), where f (x)

final

represents the solution to the differential equation. The second stack level may

contain only the value of ε, and the step Δx will be taken as a small default

value. After running function @@RKF@@, the stack will show the lines:

2: {‘x’, ‘y’, ‘f(x,y)’}

Page 16-67

The value of the solution, y , will be available in variable @@@y@@@. This function

final

is appropriate for programming since it leaves the differential equation

specifications and the tolerance in the stack ready for a new solution. Notice

that the solution uses the initial conditions x = 0 at y = 0. If, your actual initial

solutions are x = x at y = y , you can always add these values to the

init

solution provided by RKF, keeping in mind the following relationship:

RKF solution

Actual solution

init

+ x

+ y

final init final

final

init

The following screens show the RPN stack before and after applying function

RKF for the differential equation dy/dx = x+y, ε = 0.001, Δx = 0.1.

After applying function RKF, variable @@@y@@@ contains the value 4.3880...

Function RRK

This function is similar to the RKF function, except that RRK (Rosenbrock and

Runge-Kutta methods) requires as the list in stack level 3 for input not only the

names of the independent and dependent variables and the function defining

the differential equation, but also the expressions for the first and second

derivatives of the expression. Thus, the input stack for this function will look as

follows:

ꢁꢂ[[[[[ {'x', 'y', 'f(x,y)' '∂f/∂x' '∂f/∂y' }

{ ε Δx }

final

The value in the first stack level is the value of the independent variable where

you want to find your solution, i.e., you want to find, y = f (x ), where f (x)

final

represents the solution to the differential equation. The second stack level may

Page 16-68

contain only the value of ε, and the step Δx will be taken as a small default

value. After running function @@RKF@@, the stack will show the lines:

2: {‘x’, ‘y’, ‘f(x,y)’ ‘∂f/∂x’ ‘∂f/vy’ }

{ ε Δx }

The value of the solution, y , will be available in variable @@@y@@@.

final

This function can be used to solve so-called “stiff” differential equations.

The following screen shots show the RPN stack before and after application of

function RRK:

The value stored in variable y is 3.00000000004.

Function RKFSTEP

This function uses an input list similar to that of function RKF, as well as the

tolerance for the solution, and a possible step Δx, and returns the same input

list, followed by the tolerance, and an estimate of the next step in the

independent variable. The function returns the input list, the tolerance, and the

next step in the independent variable that satisfies that tolerance. Thus, the

input stack looks as follows:

ꢁꢂ[[[[[[ {‘x’, ‘y’, ‘f(x,y)’}

Δx

After running this function, the stack will show the lines:

3: {‘x’, ‘y’, ‘f(x,y)’}

(Δx)

Thus, this function is used to determine the appropriate size of a time step to

satisfy the required tolerance.

The following screen shots show the RPN stack before and after application of

function RKFSTEP:

Page 16-69

These results indicate that (Δx)

= 0.34049…

Function RRKSTEP

This function uses an input list similar to that of function RRK, as well as the

tolerance for the solution, a possible step Δx, and a number (LAST) specifying

the last method used in the solution (1, if RKF was used, or 2, if RRK was used).

Function RRKSTEP returns the same input list, followed by the tolerance, an

estimate of the next step in the independent variable, and the current method

(CURRENT) used to arrive at the next step. Thus, the input stack looks as

follows:

4: {‘x’, ‘y’, ‘f(x,y)’}

Δx

LAST

After running this function, the stack will show the lines:

4: {‘x’, ‘y’, ‘f(x,y)’}

(Δx)

CURRENT

Thus, this function is used to determine the appropriate size of a time step

((Δx) ) to satisfy the required tolerance, and the method used to arrive at that

result (CURRENT).

The following screen shots show the RPN stack before and after application of

function RRKSTEP:

Page 16-70

These results indicate that (Δx)

= 0.00558… and that the RKF method

(CURRENT = 1) should be used.

Function RKFERR

This function returns the absolute error estimate for a given step when solving a

problem as that described for function RKF. The input stack looks as follows:

2: [[[ {‘x’, ‘y’, ‘f(x,y)’}

Δx

After running this function, the stack will show the lines:

4: {‘x’, ‘y’, ‘f(x,y)’}

Δy

error

Thus, this function is used to determine the increment in the solution, Δy, as well

as the absolute error (error).

The following screen shots show the RPN stack before and after application of

function RKFERR:

-6

These result show that Δy = 0.827… and error = -1.89…×10 .

Function RSBERR

This function performs similarly to RKERR but with the input elements listed for

function RRK. Thus, the input stack for this function will look as follows:

2: {‘x’, ‘y’, ‘f(x,y)’ ‘∂f/∂x’ ‘∂f/vy’ }

Δx

After running the function, the stack will show the lines:

4: {‘x’, ‘y’, ‘f(x,y)’ ‘∂f/∂x’ ‘∂f/vy’ }:

Δy

error

Page 16-71

The following screen shots show the RPN stack before and after application of

function RSBERR:

These results indicate that Δy = 4.1514… and error = 2.762..., for Dx = 0.1.

Check that, if Dx is reduced to 0.01, Δy = -0.00307… and error = 0.000547.

Note: As you execute the commands in the DIFF menu values of x and y will

be produced and stored as variables in your calculator. The results provided

by the functions in this section will depend on the current values of x and y.

Therefore, some of the results illustrated above may differ from what you get in

your calculator.

Page 16-72

Chapter 17

Probability Applications

In this Chapter we provide examples of applications of calculator’s functions to

probability distributions.

The MTH/PROBABILITY.. sub-menu - part 1

The MTH/PROBABILITY.. sub-menu is accessible through the keystroke sequence

„´. With system flag 117 set to CHOOSE boxes, the following list of

MTH options is provided (see left-hand side figure below). We have selected

the PROBABILITY.. option (option 7), to show the following functions (see right-

hand side figure below):

In this section we discuss functions COMB, PERM, ! (factorial), RAND, and RDZ.

Factorials, combinations, and permutations

The factorial of an integer n is defined as: n! = n⋅ (n-1) ⋅ (n-2)…3⋅2⋅1. By

definition, 0! = 1. Factorials are used in the calculation of the number of

permutations and combinations of objects. For example, the number of

permutations of r objects from a set of n distinct objects is

_nP = n(n −1)(n −1)...(n − r + 1) = n!/(n − r)!

Also, the number of combinations of n objects taken r at a time is

⎛ ⎞

n(n −1)(n − 2)...(n − r +1)

⎜ ⎟

⎝ ⎠

r!(n − r)!

Page 17-1

To simplify notation, use P(n,r) for permutations, and C(n,r) for combinations.

We can calculate combinations, permutations, and factorials with functions

COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of

those functions is described next:

ꢀ COMB(n,r): Combinations of n items taken r at a time

ꢀ PERM(n,r): Permutations of n items taken r at a time

ꢀ n!: Factorial of a positive integer. For a non-integer, x! returns Γ(x+1),

where Γ(x) is the Gamma function (see Chapter 3). The factorial

symbol (!) can be entered also as the keystroke combination

~‚2.

Example of applications of these functions are shown next:

Random numbers

The calculator provides a random number generator that returns a uniformly

distributed, random real number between 0 and 1. The generator is able to

produce sequences of random numbers. However, after a certain number of

times (a very large number indeed), the sequence tends to repeat itself. For that

reason, the random number generator is more properly referred to as a pseudo-

random number generator. To generate a random number with your calculator

use function RAND from the MTH/PROBABILITY sub-menu. The following

screen shows a number of random numbers produced using RAND. The

numbers in the left-hand side figure are produced with calling function RAND

without an argument. If you place an argument list in function RAND, you get

back the list of numbers plus an additional random number attached to it as

illustrated in the right-hand side figure.

Page 17-2

Random number generators, in general, operate by taking a value, called the

“seed” of the generator, and performing some mathematical algorithm on that

“seed” that generates a new (pseudo)random number. If you want to generate

a sequence of number and be able to repeat the same sequence later, you can

change the "seed" of the generator by using function RDZ(n), where n is the

“seed,” before generating the sequence. Random number generators operate

by starting with a "seed" number that is transformed into the first random

number of the series. The current number then serves as the "seed" for the next

number and so on. By "re-seeding" the sequence with the same number you

can reproduce the same sequence more than once. For example, try the

following:

RDZ(0.25) `

RAND() `

Use 0.25 as the "seed."

First random number = 0.75285…

Second random number = 0.51109…

Third random number= 0.085429….

RAND() `

Re-start the sequence:

RDZ(0.25) `

RAND() `

Use 0.25 as the "seed."

First random number = 0.75285…

Second random number = 0.51109…

Third random number= 0.085429….

RAND() `

To generate a sequence of random numbers use function SEQ. For example, to

generate a list of 5 random numbers you can use, in ALG mode:

SEQ(RAND(),j,1,5,1). In RPN mode, use the following program:

« ꢀ n « 1 n FOR j RND NEXT n ꢀLIST » »

Store it into variable RLST (Random LiST), and use J5@RLST! to produce a

list of 5 random numbers.

Function RNDM(n,m) can be used to generate a matrix of n rows and m

columns whose elements are random integers between -1 and 1(see Chapter

10).

Discrete probability distributions

A random variable is said to be discrete when it can only take a finite number

of values. For example, the number of rainy days in a given location can be

considered a discrete random variable because we count them as integer

numbers only. Let X represent a discrete random variable, its probability mass

Page 17-3

function (pmf) is represented by f(x) = P[X=x], i.e., the probability that the

random variable X takes the value x.

The mass distribution function must satisfy the conditions that

f(x) >0, for all x,

and

f (x) =1.0

∑

all x

A cumulative distribution function (cdf) is defined as

F(x) = P[X ≤ x] =

f (k)

∑

k≤x

Next, we will define a number of functions to calculate discrete probability

distributions. We suggest that you create a sub-directory, say,

HOME\STATS\DFUN (Discrete FUNctions) where we will define the probability

mass function and cumulative distribution function for the binomial and Poisson

distributions.

Binomial distribution

The probability mass function of the binomial distribution is given by

⎛ ⎞

⋅ p ⋅(1− p)^n−x, x = 0,1,2,...,n

⎜ ⎟

f (n, p, x) =

⎝ ⎠

where ( ) = C(n,x) is the combination of n elements taken x at a time. The

values n and p are the parameters of the distribution. The value n represents

the number of repetitions of an experiment or observation that can have one of

two outcomes, e.g., success and failure. If the random variable X represents the

number of successes in the n repetitions, then p represents the probability of

getting a success in any given repetition. The cumulative distribution function

for the binomial distribution is given by

Page 17-4

F(n, p, x) =

f (n, p, x), x = 0,1,2,...,n

∑

k=0

Poisson distribution

The probability mass function of the Poisson distribution is given by

e^−λ⋅λ

f (λ, x) =

, x = 0,1,2,...,∞

In this expression, if the random variable X represents the number of

occurrences of an event or observation per unit time, length, area, volume, etc.,

then the parameter l represents the average number of occurrences per unit

time, length, area, volume, etc. The cumulative distribution function for the

Poisson distribution is given by

F(λ, x) =

f (λ, x), x = 0,1,2,...,∞

∑

k=0

Next, use function DEFINE („à) to define the following probability mass

functions (pmf) and cumulative distribution functions (cdf):

DEFINE(pmfb(n,p,x) = COMB(n,x)*p^x*(1-p)^(n-x))

DEFINE(cdfb(n,p,x) = Σ(k=0,x,pmfb(n,p,k)))

DEFINE(pmfp(λ,x) = EXP(-λ)*λ^x/x!)

DEFINE(cdfp(λ,x) = Σ(k=0,x,pmfp(λ,x)))

The function names stand for:

ꢀ pmfb: probability mass function for the binomial distribution

ꢀ cdfb: cumulative distribution function for the binomial distribution

ꢀ pmfp: probability mass function for the Poisson distribution

ꢀ cdfp: cumulative distribution function for the Poisson distribution

Examples of calculations using these functions are shown next:

Page 17-5

Continuous probability distributions

The probability distribution for a continuous random variable, X, is

characterized by a function f(x) known as the probability density function (pdf).

The pdf has the following properties: f(x) > 0, for all x, and

]

(

)

(ξ

)

∫

−∞

^+∞f (x)dx = 1.

∫

−∞

Probabilities are calculated using the cumulative distribution function (cdf), F(x),

defined by

, where P[X<x] stands for “the

P[X < x] = F(x) =

f (ξ )dξ

∫

−∞

probability that the random variable X is less than the value x”.

In this section we describe several continuous probability distributions including

the gamma, exponential, beta, and Weibull distributions. These distributions

are described in any statistics textbook. Some of these distributions make use of

a the Gamma function defined earlier, which is calculated in the calculator by

using the factorial function as Γ(x) = (x-1)!, for any real number x.

The gamma distribution

The probability distribution function (pdf) for the gamma distribution is given by

f (x) =

⋅ x^{α −1}⋅exp(− ), for x > 0,α > 0, β > 0;

β ^αΓ(α)

The corresponding (cumulative) distribution function (cdf) would be given by an

integral that has no closed-form solution.

The exponential distribution

The exponential distribution is the gamma distribution with a = 1. Its pdf is

given by

Page 17-6

f (x) = ⋅exp(− ), for x > 0, β > 0

while its cdf is given by F(x) = 1 - exp(-x/β), for x>0, β >0.

The beta distribution

The pdf for the gamma distribution is given by

Γ(α + β )

Γ(α) ⋅Γ(β )

f (x) =

⋅ x^{α −1}⋅(1− x)^{β −1}, for 0 < x < 1,α > 0, β > 0

As in the case of the gamma distribution, the corresponding cdf for the beta

distribution is also given by an integral with no closed-form solution.

The Weibull distribution

The pdf for the Weibull distribution is given by

f (x) =α ⋅ β ⋅ x^{β −1}⋅exp(−α ⋅ x^β), for x > 0,α > 0, β > 0

While the corresponding cdf is given by

F(x) = 1− exp(−α ⋅ x^β), for x > 0,α > 0, β > 0

Functions for continuous distributions

To define a collection of functions corresponding to the gamma, exponential,

beta, and Weibull distributions, first create a sub-directory called CFUN

(Continuous FUNctions) and define the following functions (change to Approx

mode):

Gamma pdf:

Gamma cdf:

Beta pdf:

'gpdf(x) = x^(α-1)*EXP(-x/β)/(β^α*GAMMA(α))'

'gcdf(x) = ∫(0,x,gpdf(t),t)'

' βpdf(x)= GAMMA(α+β)*x^(α-1)*(1-x)^(β-1)/(GAMMA(α)*GAMMA(β))'

Beta cdf: 'βcdf(x)= ∫(0,x,βpdf(t),t)'

Page 17-7

Exponential pdf:

Exponential cdf:

Weibull pdf:

'epdf(x) = EXP(-x/β)/β'

'ecdf(x) = 1 - EXP(-x/β)'

'Wpdf(x) = α*β*x^(β-1)*EXP(-α*x^β)'

'Wcdf(x) = 1 - EXP(-α*x^β)'

Weibull cdf:

Use function DEFINE to define all these functions. Next, enter the values of α

and β, e.g., 1K~‚a` 2K ~‚b`

Finally, for the cdf for Gamma and Beta cdf’s, you need to edit the program

definitions to add ꢀNUM to the programs produced by function DEFINE. For

example, the Gamma cdf, i.e., the function gcdf, should be modified to read: «

ꢀ x'ꢀNUM( ∫ (0,x,gpdf(t),t))' » and stored back into @gcdf. Repeat

the procedure for βcdf. Use RPN mode to perform these changes.

Unlike the discrete functions defined earlier, the continuous functions defined in

this section do not include their parameters (α and/or β) in their definitions.

Therefore, you don't need to enter them in the display to calculate the functions.

However, those parameters must be previously defined by storing the

corresponding values in the variables α and β. Once all functions and the

values α and β have been stored, you can order the menu labels by using

function ORDER. The call to the function will be the following:

ORDER({‘α’,’β’,’gpdf’,’gcdf’,’βpdf’,’βcdf’,’epdf’,’ecdf’,’Wpdf’,’Wcdf’})

Following this command the menu labels will show as follows (Press L to

move to the second list. Press L once more to move to the first list):

Some examples of application of these functions, for values of α = 2, β = 3, are

shown below. Notice the variable IERR that shows up in the second screen shot.

This results from a numerical integration for function gcdf.

Page 17-8

Continuous distributions for statistical inference

In this section we discuss four continuous probability distributions that are

commonly used for problems related to statistical inference. These distributions

are the normal distribution, the Student’s t distribution, the Chi-square (χ²)

distribution, and the F-distribution. The functions provided by the calculator to

evaluate probabilities for these distributions are contained in the MTH/

PROBABILITY menu introduced earlier in this chapter. The functions are NDIST,

UTPN, UTPT, UTPC, and UTPF. Their application is described in the following

sections. To see these functions activate the MTH menu: „´ and select

the PROBABILITY option:

Normal distribution pdf

The expression for the normal distribution pdf is:

(x − μ)²

2σ

f (x) =

exp[−

σ 2π

Page 17-9

where μ is the mean, and σ²is the variance of the distribution. To calculate the

value of f(μ,σ²,x) for the normal distribution, use function NDIST with the

following arguments: the mean, μ, the variance, σ², and, the value x , i.e.,

NDIST(μ,σ²,x). For example, check that for a normal distribution, f(1.0,0.5,2.0)

= 0.20755374.

Normal distribution cdf

The calculator has a function UTPN that calculates the upper-tail normal

distribution, i.e., UTPN(x) = P(X>x) = 1 - P(X<x). To obtain the value of the

upper-tail normal distribution UTPN we need to enter the following values: the

mean, μ; the variance, σ²; and, the value x, e.g., UTPN((μ,σ²,x)

For example, check that for a normal distribution, with μ = 1.0, σ²= 0.5,

UTPN(0.75) = 0.638163. Use UTPN(1.0,0.5,0.75) = 0.638163.

Different probability calculations for normal distributions [X is N(μ,σ²)] can be

defined using the function UTPN, as follows:

ꢀ P(X<a) = 1 - UTPN(μ, σ²,a)

ꢀ P(a<X<b) = P(X<b) - P(X<a) = 1 - UTPN(μ, σ²,b) - (1 - UTPN(μ, σ²,a)) =

UTPN(μ, σ²,a) - UTPN(μ, σ²,b)

ꢀ P(X>c) = UTPN(μ, σ²,c)

Examples: Using μ = 1.5, and σ²= 0.5, find:

P(X<1.0) = 1 - P(X>1.0) = 1 - UTPN(1.5, 0.5, 1.0) = 0.239750.

P(X>2.0) = UTPN(1.5, 0.5, 2.0) = 0.239750.

P(1.0<X<2.0) = F(1.0) - F(2.0) = UTPN(1.5,0.5,1.0) - UTPN(1.5,0.5,2.0) =

0.7602499 - 0.2397500 = 0.524998.

The Student-t distribution

The Student-t, or simply, the t-, distribution has one parameter ν, known as the

degrees of freedom of the distribution. The probability distribution function (pdf)

is given by

Page 17-10

ν +1

Γ(

)

ν +1

t²

−

f (t) =

⋅(1+

)

,−∞ < t < ∞

Γ( ) ⋅ πν

where Γ(α) = (α-1)! is the GAMMA function defined in Chapter 3.

The calculator provides for values of the upper-tail (cumulative) distribution

function for the t-distribution, function UTPT, given the parameter ν and the value

of t, i.e., UTPT(ν,t). The definition of this function is, therefore,

∞

UTPT(ν,t) =

f (t)dt =1− f (t)dt =1− P(T ≤ t)

∫

−∞

For example, UTPT(5,2.5) = 2.7245…E-2. Other probability calculations for the

t-distribution can be defined using the function UTPT, as follows:

ꢀ P(T<a) = 1 - UTPT(ν,a)

ꢀ P(a<T<b) = P(T<b) - P(T<a) = 1 - UTPT(ν,b) - (1 - UTPT(ν,a)) =

UTPT(ν,a) - UTPT(ν,b)

ꢀ P(T>c) = UTPT(ν,c)

Examples: Given ν = 12, determine:

P(T<0.5) = 1-UTPT(12,0.5) = 0.68694..

P(-0.5<T<0.5) = UTPT(12,-0.5)-UTPT(12,0.5) = 0.3738…

P(T> -1.2) = UTPT(12,-1.2) = 0.8733…

The Chi-square distribution

The Chi-square (χ²) distribution has one parameter ν, known as the degrees of

freedom. The probability distribution function (pdf) is given by

−1

−

f (x) =

⋅ x ²⋅ e ,ν > 0, x > 0

2²⋅ Γ( )

Page 17-11

The calculator provides for values of the upper-tail (cumulative) distribution

function for the χ²-distribution using [UTPC] given the value of x and the

parameter ν. The definition of this function is, therefore,

∞

UTPC(ν, x) =

f (x)dx =1−

f (x)dx =1− P(X ≤ x)

∫

−∞

To use this function, we need the degrees of freedom, ν, and the value of the

chi-square variable, x, i.e., UTPC(ν,x). For example, UTPC(5, 2.5) =

0.776495…

Different probability calculations for the Chi-squared distribution can be defined

using the function UTPC, as follows:

ꢀ P(X<a) = 1 - UTPC(ν,a)

ꢀ P(a<X<b) = P(X<b) - P(X<a) = 1 - UTPC(ν,b) - (1 - UTPC(ν,a)) =

UTPC(ν,a) - UTPC(ν,b)

ꢀ P(X>c) = UTPC(ν,c)

Examples: Given ν = 6, determine:

P(X<5.32) = 1-UTPC(6,5.32) = 0.4965..

P(1.2<X<10.5) = UTPC(6,1.2)-UTPC(6,10.5) = 0.8717…

P(X> 20) = UTPC(6,20) = 2.769..E-3

The F distribution

The F distribution has two parameters νN = numerator degrees of freedom, and

νD = denominator degrees of freedom. The probability distribution

function (pdf) is given by

νN

−1

νN +νD νN

Γ(

)⋅( ) ²⋅ F

νD

f (x) =

νN +νD

(

)

νN

νD

νN ⋅ F

νD

Γ( )⋅Γ( )⋅(1−

)

Page 17-12

The calculator provides for values of the upper-tail (cumulative) distribution

function for the F distribution, function UTPF, given the parameters νN and νD,

and the value of F. The definition of this function is, therefore,

∞

UTPF(νN,νD, F) =

f (F)dF = 1−

f (F)dF = 1− P(ℑ ≤ F)

∫

−∞

For example, to calculate UTPF(10,5, 2.5) = 0.161834…

Different probability calculations for the F distribution can be defined using the

function UTPF, as follows:

ꢀ P(F<a) = 1 - UTPF(νN, νD,a)

ꢀ P(a<F<b) = P(F<b) - P(F<a) = 1 -UTPF(νN, νD,b)- (1 - UTPF(νN, νD,a))

= UTPF(νN, νD,a) - UTPF(νN, νD,b)

ꢀ P(F>c) = UTPF(νN, νD,a)

Example: Given νN = 10, νD = 5, find:

P(F<2) = 1-UTPF(10,5,2) = 0.7700…

P(5<F<10) = UTPF(10,5,5) – UTPF(10,5,10) = 3.4693..E-2

P(F>5) = UTPF(10,5,5) = 4.4808..E-2

Inverse cumulative distribution functions

For a continuous random variable X with cumulative density function (cdf) F(x) =

P(X<x) = p, to calculate the inverse cumulative distribution function we need to

-1

find the value of x, such that x = F (p). This value is relatively simple to find for

the cases of the exponential and Weibull distributions since their cdf’s have a

closed form expression:

ꢀ Exponential, F(x) = 1 - exp(-x/β)

ꢀ Weibull, F(x) = 1-exp(-αx )

(Before continuing, make sure to purge variables α and β). To find the inverse

cdf’s for these two distributions we need just solve for x from these expressions,

i.e.,

Page 17-13

Exponential:

Weibull:

For the Gamma and Beta distributions the expressions to solve will be more

complicated due to the presence of integrals, i.e.,

⋅ z^{α −1}⋅exp(− )dz

p =

•

Gamma,

Beta,

∫

β Γ(α)

Γ(α + β )

Γ(α)⋅Γ(β )

p =

⋅ z^{α −1}⋅(1− z)^{β −1}dz

∫

A numerical solution with the numerical solver will not be feasible because of

the integral sign involved in the expression. However, a graphical solution is

possible. Details on how to find the root of a graph are presented in Chapter

12. To ensure numerical results, change the CAS setting to Approx. The

function to plot for the Gamma distribution is

Y(X) = ∫(0,X,z^(α-1)*exp(-z/β)/(β^α*GAMMA(α)),z)-p

For the Beta distribution, the function to plot is

Y(X) =

∫(0,X,z^(α-1)*(1-z)^(β-1)*GAMMA(α+β)/(GAMMA(α)*GAMMA(β)),z)-p

To produce the plot, it is necessary to store values of α, β, and p, before

attempting the plot. For example, for α = 2, β = 3, and p = 0.3, the plot of

Y(X) for the Gamma distribution is shown below. (Please notice that, because of

the complicated nature of function Y(X), it will take some time before the graph

is produced. Be patient.)

Page 17-14

There are two roots of this function found by using function @ROOT within the plot

environment. Because of the integral in the equation, the root is approximated

and will not be shown in the plot screen. You will only get the message

Constant? Shown in the screen. However, if you press ` at this point, the

approximate root will be listed in the display. Two roots are shown in the right-

hand figure below.

Alternatively, you can use function @TRACE @(X,Y)@ to estimate the roots by tracing

the curve near its intercepts with the x-axis. Two estimates are shown below:

These estimates suggest solutions x = -1.9 and x = 3.3. You can verify these

“solutions” by evaluating function Y1(X) for X = -1.9 and X = 3.3, i.e.,

For the normal, Student’s t, Chi-square (χ²), and F distributions, which are

represented by functions UTPN, UTPT, UPTC, and UTPF in the calculator, the

inverse cuff can be found by solving one of the following equations:

ꢀ Normal, p = 1 – UTPN(μ,σ2,x)

ꢀ Student’s t, p = 1 – UTPT(ν,t)

ꢀ Chi-square, p = 1 – UTPC(ν,x)

ꢀ F distribution: p = 1 – UTPF(νN,νD,F)

Page 17-15

Notice that the second parameter in the UTPN function is σ2, not σ²,

representing the variance of the distribution. Also, the symbol ν (the lower-case

Greek letter no) is not available in the calculator. You can use, for example, γ

(gamma) instead of ν. The letter γ is available thought the character set

(‚

For example, to obtain the value of x for a normal distribution, with μ = 10, σ²

= 2, with p = 0.25, store the equation ‘p=1-UTPN(μ,σ2,x)’ into variable

EQ (figure in the left-hand side below). Then, launch the numerical solver, to

get the input form in the right-hand side figure:

The next step is to enter the values of μ, σ², and p, and solve for x:

This input form can be used to solve for any of the four variables involved in the

equation for the normal distribution.

To facilitate solution of equations involving functions UTPN, UTPT, UTPC, and

UTPF, you may want to create a sub-directory UTPEQ were you will store the

equations listed above:

Page 17-16

Thus, at this point, you will have the four equations available for solution. You

needs just load one of the equations into the EQ field in the numerical solver

and proceed with solving for one of the variables. Examples of the UTPT, UTPC,

and UPTF are shown below:

Notice that in all the examples shown above, we are working with p = P(X<x).

In many statistical inference problems we will actually try to find the value of x

for which P(X>x) = α. Furthermore, for the normal distribution, we most likely

will be working with the standard normal distribution in which μ =0, and σ²=

1. The standard normal variable is typically referred to as Z, so that the

problem to solve will be P(Z>z) = α. For these cases of statistical inference

problems, we could store the following equations:

Page 17-17

With these four equations, whenever you launch the numerical solver you have

the following choices:

Examples of solution of equations EQNA, EQTA, EQCA, and EQFA are shown

below:

[[[[[

Page 17-18

Chapter 18

Statistical Applications

In this Chapter we introduce statistical applications of the calculator including

statistics of a sample, frequency distribution of data, simple regression,

confidence intervals, and hypothesis testing.

Pre-programmed statistical features

The calculator provides pre-programmed statistical features that are accessible

through the keystroke combination ‚Ù (same key as the number 5 key).

The statistical applications available in the calculator are:

These applications are presented in detail in this Chapter. First, however, we

demonstrate how to enter data for statistical analysis.

Entering data

For the analysis of a single set of data (a sample) we can use applications

number 1, 2, and 4 from the list above. All of these applications require that

the data be available as columns of the matrix ΣDAT. This can be

accomplished by entering the data in columns using the matrix writer, „².

This operation may become tedious for large number of data points. Instead,

you may want to enter the data as a list (see Chapter 8) and convert the list into

a column vector by using program CRMC (see Chapter 10). Alternatively, you

can enter the following program to convert a list into a column vector. Type the

program in RPN mode:

« OBJꢀ 1 2 ꢀLIST ꢀARRY »

Page 18-1

Store the program in a variable called LXC. After storing this program in RPN

mode you can also use it in ALG mode.

To store a column vector into variable ΣDAT use function STOΣ, available

through the catalog (‚N), e.g., STOΣ (ANS(1)) in ALG mode.

Example 1 – Using the program LXC, defined above, create a column vector

using the following data: 2.1 1.2 3.1 4.5 2.3 1.1 2.3 1.5 1.6 2.2

1.2 2.5.

In RPG mode, type in the data in a list:

{2.1 1.2 3.1 4.5 2.3 1.1 2.3 1.5 1.6 2.2 1.2 2.5 } `@LXC

Use function STOΣ to store the data into ΣDAT.

Note: You can also enter statistical data by launching a statistics application

(such as Single-var, Frequenciesor Summary stats) and

pressing #EDIT#. This launches the Matrix Writer. Enter the data as you usually

do. In this case, when you exit the Matrix Writer, the data you have entered is

automatically saved in ΣDAT.

Calculating single-variable statistics

Assuming that the single data set was stored as a column vector in variable

ΣDAT. To access the different STAT programs, press ‚Ù. Press @@@OK@@ to

select ^{1. Single-var..}There will be available to you an input form labeled

SINGLE-VARIABLE STATISTICS, with the data currently in your ΣDAT variable

listed in the form as a vector. Since you only have one column, the field ^Col:

should have the value 1 in front of it. The ^Typefield determines whether you

are working with a sample or a population, the default setting is Sample. Move

the cursor to the horizontal line preceding the fields ^Mean, Std Dev, Variance,

Total, Maximum, Minimum, pressing the @ꢀCHK@ soft menu key to select those

measures that you want as output of this program. When ready, press @@@OK@@.

The selected values will be listed, appropriately labeled, in the screen of your

calculator.

Page 18-2

Example 1 -- For the data stored in the previous example, the single-variable

statistics results are the following:

Mean: 2.13333333333, Std Dev: 0.964207949406,

Variance: 0.929696969697

Total: 25.6, Maximum: 4.5, Minimum: 1.1

Definitions

The definitions used for these quantities are the following:

Suppose that you have a number data points x , x , x , …, representing

different measurements of the same discrete or continuous variable x. The set of

all possible values of the quantity x is referred to as the population of x. A

finite population will have only a fixed number of elements x. If the quantity x

represents the measurement of a continuous quantity, and since, in theory, such

a quantity can take an infinite number of values, the population of x in this case

is infinite. If you select a sub-set of a population, represented by the n data

values {x , x , …, x }, we say you have selected a sample of values of x.

Samples are characterized by a number of measures or statistics. There are

measures of central tendency, such as the mean, median, and mode, and

measures of spreading, such as the range, variance, and standard deviation.

Measures of central tendency

The mean (or arithmetic mean) of the sample, ⎯x, is defined as the average

value of the sample elements,

x =

⋅

x .

∑

i=1

The value labeled Totalobtained above represents the summation of the

values of x, or Σx = n⋅⎯x. This is the value provided by the calculator under the

heading Mean. Other mean values used in certain applications are the

geometric mean, x , or the harmonic mean, x , defined as:

x_g= x₁⋅ x₂L x_n,

∑

x_h

x_i

i=1

Page 18-3

Examples of calculation of these measures, using lists, are available in Chapter

The median is the value that splits the data set in the middle when the elements

are placed in increasing order. If you have an odd number, n, of ordered

elements, the median of this sample is the value located in position (n+1)/2. If

you have an even number, n, of elements, the median is the average of the

elements located in positions n/2 and (n+1)/2. Although the pre-programmed

statistical features of the calculator do not include the calculation of the median,

it is very easily to write a program to calculate such quantity by working with

lists. For example, if you want to use the data in ΣDAT to find the median, type

the following program in RPN mode (see Chapter 21 for more information on

programming in User RPL language).:

« ꢀ nC « RCLΣ DUP SIZE 2 GET IF 1 > THEN nC COL− SWAP DROP OBJꢀ

1 + ꢀARRY END OBJꢀ OBJꢀ DROP DROP DUP ꢀ n « ꢀLIST SORT IF ‘n

MOD 2 == 0’ THEN DUP ‘n/2’ EVAL GET SWAP ‘(n+1)/2’ EVAL GET + 2 /

ELSE ‘(n+1)/2’ EVAL GET END “Median” ꢀTAG » » »

Store this program under the name MED. An example of application of this

program is shown next.

Example 2 – To run the program, first you need to prepare your ΣDAT matrix.

Then, enter the number of the column in ΣDAT whose median you want to find,

and press @@MED@@. For the data currently in ΣDAT (entered in an earlier example),

use program MED to show that Median: 2.15.

The mode of a sample is better determined from histograms, therefore, we leave

its definition for a later section.

Measures of spread

s_x²=

⋅

(x − x)²

The variance (Var) of the sample is defined as

∑

n −1

i=1

The standard deviation (St Dev) of the sample is just the square root of the

variance, i.e., s .

Page 18-4

The range of the sample is the difference between the maximum and minimum

values of the sample. Since the calculator, through the pre-programmed

statistical functions provides the maximum and minimum values of the sample,

you can easily calculate the range.

Coefficient of variation

The coefficient of variation of a sample combines the mean, a measure of

central tendency, with the standard deviation, a measure of spreading, and is

defined, as a percentage, by: V = (s /⎯x)100.

Sample vs. population

The pre-programmed functions for single-variable statistics used above can be

applied to a finite population by selecting the Type: Populationin the

SINGLE-VARIABLE STATISTICSscreen. The main difference is in the values of

the variance and standard deviation which are calculated using n in the

denominator of the variance, rather than (n-1).

Example 3 -- If you were to repeat the exercise in Example 1 of this section,

using Populationrather than Sampleas the Type, you will get the same

values for the mean, total, maximum, and minimum. The variance and

standard deviation, however, will be given by: Variance: 0.852, Std Dev:

0.923.

Obtaining frequency distributions

The application 2. Frequencies.. in the STAT menu can be used to obtain

frequency distributions for a set of data. Again, the data must be present in the

form of a column vector stored in variable ΣDAT. To get started, press

‚Ù˜ @@@OK@@@. The resulting input form contains the following fields:

ΣDAT:

Col:

X-Min:

Bin Count:

Bin Width:

the matrix containing the data of interest.

the column of ΣDAT that is under scrutiny.

the minimum class boundary (default = -6.5).

the number of classes(default = 13).

the uniform width of each class (default = 1).

Page 18-5

Definitions

To understand the meaning of these parameters we present the following

definitions: Given a set of n data values: {x , x , …, x } listed in no particular

order, it is often required to group these data into a series of classes by

counting the frequency or number of values corresponding to each class.

(Note: the calculators refers to classes as bins).

Suppose that the classes, or bins, will be selected by dividing the interval (x

bot

), into k = Bin Count classes by selecting a number of class boundaries, i.e.,

top

{xB , xB , …, xB }, so that class number 1 is limited by xB -xB , class

k+1

number 2 by xB - xB , and so on. The last class, class number k, will be limited

by xB - xB

k +1

The value of x corresponding to the middle of each class is known as the class

mark, and is defined as xM = (xB + xB )/2, for i = 1, 2, …, k.

i+1

If the classes are chosen such that the class size is the same, then we can define

the class size as the value Bin Width = Δx = (x - x ) / k,

max

min

and the class boundaries can be calculated as xB = x + (i - 1) * Δx.

bot

Any data point, x , j = 1, 2, …, n, belongs to the i-th class, if xB ≤ x < xB

i+1

The application 2. Frequencies.. in the STAT menu will perform this frequency

count, and will keep track of those values that may be below the minimum and

above the maximum class boundaries (i.e., the outliers).

Example 1 -- In order to better illustrate obtaining frequency distributions, we

want to generate a relatively large data set, say 200 points, by using the

following:

ꢀ First, seed the random number generator using: RDZ(25) in ALG mode,

or 25 ` RDZ in RPN mode (see Chapter 17).

ꢀ Type in the following program in RPN mode:

« ꢀ n « 1 n FOR j RAND 100 * 2 RND NEXT n ꢀLIST » »

and save it under the name RDLIST (RanDom number LIST generator).

Page 18-6

ꢀ Generate the list of 200 number by using RDLIST(200) in ALG mode, or

200 ` @RDLIST@ in RPN mode.

ꢀ Use program LXC (see above) to convert the list thus generated into a

column vector.

ꢀ Store the column vector into ΣDAT, by using function STOΣ.

ꢀ Obtain single-variable information using: ‚Ù@@@OK@@@. Use Sample for

the Type of data set, and select all options as results. The results for this

example were:

Mean: 51.0406, Std Dev: 29.5893…, Variance: 875.529…

Total: 10208.12, Maximum: 99.35, Minimum: 0.13

This information indicates that our data ranges from values close to zero to

values close to 100. Working with whole numbers, we can select the range of

variation of the data as (0,100). To produce a frequency distribution we will

use the interval (10,90) dividing it into 8 bins of width 10 each.

ꢀ Select the program ^{2. Frequencies..}by using ‚Ù˜ @@@OK@@@. The

data is already loaded in ΣDAT, and the option Col should hold the value 1

since we have only one column in ΣDAT.

ꢀ Change X-Min to 10, Bin Count to 8, and Bin Width to 10, then press

@@@OK@@@.

Using the RPN mode, the results are shown in the stack as a column vector in

stack level 2, and a row vector of two components in stack level 1. The vector

in stack level 1 is the number of outliers outside of the interval where the

frequency count was performed. For this case, I get the values [ 25. 22.]

indicating that there are, in my ΣDAT vector, 25 values smaller than 10 and 22

larger than 90.

ꢀ Press ƒ to drop the vector of outliers from the stack. The remaining result

is the frequency count of data. This can be translated into a table as shown

below.

This table was prepared from the information we provided to generate the

frequency distribution, although the only column returned by the calculator is

the Frequency column (f ). The class numbers, and class boundaries are easy

Page 18-7

to calculate for uniform-size classes (or bins), and the class mark is just the

average of the class boundaries for each class. Finally, the cumulative

frequency is obtained by adding to each value in the last column, except the

first, the frequency in the next row, and replacing the result in the last column of

the next row. Thus, for the second class, the cumulative frequency is 18+15 =

33, while for class number 3, the cumulative frequency is 33 + 16 = 49, and so

on. The cumulative frequency represents the frequency of those numbers that

are smaller than or equal to the upper boundary of any given class.

Class Class Bound.

No.

Class

mark.

Frequency Cumulative

frequency

i+1

< XB

outlier below

range

110

134

153

k = 8

>XB outliers above

range

Given the (column) vector of frequencies generated by the calculator, you can

obtain a cumulative frequency vector by using the following program in RPN

mode:

Page 18-8

« DUP SIZE 1 GET ꢀ freq k « {k 1} 0 CON ꢀ cfreq « ‘freq(1,1)’ EVAL

‘cfreq(1,1)’ STO 2 k FOR j ‘cfreq(j-1,1) +freq(j,1)’ EVAL ‘cfreq (j,1)’ STO NEXT

cfreq » » »

Save it under the name CFREQ. Use this program to generate the list of

cumulative frequencies (press @CFREQ with the column vector of frequencies in the

stack). The result, for this example, is a column vector representing the last

column of the table above.

Histograms

A histogram is a bar plot showing the frequency count as the height of the bars

while the class boundaries shown the base of the bars. If you have your raw

data (i.e., the original data before the frequency count is made) in the variable

ΣDAT, you can select Histogramas your graph type and provide information

regarding the initial value of x, the number of bins, and the bin width, to

generate the histogram. Alternatively, you can generate the column vector

containing the frequency count, as performed in the example above, store this

vector into ΣDAT, and select Barplotas your graph type. In the next example,

we show you how to use the first method to generate a histogram.

Example 1 – Using the 200 data points generated in the example above

(stored as a column vector in ΣDAT), generate a histogram plot of the data

using X-Min = 10, Bin Count = 16, and Bin Width = 5.

ꢀ First, press „ô (simultaneously, if in RPN mode) to enter the PLOT

SETUP screen. Within this screen, change Type: to Histogram, and check

that the option Col: 1 is selected. Then, press L@@@OK@@@.

ꢀ Next, press „ò (simultaneously, if in RPN mode) to enter the PLOT

WINDOW – HISTOGRAM screen. Within that screen modify the

information to H-View: 10 90, V-View: 0 15, Bar Width: 5.

ꢀ Press @ERASE @DRAW@ to generate the following histogram:

Page 18-9

ꢀ Press @CANCEL to return to the previous screen. Change the V-view and Bar

Width once more, now to read V-View: 0 30, Bar Width: 10. The new

histogram, based on the same data set, now looks like this:

A plot of frequency count, f , vs. class marks, xM , is known as a frequency

polygon. A plot of the cumulative frequency vs. the upper boundaries is

known as a cumulative frequency ogive. You can produce scatterplots that

simulate these two plots by entering the proper data in columns 1 and 2 of a

new ΣDAT matrix and changing the Type: to SCATTERin the PLOT SETUP

window.

Fitting data to a function y = f(x)

The program 3. Fit data.., available as option number 3 in the STAT menu, can

be used to fit linear, logarithmic, exponential, and power functions to data sets

(x,y), stored in columns of the ΣDAT matrix. In order for this program to be

effective, you need to have at least two columns in your ΣDAT variable.

Example 1 – Fit a linear relationship to the data shown in the table below:

0.5

2.3

3.6

6.7

7.2

Page 18-10

ꢀ First, enter the two rows of data into column in the variable ΣDAT by using

the matrix writer, and function STOΣ.

ꢀ To access the program ^3.Fit data.., use the following keystrokes:

‚Ù˜˜@@@OK@@@ The input form will show the current ΣDAT, already

loaded. If needed, change your set up screen to the following parameters

for a linear fitting:

ꢀ To obtain the data fitting press @@OK@@. The output from this program, shown

below for our particular data set, consists of the following three lines in RPN

mode:

3: '0.195238095238 + 2.00857142857*X'

2: Correlation: 0.983781424465

1: Covariance: 7.03

Level 3 shows the form of the equation. In this case, y = 0.06924 + 0.00383 x.

Level 2 shows the sample correlation coefficient, and level 1 shows the

covariance of x-y.

Definitions

For a sample of data points (x,y), we define the sample covariance as

s_xy=

(x − x)(y − y)

∑

n −1

i=1

The sample correlation coefficient for x,y is defined as

s_xy

r_xy=

s_x⋅ s_y

Page 18-11

Where s , s are the standard deviations of x and y, respectively, i.e.

s_x²=

(x − x)²

s_y²=

(y − y)²

∑

n −1

n −1 _i=1

i=1

The values s and r are the "Covariance" and "Correlation," respectively,

obtained by using the "Fit data" feature of the calculator.

Linearized relationships

Many curvilinear relationships "straighten out" to a linear form. For example,

the different models for data fitting provided by the calculator can be linearized

as described in the table below.

Indep.

Depend.

Type of

Fitting

Actual

Model

Linearized

Model

variable Variable

Covar.

ξη

Linear

Log.

y = a + bx

[same]

y = a + b ln(x)

ln(x)

ln(x),y

x,ln(y)

Exp.

ln(y) = ln(a) + bx

ln(y) = ln(a) + b ln(x)

ln(y)

y = a e

Power

ln(x)

ln(x),ln(y)

y = a x

s_ξη

(ξ −ξ )(η −η )

The sample covariance of ξ,η is given by

∑

n −1

Also, we define the sample variances of ξ and η, respectively, as

s_ξ²=

(ξ −ξ )²

s_η²=

(η −η )²

∑

n −1 _i=1

s_ξη

The sample correlation coefficient r is

r =

ξη

s_ξ⋅ s_η

Page 18-12

The general form of the regression equation is η = A + Bξ.

Best data fitting

The calculator can determine which one of its linear or linearized relationship

offers the best fitting for a set of (x,y) data points. We will illustrate the use of

this feature with an example. Suppose you want to find which one of the data

fitting functions provides the best fit for the following data:

0.2

3.16 2.73 2.12 1.65 1.29 0.47 0.29 0.01

0.5

1.5

First, enter the data as a matrix, either by using the Matrix Writer and entering

the data, or by entering two lists of data corresponding to x and y and using

the program CRMC developed in Chapter 10. Next, save this matrix into the

statistical matrix ΣDAT, by using function STOΣ.

Finally, launch the data fit application by using: ‚Ù˜˜@@@OK@@@ . The

display shows the current ΣDAT, already loaded. Change your set up screen to

the following parameters if needed:

Press @@@OK@@@, to get:

3: '3.99504833324*EXP(-.579206831203*X)'

2: Correlation: -0.996624999526

1: Covariance: -6.23350666124

-0.58⋅x

The best fit for the data is, therefore, y = 3.995 e

Obtaining additional summary statistics

The application 4. Summary stats.. in the STAT menu can be useful in some

calculations for sample statistics. To get started, press ‚Ù once more,

move to the fourth option using the down-arrow key ˜, and press @@@OK@@@. The

resulting input form contains the following fields:

ΣDAT:

the matrix containing the data of interest.

Page 18-13

X-Col, Y-Col:

these options apply only when you have more than two

columns in the matrix ΣDAT. By default, the x column is

column 1, and the y column is column 2.

_ΣX _ ΣY…:

summary statistics that you can choose as results of this

program by checking the appropriate field using [ꢀCHK]

when that field is selected.

Many of these summary statistics are used to calculate statistics of two variables

(x,y) that may be related by a function y = f(x). Therefore, this program can be

thought off as a companion to program ^{3. Fit data..}

Example 1 – For the x-y data currently in ΣDAT, obtain all the summary statistics.

ꢀ To access the summary stats… option, use: ‚Ù˜˜˜@@@OK@@@

ꢀ Select the column numbers corresponding to the x- and y-data, i.e., X-Col:

1, and Y-Col: 2.

ꢀ Using the @ꢀCHK@ key select all the options for outputs, i.e., _ΣX, _ΣY, etc.

ꢀ Press @@@OK@@@ to obtain the following results:

ΣX: 24.2, ΣY: 11.72, ΣX2: 148.54, ΣY2: 26.6246, ΣXY: 12.602, NΣ:8

Note: There are two other applications under the STAT menu, namely, ^5.

^{Hypth. tests..}and 6. Conf. Interval.. These two applications will be discussed

later in the chapter.

Calculation of percentiles

Percentiles are measures that divide a data set into 100 parts. The basic

procedure to calculate the 100⋅p-th Percentile (0 < p < 1) in a sample of size n

is as follows:

1. Order the n observations from smallest to largest.

2. Determine the product n⋅p

A. If n⋅p is not an integer, round it up to the next integer and find the

corresponding ordered value.

Page 18-14

B. If n⋅p is an integer, say k, calculate the mean of the k-th and (k-1) th

ordered observations.

Note: Integer rounding rule, for a non-integer x.yz…, if y ≥ 5, round up to

x+1; if y < 5, round up to x.

This algorithm can be implemented in the following program typed in RPN

mode (See Chapter 21 for programming information):

« SORT DUP SIZE ꢀ p X n « n p * ꢀ k « IF k CEIL k FLOOR - NOT THEN X k

GET X k 1 + GET + 2 / ELSE k 0 RND X SWAP GET END » » »

which we’ll store in variable %TILE (percent-tile). This program requires as

input a value p within 0 and 1, representing the 100p percentile, and a list of

values. The program returns the 100p percentile of the list.

Example 1 - Determine the 27% percentile of the list { 2 1 0 1 3 5 1 2 3 6 7 9}.

In RPN mode, enter 0.27 ` { 2 1 0 1 3 5 1 2 3 6 7 9} ` @%TILE. In ALG

mode, enter %TILE(0.27,{2,1,0,1,3,5,1,2,3,6,7,9}. The result is 1.

The STAT soft menu

All the pre-programmed statistical functions described above are accessible

through a STAT soft menu. The STAT soft menu can be accessed by using, in

RPN mode, the command: 96 MENU

You can create your own program, say @STATm, to activate the STAT soft menu

directly. The contents of this program are simply: « 96 MENU ».

The STAT soft menu contains the following functions:

Pressing the key corresponding to any of these menus provides access to

different functions as described below.

Page 18-15

The DATA sub-menu

The DATA sub-menu contains functions used to manipulate the statistics matrix

ΣDATA:

The operation of these functions is as follows:

Σ+ : add row in level 1 to bottom of ΣDATA matrix.

Σ- : removes last row in ΣDATA matrix and places it in level of 1 of the stack.

The modified ΣDATA matrix remains in memory.

CLΣ : erases current ΣDATA matrix.

ΣDAT: places contents of current ΣDATA matrix in level 1 of the stack.

„ΣDAT: stores matrix in level 1 of stack into ΣDATA matrix.

The ΣPAR sub-menu

The ΣPAR sub-menu contains functions used to modify statistical parameters.

The parameters shown correspond to the last example of data fitting.

The parameters shown in the display are:

Xcol: indicates column of ΣDATA representing x (Default: 1)

Ycol: indicates column of ΣDATA representing y (Default: 2)

Intercept: shows intercept of most recent data fitting (Default: 0)

Slope: shows slope of most recent data fitting (Default: 0)

Model: shows current data fit model (Default: LINFIT)

The functions listed in the soft menu keys operate as follows:

XCOL: entered as n @XCOL, changes Xcol to n.

YCOL: entered as n @YCOL, changes Ycol to n.

Page 18-16

ΣPAR: shows statistical parameters.

RESET: reset parameters to default values

INFO: shows statistical parameters

The MODL sub-menu within ΣPAR

This sub-menu contains functions that let you change the data-fitting model to

LINFIT, LOGFIT, EXPFIT, PWRFIT or BESTFIT by pressing the appropriate button.

The 1VAR sub menu

The 1VAR sub menu contains functions that are used to calculate statistics of

columns in the ΣDATA matrix.

The functions available are the following:

TOT: show sum of each column in ΣDATA matrix.

MEAN: shows average of each column in ΣDATA matrix.

SDEV: shows standard deviation of each column in ΣDATA matrix.

MAXΣ: shows maximum value of each column in ΣDATA matrix.

MINΣ: shows average of each column in ΣDATA matrix.

BINS: used as x , Δx, n [BINS], provides frequency distribution for data in

Xcol column in ΣDATA matrix with the frequency bins defined as

[x ,x +Δx], [x ,x +2Δx],…, [x ,x +nΔx].

VAR: shows variance of each column in ΣDATA matrix.

PSDEV: shows population standard deviation (based on n rather than on (n-1))

of each column in ΣDATA matrix.

PVAR: shows population variance of each column in ΣDATA matrix.

MINΣ: shows average of each column in ΣDATA matrix.

The PLOT sub-menu

The PLOT sub-menu contains functions that are used to produce plots with the

data in the ΣDATA matrix.

Page 18-17

The functions included are:

BARPL: produces a bar plot with data in Xcol column of the ΣDATA matrix.

HISTP: produces histogram of the data in Xcol column in the ΣDATA matrix,

using the default width corresponding to 13 bins unless the bin size is

modified using function BINS in the 1VAR sub-menu (see above).

SCATR: produces a scatterplot of the data in Ycol column of the ΣDATA matrix

vs. the data in Xcol column of the ΣDATA matrix. Equation fitted will be

stored in the variable EQ.

The FIT sub-menu

The FIT sub-menu contains functions used to fit equations to the data in columns

Xcol and Ycol of the ΣDATA matrix.

The functions available in this sub-menu are:

ΣLINE: provides the equation corresponding to the most recent fitting.

LR: provides intercept and slope of most recent fitting.

PREDX: used as y @PREDX, given y find x for the fitting y = f(x).

PREDY: used as x @PREDY, given x find y for the fitting y = f(x).

CORR: provides the correlation coefficient for the most recent fitting.

COV: provides sample co-variance for the most recent fitting

PCOV: shows population co-variance for the most recent fitting.

The SUMS sub-menu

The SUMS sub-menu contains functions used to obtain summary statistics of the

data in columns Xcol and Ycol of the ΣDATA matrix.

ΣX : provides the sum of values in Xcol column.

ΣY : provides the sum of values in Ycol column.

Page 18-18

ΣX^2 : provides the sum of squares of values in Xcol column.

ΣY^2 : provides the sum of squares of values in Ycol column.

ΣX*Y: provides the sum of x⋅y, i.e., the products of data in columns Xcol and

Ycol.

NΣ : provides the number of columns in the ΣDATA matrix.

Example of STAT menu operations

Let ΣDATA be the matrix shown in next page.

ꢀ Type the matrix in level 1 of the stack by using the Matrix Writer.

ꢀ To store the matrix into ΣDATA, use: @DATA „ @£DAT

ꢀ Calculate statistics of each column: @STAT @1VAR:

@TOT

@MEAN

@SDEV

@MAX£

@MIN£

L @VAR

@PSDEV

@PVAR

produces [38.5 87.5 82799.8]

produces [5.5. 12.5 11828.54…]

produces [3.39… 6.78… 21097.01…]

produces [10 21.5 55066]

produces [1.1 3.7 7.8]

produces [11.52 46.08 445084146.33]

produces [3.142… 6.284… 19532.04…]

produces [9.87… 39.49… 381500696.85…]

ꢀ Data:

1.1 3.7

3.7 8.9

2.2 5.9

7.8

101

⎡

⎢

⎤

⎥

5.5 12.5 612

6.8 15.1 2245

9.2 19.9 24743

⎥

10.0 21.5 55066

⎣

⎦

ꢀ Generate a scatterplot of the data in columns 1 and 2 and fit a straight line

to it:

Page 18-19

@STAT @£PAR @RESET

resets statistical parameters

L@STAT @PLOT @SCATR

@STATL

produces scatterplot

draws data fit as a straight line

@CANCL

returns to main display

ꢀ Determine the fitting equation and some of its statistics:

@STAT @FIT@ @£LINE

@@@LR@@@

3 @PREDX

1 @PREDY

@CORR

produces '1.5+2*X'

produces Intercept: 1.5, Slope: 2

produces 0.75

produces 3. 50

produces 1.0

@@COV@@

L@PCOV

produces 23.04

produces 19.74…

ꢀ Obtain summary statistics for data in columns 1 and 2: @STAT @SUMS:

@@@£X@@

@@@£Y@@

@@£X2@

@@£Y2@

@@£XY@

@@@N£@@

produces 38.5

produces 87.5

produces 280.87

produces 1370.23

produces 619.49

produces 7

Page 18-20

ꢀ Fit data using columns 1 (x) and 3 (y) using a logarithmic fitting:

@STAT @£PAR 3 @YCOL

select Ycol = 3, and

select Model = Logfit

@MODL @LOGFI

L@STAT @PLOT @SCATR

@STATL

produce scattergram of y vs. x

show line for log fitting

Obviously, the log-fit is not a good choice.

@CANCL

returns to normal display.

•

Select the best fitting by using:

@STAT @£PAR @MODL @BESTF

shows EXPFIT as the best fit for these data

L@STAT @FIT @£LINE

@CORR

2300 @PREDX

5.2 @PREDY

produces '2.6545*EXP(0.9927*X)'

produces 0.99995… (good correlation)

produces 6.8139

produces 463.33

Page 18-21

L@STAT @PLOT @SCATR

@STATL

produce scattergram of y vs. x

show line for log fitting

ꢀ To return to STAT menu use: L@STAT

ꢀ To get your variable menu back use: J.

Confidence intervals

Statistical inference is the process of making conclusions about a population

based on information from sample data. In order for the sample data to be

meaningful, the sample must be random, i.e., the selection of a particular

sample must have the same probability as that of any other possible sample out

of a given population. The following are some terms relevant to the concept of

random sampling:

ꢀ Population: collection of all conceivable observations of a process or

attribute of a component.

ꢀ Sample: sub-set of a population.

ꢀ Random sample: a sample representative of the population.

ꢀ Random variable: real-valued function defined on a sample space. Could

be discrete or continuous.

If the population follows a certain probability distribution that depends on a

parameter θ, a random sample of observations (X ,X ,X ,... , X ), of size n,

can be used to estimate θ.

ꢀ Sampling distribution: the joint probability distribution of X ,X ,X ,... , X .

ꢀ A statistic: any function of the observations that is quantifiable and does not

contain any unknown parameters. A statistic is a random variable that

provides a means of estimation.

Page 18-22

ꢀ Point estimation: when a single value of the parameter θ is provided.

ꢀ Confidence interval: a numerical interval that contains the parameter θ at a

given level of probability.

ꢀ Estimator: rule or method of estimation of the parameter θ.

ꢀ Estimate: value that the estimator yields in a particular application.

Example 1 -- Let X represent the time (hours) required by a specific

manufacturing process to be completed. Given the following sample of values

of X: 2.2 2.5 2.1 2.3 2.2. The population from where this sample is

taken is the collection of all possible values of the process time, therefore, it is

an infinite population. Suppose that the population parameter we are trying to

estimate is its mean value, μ. We will use as an estimator the mean value of

X =

⋅

X .

∑

i=1

the sample, ⎯X, defined by (a rule):

For the sample under consideration, the estimate of μ is the sample statistic ⎯x =

(2.2+2.5+2.1+2.3+2.2)/5 = 2.26. This single value of ⎯X, namely ⎯x = 2.26,

constitutes a point estimation of the population parameter μ.

Estimation of Confidence Intervals

The next level of inference from point estimation is interval estimation, i.e.,

instead of obtaining a single value of an estimator we provide two statistics, a

and b, which define an interval containing the parameter θ with a certain level

of probability. The end points of the interval are known as confidence limits,

and the interval (a,b) is known as the confidence interval.

Definitions

Let (C,C ) be a confidence interval containing an unknown parameter θ.

ꢀ Confidence level or confidence coefficient is the quantity (1-α), where 0 < α

< 1, such that P[C < θ < C ] = 1 - α, where P[ ] represents a probability

(see Chapter 17). The previous expression defines the so-called two-sided

confidence limits.

ꢀ A lower one-sided confidence interval is defined by Pr[C < θ] = 1 - α.

ꢀ An upper one-sided confidence interval is defined by Pr[θ < C ] = 1 - α.

Page 18-23

ꢀ The parameter α is known as the significance level. Typical values of α are

0.01, 0.05, 0.1, corresponding to confidence levels of 0.99, 0.95, and

0.90, respectively.

Confidence intervals for the population mean when the

population variance is known

Let ⎯X be the mean of a random sample of size n, drawn from an infinite

population with known standard deviation σ. The 100(1-α) % [i.e., 99%, 95%,

90%, etc.], central, two-sided confidence interval for the population mean μ is

(⎯X−z ⋅σ/√n , ⎯X+z ⋅σ/√n ), where z is a standard normal variate that

α/2

is exceeded with a probability of α /2. The standard error of the sample

mean, ⎯X, is ⋅σ/√n.

The one-sided upper and lower 100(1-α) % confidence limits for the population

mean μ are, respectively, X+z ⋅σ/√n , and ⎯X−z ⋅σ/√n . Thus, a lower, one-

sided, confidence interval is defined as (-∞ , X+z ⋅σ/√n), and an upper, one-

sided, confidence interval as (X−z ⋅σ/√n,+∞). Notice that in these last two

intervals we use the value z , rather than z .

α/2

In general, the value z in the standard normal distribution is defined as that

value of z whose probability of exceedence is k, i.e., Pr[Z>z ] = k, or Pr[Z<z ]

= 1 – k. The normal distribution was described in Chapter 17.

Confidence intervals for the population mean when the

population variance is unknown

Let ⎯X and S, respectively, be the mean and standard deviation of a random

sample of size n, drawn from an infinite population that follows the normal

distribution with unknown standard deviation σ. The 100⋅(1−α) % [i.e., 99%,

95%, 90%, etc.] central two-sided confidence interval for the population mean

μ, is (⎯X− t

⋅S /√n , ⎯X+ t

⋅S/√n ), where t

is Student's t

n-1, α/2

variate with ν = n-1 degrees of freedom and probability α/2 of exceedence.

The one-sided upper and lower 100⋅ (1-α) % confidence limits for the

population mean μ are, respectively,

X + t

⋅S/√n , and ⎯X− t

⋅S /√n.

n-1, α/2

Page 18-24

Small samples and large samples

The behavior of the Student’s t distribution is such that for n>30, the distribution

is indistinguishable from the standard normal distribution. Thus, for samples

larger than 30 elements when the population variance is unknown, you can use

the same confidence interval as when the population variance is known, but

replacing σ with S. Samples for which n>30 are typically referred to as large

samples, otherwise they are small samples.

Confidence interval for a proportion

A discrete random variable X follows a Bernoulli distribution if X can take only

two values, X = 0 (failure), and X = 1 (success). Let X ~ Bernoulli(p), where p

is the probability of success, then the mean value, or expectation, of X is E[X] =

p, and its variance is Var[X] = p(1-p).

If an experiment involving X is repeated n times, and k successful outcomes are

recorded, then an estimate of p is given by p’= k/n, while the standard error of

p’ is σ = √(p⋅(1-p)/n) . In practice, the sample estimate for p, i.e., p’ replaces

p’

p in the standard error formula.

For a large sample size, n>30, and n⋅p > 5 and n⋅(1-p)>5, the sampling

distribution is very nearly normal. Therefore, the 100(1-α) % central two-sided

confidence interval for the population mean p is (p’+z ⋅σ , p’+z ⋅σ ).

α/2 p’

For a small sample (n<30), the interval can be estimated as (p’-t

⋅σ ,p’+t

n-1,α/2 p’

⋅σ ).

1,α/2 p’

Sampling distribution of differences and sums of statistics

Let S and S be independent statistics from two populations based on samples

of sizes n and n , respectively. Also, let the respective means and standard

errors of the sampling distributions of those statistics be μ and μ , and σ

and σ , respectively. The differences between the statistics from the two

populations, S -S , have a sampling distribution with mean μ

= μ - μ ,

S1−S2

S1 S2

2 1/2

and standard error σ

= (σ

+ σ

)

. Also, the sum of the statistics

S1−S2

T +T has a mean μ

= μ +μ , and standard error σ

= (σ

S1+S2

2 1/2

)

Page 18-25

Estimators for the mean and standard deviation of the difference and sum of the

statistics S and S are given by:

σ _S1σ _{S 2}

μ_S

= X₁X ₂,

σ _S

S₂

n₁

n₂

In these expressions, ⎯X and ⎯X are the values of the statistics S and S from

samples taken from the two populations, and σ

and σ

are the variances

of the populations of the statistics S and S from which the samples were

taken.

Confidence intervals for sums and differences of mean values

If the population variances σ and σ are known, the confidence intervals for

the difference and sum of the mean values of the populations, i.e., μ μ , are

given by:

⎛

⎜

⎞

⎟

σ₁σ ₂

n₁n₂

σ₁σ ₂

n₁n₂

(X₁X ₂) − z_{α / 2}

⋅

, (X₁X ₂) + z_{α / 2}

⋅

⎜

⎟

⎝

⎠

For large samples, i.e., n > 30 and n > 30, and unknown, but equal,

population variances σ = σ , the confidence intervals for the difference and

sum of the mean values of the populations, i.e., μ μ , are given by:

S₁²S₂²

⎛

⎜

⎞

⎟

(X₁X ₂) − z_{α / 2}

⋅

, (X₁X ₂) + z_{α / 2}

⋅

⎜

⎟

n₁n₂

⎝

⎠

If one of the samples is small, i.e., n < 30 or n < 30, and with unknown, but

equal, population variances σ

= σ , we can obtain a “pooled” estimate of

the variance of μ μ , as s = [(n -1)⋅s +(n -1)⋅s ]/( n +n -2).

Page 18-26

In this case, the centered confidence intervals for the sum and difference of the

mean values of the populations, i.e., μ μ , are given by:

(

(X₁X ₂) − t_{ν ,α / 2}⋅ s²_p, (X₁X ₂) + t_{ν ,α / 2}⋅ s²_p

)

where ν = n +n -2 is the number of degrees of freedom in the Student’s t

distribution.

In the last two options we specify that the population variances, although

unknown, must be equal. This will be the case in which the two samples are

taken from the same population, or from two populations about which we

suspect that they have the same population variance. However, if we have

reason to believe that the two unknown population variances are different, we

can use the following confidence interval

(

(X₁X ₂) − t_{ν ,α / 2}⋅ s²

, (X₁X ₂) + t_{ν ,α / 2}⋅ s²

)

X₁X₂

where the estimated standard deviation for the sum or difference is

s₁²s₂²

s_X

X₂

n₁n₂

and n, the degrees of freedom of the t variate, are calculated using the integer

value closest to

[(S₁²/ n₁) + (S₂²/ n₂)]²

ν =

[(S₁²/ n₁) /(n₁−1)] +[(S₂²/ n₂) /(n₂−1)]

Determining confidence intervals

The application 6. Conf Interval can be accessed by using ‚Ù—

@@@OK@@@. The application offers the following options:

Page 18-27

These options are to be interpreted as follows:

1. Z-INT: 1 μ.: Single sample confidence interval for the population mean, μ,

with known population variance, or for large samples with unknown

population variance.

2. Z-INT: μ1−μ2.: Confidence interval for the difference of the population

means, μ - μ , with either known population variances, or for large

samples with unknown population variances.

3. Z-INT: 1 p.: Single sample confidence interval for the proportion, p, for

large samples with unknown population variance.

4. Z-INT: p1− p2.: Confidence interval for the difference of two proportions,

p -p , for large samples with unknown population variances.

5. T-INT: 1 μ.: Single sample confidence interval for the population mean, μ,

for small samples with unknown population variance.

6. T-INT: μ1−μ2.: Confidence interval for the difference of the population

means, μ - μ , for small samples with unknown population variances.

Example 1 – Determine the centered confidence interval for the mean of a

population if a sample of 60 elements indicate that the mean value of the

sample is ⎯x = 23.3, and its standard deviation is s = 5.2. Use α = 0.05. The

confidence level is C = 1-α = 0.95.

Select case 1 from the menu shown above by pressing @@@OK@@@. Enter the values

required in the input form as shown:

Page 18-28

Press @HELP to obtain a screen explaining the meaning of the confidence interval

in terms of random numbers generated by a calculator. To scroll down the

resulting screen use the down-arrow key ˜. Press @@@OK@@@ when done with the

help screen. This will return you to the screen shown above.

To calculate the confidence interval, press @@@OK@@@. The result shown in the

calculator is:

The result indicates that a 95% confidence interval has been calculated. The

Critical z value shown in the screen above corresponds to the values z in

α/2

the confidence interval formula (⎯X−z ⋅σ/√n , ⎯X+z ⋅σ/√n ). The values μ

α/2

Min and μ Max are the lower and upper limits of this interval, i.e., μ Min = ⎯X−

⋅σ/√n, and μ Max = ⎯X+z ⋅σ/√n.

α/2

Press @GRAPH to see a graphical display of the confidence interval information:

The graph shows the standard normal distribution pdf (probability density

function), the location of the critical points z , the mean value (23.3) and the

α/2

corresponding interval limits (21.98424 and 24.61576). Press @TEXT to return

to the previous results screen, and/or press @@@OK@@@ to exit the confidence interval

environment. The results will be listed in the calculator’s display.

Page 18-29

Example 2 -- Data from two samples (samples 1 and 2) indicate that ⎯x =

57.8 and ⎯x = 60.0. The sample sizes are n = 45 and n = 75. If it is

known that the populations’ standard deviations are σ = 3.2, and σ = 4.5,

determine the 90% confidence interval for the difference of the population

means, i.e., μ - μ .

Press ‚Ù—@@@OK@@@to access the confidence interval feature in the

calculator. Press ˜@@@OK@@@ to select option 2. Z-INT: μ 1 – μ2.. Enter the

following values:

When done, press @@@OK@@@. The results, as text and graph, are shown below:

The variable Δμ represents μ 1 – μ2.

Example 3 – A survey of public opinion indicates that in a sample of 150

people 60 favor increasing property taxes to finance some public projects.

Determine the 99% confidence interval for the population proportion that would

favor increasing taxes.

Press ‚Ù—@@@OK@@@to access the confidence interval feature in the

calculator. Press ˜˜ @@@OK@@@ to select option 3. Z-INT: μ 1 – μ2.. Enter the

following values:

Page 18-30

When done, press @@@OK@@@. The results, as text and graph, are shown below:

Example 4 -- Determine a 90% confidence interval for the difference between

two proportions if sample 1 shows 20 successes out of 120 trials, and sample

2 shows 15 successes out of 100 trials.

Press ‚Ù—@@@OK@@@ to access the confidence interval feature in the

calculator. Press ˜˜˜@@@OK@@@ to select option 4. Z-INT: p1 – p2.. Enter

the following values:

When done, press @@@OK@@@. The results, as text and graph, are shown below:

Page 18-31

Example 5 – Determine a 95% confidence interval for the mean of the

population if a sample of 50 elements has a mean of 15.5 and a standard

deviation of 5. The population’s standard deviation is unknown.

Press ‚Ù—@@@OK@@@ to access the confidence interval feature in the

calculator. Press — — @@@OK@@@ to select option 5. T-INT: μ. Enter the

following values:

When done, press @@@OK@@@. The results, as text and graph, are shown below:

The figure shows the Student’s t pdf for ν = 50 – 1 = 49 degrees of freedom.

Example 6 -- Determine the 99% confidence interval for the difference in means

of two populations given the sample data:⎯x = 157.8 ,⎯x = 160.0, n = 50,

n = 55. The populations standard deviations are s = 13.2, s = 24.5.

Press ‚Ù—@@@OK@@@ to access the confidence interval feature in the

calculator. Press —@@@OK@@@ to select option 6. T-INT: μ1−μ2.. Enter the

following values:

hen done, press @@@OK@@@. The results, as text and graph, are shown below:

Page 18-32

These results assume that the values s and s are the population standard

deviations. If these values actually represent the samples’ standard deviations,

you should enter the same values as before, but with the option _pooled

selected. The results now become:

Confidence intervals for the variance

To develop a formula for the confidence interval for the variance, first we

introduce the sampling distribution of the variance: Consider a random sample

X , X ..., X of independent normally-distributed variables with mean μ,

variance σ , and sample mean ⎯X. The statistic

S =

⋅

(X − X )²,

∑

n −1

i=1

is an unbiased estimator of the variance σ .

(X − X )²,

The quantity

has a χ

(chi-square)

(n −1)⋅

n-1

∑

i=1

distribution with ν = n-1 degrees of freedom. The (1-α)⋅100 % two-sided

confidence interval is found from

Pr[χ

< (n-1)⋅S /σ < χ

] = 1- α.

n-1,α/2

n-1,1-α/2

Page 18-33

The confidence interval for the population variance σ is therefore,

[(n-1)⋅S / χ

; (n-1)⋅S / χ

n-1,1-α/2

n-1,α/2

where χ

, and χ

are the values that a χ variable, with ν = n-1

n-1,α/2

n-1,1-α/2

degrees of freedom, exceeds with probabilities α/2 and 1- α /2, respectively.

The one-sided upper confidence limit for σ is defined as (n-1)⋅S / χ

n-1,1-α

Example 1 – Determine the 95% confidence interval for the population variance

σ based on the results from a sample of size n = 25 that indicates that the

sample variance is s = 12.5.

In Chapter 17 we use the numerical solver to solve the equation α = UTPC(γ,x).

In this program, γ represents the degrees of freedom (n-1), and α represents the

probability of exceeding a certain value of x (χ ), i.e., Pr[χ > χ ] = α.

For the present example, α = 0.05, γ = 24 and α = 0.025. Solving the

equation presented above results in χ

= χ

= 39.3640770266.

24,0.025

n-1,α/2

On the other hand, the value χ

= χ

is calculated by using the

n-1,α/2

24,0.975

values γ = 24 and α = 0.975. The result is χ

= χ

n-1,1-α/2

24,0.975

12.4011502175.

The lower and upper limits of the interval will be (Use ALG mode for these

calculations):

(n-1)⋅S / χ

= (25-1)⋅12.5/39.3640770266 = 7.62116179676

n-1,α/2

(n-1)⋅S / χ

= (25-1)⋅12.5/12.4011502175 = 24.1913044144

n-1,1-α/2

Thus, the 95% confidence interval for this example is:

7.62116179676 < σ < 24.1913044144.

Page 18-34

Hypothesis testing

A hypothesis is a declaration made about a population (for instance, with

respect to its mean). Acceptance of the hypothesis is based on a statistical test

on a sample taken from the population. The consequent action and decision-

making are called hypothesis testing.

The process of hypothesis testing consists on taking a random sample from the

population and making a statistical hypothesis about the population. If the

observations do not support the model or theory postulated, the hypothesis is

rejected. However, if the observations are in agreement, then hypothesis is not

rejected, but it is not necessarily accepted. Associated with the decision is a

level of significance α.

Procedure for testing hypotheses

The procedure for hypothesis testing involves the following six steps:

1. Declare a null hypothesis, H . This is the hypothesis to be tested. For

example, H : μ -μ = 0, i.e., we hypothesize that the mean value of

population 1 and the mean value of population 2 are the same. If H is

true, any observed difference in means is attributed to errors in random

sampling.

2. Declare an alternate hypothesis, H . For the example under consideration,

it could be H : μ -μ ≠ 0 [Note: this is what we really want to test.]

3. Determine or specify a test statistic, T. In the example under consideration, T

will be based on the difference of observed means, ⎯X -⎯X .

4. Use the known (or assumed) distribution of the test statistic, T.

5. Define a rejection region (the critical region, R) for the test statistic based on

a pre-assigned significance level α.

6. Use observed data to determine whether the computed value of the test

statistic is within or outside the critical region. If the test statistic is within the

critical region, then we say that the quantity we are testing is significant at

the 100α percent level.

Page 18-35

Notes:

1. For the example under consideration, the alternate hypothesis H : μ -μ ≠ 0

produces what is called a two-tailed test. If the alternate hypothesis is H : μ -μ

> 0 or H : μ -μ < 0, then we have a one-tailed test.

2. The probability of rejecting the null hypothesis is equal to the level of

significance, i.e., Pr[T∈R|H ]=α. The notation Pr[A|B] represents the

conditional probability of event A given that event B occurs.

Errors in hypothesis testing

In hypothesis testing we use the terms errors of Type I and Type II to define the

cases in which a true hypothesis is rejected or a false hypothesis is accepted

(not rejected), respectively. Let T = value of test statistic, R = rejection region, A

= acceptance region, thus, R∩A = ∅, and R∪A = Ω, where Ω = the parameter

space for T, and ∅ = the empty set. The probabilities of making an error of

Type I or of Type II are as follows:

Rejecting a true hypothesis,

Pr[Type I error] = Pr[T∈R|H ] = α

Not rejecting a false hypothesis, Pr[Type II error] = Pr[T∈A|H ] = β

Now, let's consider the cases in which we make the correct decision:

Not rejecting a true hypothesis, Pr[Not(Type I error)] = Pr[T∈A|H ] = 1 - α

Rejecting a false hypothesis, Pr[Not(Type II error)] = Pr [T∈R|H ] = 1 - β

The complement of β is called the power of the test of the null hypothesis H vs.

the alternative H . The power of a test is used, for example, to determine a

minimum sample size to restrict errors.

Selecting values of α and β

A typical value of the level of significance (or probability of Type I error) is α =

0.05, (i.e., incorrect rejection once in 20 times on the average). If the

consequences of a Type I error are more serious, choose smaller values of α,

say 0.01 or even 0.001.

Page 18-36

The value of β, i.e., the probability of making an error of Type II, depends on α,

the sample size n, and on the true value of the parameter tested. Thus, the

value of β is determined after the hypothesis testing is performed. It is

customary to draw graphs showing β, or the power of the test (1- β), as a

function of the true value of the parameter tested. These graphs are called

operating characteristic curves or power function curves, respectively.

Inferences concerning one mean

Two-sided hypothesis

The problem consists in testing the null hypothesis H : μ = μ , against the

alternative hypothesis, H : μ≠ μ at a level of confidence (1-α)100%, or

significance level α, using a sample of size n with a mean ⎯x and a standard

deviation s. This test is referred to as a two-sided or two-tailed test. The

procedure for the test is as follows:

First, we calculate the appropriate statistic for the test (t or z ) as follows:

ꢀ If n < 30 and the standard deviation of the population, σ, is known, use the

x − μ_o

z-statistic:

z_o=

σ / n

ꢀ If n > 30, and σ is known, use z as above. If σ is not known, replace s for

x − μ_o

s / n

σ in z , i.e., use

z_o=

x − μ_o

s/ n

ꢀ If n < 30, and σ is unknown, use the t-statistic

, with ν = n - 1

t_o=

degrees of freedom.

Then, calculate the P-value (a probability) associated with either z or t , and

compare it to α to decide whether or not to reject the null hypothesis. The P-

value for a two-sided test is defined as either

P-value = P(|z|>|z |), or, P-value = P(|t|>|t |).

Page 18-37

The criteria to use for hypothesis testing is:

ꢀ Reject H if P-value < α

ꢀ Do not reject H if P-value > α.

The P-value for a two-sided test can be calculated using the probability functions

in the calculator as follows:

ꢀ If using z, P-value = 2⋅UTPN(0,1,|z |)

ꢀ If using t, P-value = 2⋅UTPT(ν,|t |)

Example 1 -- Test the null hypothesis H : μ = 22.5 ( = μ ), against the

alternative hypothesis, H : μ ≠22.5, at a level of confidence of 95% i.e., α =

0.05, using a sample of size n = 25 with a mean ⎯x = 22.0 and a standard

deviation s = 3.5. We assume that we don't know the value of the population

standard deviation, therefore, we calculate a t statistic as follows:

x − μ_o22.0 − 22.5

t_o=

= −0.7142

s / n

3.5/ 25

The corresponding P-value, for n = 25 - 1 = 24 degrees of freedom is

P-value = 2⋅UTPT(24,-0.7142) = 2⋅0.7590 = 1.518,

since 1.518 > 0.05, i.e., P-value > α, we cannot reject the null hypothesis H : μ

= 22.0.

One-sided hypothesis

The problem consists in testing the null hypothesis H : μ = μ , against the

alternative hypothesis, H : μ > μ or H : μ < μ at a level of confidence (1-

α)100%, or significance level α, using a sample of size n with a mean ⎯x and a

standard deviation s. This test is referred to as a one-sided or one-tailed test.

The procedure for performing a one-side test starts as in the two-tailed test by

calculating the appropriate statistic for the test (t or z ) as indicated above.

Page 18-38

Next, we use the P-value associated with either z or t , and compare it to α to

decide whether or not to reject the null hypothesis. The P-value for a two-sided

test is defined as either

P-value = P(z > |z |), or, P-value = P(t > |t |).

The criteria to use for hypothesis testing is:

ꢀ Reject H if P-value < α

ꢀ Do not reject H if P-value > α.

Notice that the criteria are exactly the same as in the two-sided test. The main

difference is the way that the P-value is calculated. The P-value for a one-sided

test can be calculated using the probability functions in the calculator as

follows:

ꢀ If using z, P-value = UTPN(0,1,z )

ꢀ If using t, P-value = UTPT(ν,t )

Example 2 -- Test the null hypothesis H : μ = 22.0 ( = μ ), against the

alternative hypothesis, H : μ >22.5 at a level of confidence of 95% i.e., α =

0.05, using a sample of size n = 25 with a mean ⎯x = 22.0 and a standard

deviation s = 3.5. Again, we assume that we don't know the value of the

population standard deviation, therefore, the value of the t statistic is the same

as in the two-sided test case shown above, i.e., t = -0.7142, and P-value, for ν

= 25 - 1 = 24 degrees of freedom is

P-value = UTPT(24, |-0.7142|) = UTPT(24,0.7142) = 0.2409,

since 0.2409 > 0.05, i.e., P-value > α, we cannot reject the null hypothesis H :

μ = 22.0.

Inferences concerning two means

The null hypothesis to be tested is H : μ -μ = δ, at a level of confidence (1-

α)100%, or significance level α, using two samples of sizes, n and n , mean

Page 18-39

values ⎯x and ⎯x , and standard deviations s and s . If the populations

standard deviations corresponding to the samples, σ and σ , are known, or if

n > 30 and n > 30 (large samples), the test statistic to be used is

(x₁− x₂) −δ

z_o=

σ₁σ₂

n₁

n₂

If n < 30 or n < 30 (at least one small sample), use the following test statistic:

(x₁− x₂) −δ

(n₁−1)s₁²+ (n₂−1)s₂²

n₁n₂(n₁+ n₂− 2)

t =

n₁+ n₂

Two-sided hypothesis

If the alternative hypothesis is a two-sided hypothesis, i.e., H : μ -μ ≠ δ, The P-

value for this test is calculated as

ꢀ If using z,

P-value = 2⋅UTPN(0,1, |z |)

ꢀ If using t,

P-value = 2⋅UTPT(ν,|t |)

with the degrees of freedom for the t-distribution given by ν = n + n - 2. The

test criteria are

ꢀ Reject H if P-value < α

ꢀ Do not reject H if P-value > α.

One-sided hypothesis

If the alternative hypothesis is a two-sided hypothesis, i.e., H : μ -μ < δ, or,

H : μ -μ < δ,, the P-value for this test is calculated as:

ꢀ If using z,

P-value = UTPN(0,1, |z |)

ꢀ If using t,

P-value = UTPT(ν,|t |)

Page 18-40

The criteria to use for hypothesis testing is:

ꢀ Reject H if P-value < α

ꢀ Do not reject H if P-value > α.

Paired sample tests

When we deal with two samples of size n with paired data points, instead of

testing the null hypothesis, H : μ -μ = δ, using the mean values and standard

deviations of the two samples, we need to treat the problem as a single sample

of the differences of the paired values. In other words, generate a new random

variable X = X -X , and test H : μ = δ, where μ represents the mean of the

population for X. Therefore, you will need to obtain⎯x and s for the sample of

values of x. The test should then proceed as a one-sample test using the

methods described earlier.

Inferences concerning one proportion

Suppose that we want to test the null hypothesis, H : p = p , where p represents

the probability of obtaining a successful outcome in any given repetition of a

Bernoulli trial. To test the hypothesis, we perform n repetitions of the

experiment, and find that k successful outcomes are recorded. Thus, an

estimate of p is given by p’ = k/n.

The variance for the sample will be estimated as s = p’(1-p’)/n = k⋅(n-k)/n .

Assume that the Z score, Z = (p-p )/s , follows the standard normal distribution,

i.e., Z ~ N(0,1). The particular value of the statistic to test is z = (p’-p )/s .

Instead of using the P-value as a criterion to accept or not accept the hypothesis,

we will use the comparison between the critical value of z0 and the value of z

corresponding to α or α/2.

Two-tailed test

If using a two-tailed test we will find the value of z

, from

α/2

Pr[Z> z ] = 1-Φ(z ) = α/2, or Φ(z ) = 1- α/2,

α/2

Page 18-41

where Φ(z) is the cumulative distribution function (CDF) of the standard normal

distribution (see Chapter 17).

Reject the null hypothesis, H , if z >z , or if z < - z .

α/2

In other words, the rejection region is R = { |z | > z

}, while the acceptance

α/2

region is A = {|z | < z

α/2

One-tailed test

If using a one-tailed test we will find the value of S , from

Pr[Z> z ] = 1-Φ(z ) = α, or Φ(z ) = 1- α,

Reject the null hypothesis, H , if z >z , and H : p>p , or if z < - z , and H :

p<p .

Testing the difference between two proportions

Suppose that we want to test the null hypothesis, H : p -p = p , where the p's

represents the probability of obtaining a successful outcome in any given

repetition of a Bernoulli trial for two populations 1 and 2. To test the

hypothesis, we perform n repetitions of the experiment from population 1, and

find that k successful outcomes are recorded. Also, we find k successful

outcomes out of n trials in sample 2. Thus, estimates of p and p are given,

respectively, by p ’ = k /n , and p ’ = k /n .

The variances for the samples will be estimated, respectively, as

= p ’(1-p ’)/n = k ⋅(n -k )/n , and s = p ’(1-p ’)/n = k ⋅(n -k )/n .

1 1

2 2

And the variance of the difference of proportions is estimated from: s = s

Assume that the Z score, Z = (p -p -p )/s , follows the standard normal

distribution, i.e., Z ~ N(0,1). The particular value of the statistic to test is z =

(p ’-p ’-p )/s .

Page 18-42

Two-tailed test

If using a two-tailed test we will find the value of z

, from

α/2

Pr[Z> z ] = 1-Φ(z ) = α/2, or Φ(z ) = 1- α/2,

α/2

where Φ(z) is the cumulative distribution function (CDF) of the standard normal

distribution.

Reject the null hypothesis, H , if z >z , or if z < - z .

α/2

In other words, the rejection region is R = { |z | > z

}, while the acceptance

α/2

region is A = {|z | < z

α/2

One-tailed test

If using a one-tailed test we will find the value of z , from

Pr[Z> z ] = 1-Φ(z ) = α, or Φ(z ) = 1- α,

Reject the null hypothesis, H , if z >z , and H : p -p > p , or if z < - z , and

H : p -p <p .

Hypothesis testing using pre-programmed features

The calculator provides with hypothesis testing procedures under application 5.

Hypoth. tests.. can be accessed by using ‚Ù—— @@@OK@@@.

As with the calculation of confidence intervals, discussed earlier, this program

offers the following 6 options:

These options are interpreted as in the confidence interval applications:

Page 18-43

1. Z-Test: 1 μ.: Single sample hypothesis testing for the population mean, μ,

with known population variance, or for large samples with unknown

population variance.

2. Z-Test: μ1−μ2.: Hypothesis testing for the difference of the population

means, μ - μ , with either known population variances, or for large

samples with unknown population variances.

3. Z-Test: 1 p.: Single sample hypothesis testing for the proportion, p, for

large samples with unknown population variance.

4. Z-Test: p1− p2.: Hypothesis testing for the difference of two proportions, p -

p , for large samples with unknown population variances.

5. T-Test: 1 μ.: Single sample hypothesis testing for the population mean, μ,

for small samples with unknown population variance.

6. T-Test: μ1−μ2.: Hypothesis testing for the difference of the population

means, μ - μ , for small samples with unknown population variances.

Try the following exercises:

Example 1 – For μ = 150, σ = 10, ⎯x = 158, n = 50, for α = 0.05, test the

hypothesis H : μ = μ , against the alternative hypothesis, H : μ ≠ μ .

Press ‚Ù—— @@@OK@@@ to access the hypothesis testing feature in the

calculator. Press @@@OK@@@ to select option 1. Z-Test: 1 μ.

Enter the following data and press @@@OK@@@:

You are then asked to select the alternative hypothesis. Select μ ≠150, and

press @@OK@@. The result is:

Page 18-44

Then, we reject H : μ = 150, against H : μ ≠ 150. The test z value is z =

-8

5.656854. The P-value is 1.54×10 . The critical values of z

α/2

1.959964, corresponding to critical ⎯x range of {147.2 152.8}.

This information can be observed graphically by pressing the soft-menu key

@GRAPH:

Example 2 -- For μ = 150, ⎯x = 158, s = 10, n = 50, for α = 0.05, test the

hypothesis H : μ = μ , against the alternative hypothesis, H : μ > μ . The

population standard deviation, σ, is not known.

Press ‚Ù—— @@@OK@@@ to access the hypothesis testing feature in the

calculator. Press ——@@@OK@@@ to select option 5. T-Test: 1 μ.:

Enter the following data and press @@@OK@@@:

Select the alternative hypothesis, H : μ > 150, and press @@@OK@@@. The result is:

Page 18-45

We reject the null hypothesis, H : μ = 150, against the alternative hypothesis,

H : μ > 150. The test t value is t = 5.656854, with a P-value =

0.000000393525. The critical value of t is t = 1.676551, corresponding to a

critical ⎯x = 152.371.

Press @GRAPH to see the results graphically as follows:

Example 3 – Data from two samples show that⎯x = 158, ⎯x = 160, s = 10,

s = 4.5, n1 = 50, and n = 55. For α = 0.05, and a “pooled” variance, test

the hypothesis H : μ −μ = 0, against the alternative hypothesis, H : μ −μ < 0.

Press ‚Ù—— @@@OK@@@ to access the hypothesis testing feature in the

calculator. Press —@@@OK@@@ to select option 6. T-Test: μ1−μ2.: Enter the

following data and press @@@OK@@@:

Select the alternative hypothesis μ1< μ2, and press @@@OK@@@. The result is

Page 18-46

Thus, we accept (more accurately, we do not reject) the hypothesis: H : μ −μ =

0, or H : μ =μ , against the alternative hypothesis H : μ −μ < 0, or H : μ =μ .

The test t value is t = -1.341776, with a P-value = 0.09130961, and critical t is

–t = -1.659782. The graphical results are:

These three examples should be enough to understand the operation of the

hypothesis testing pre-programmed feature in the calculator.

Inferences concerning one variance

The null hypothesis to be tested is , H : σ = σ , at a level of confidence (1-

α)100%, or significance level α, using a sample of size n, and variance s . The

test statistic to be used is a chi-squared test statistic defined as

(n −1)s²

χ_o=

σ ₀

Depending on the alternative hypothesis chosen, the P-value is calculated as

follows:

ꢀ H : σ < σ ,

P-value = P(χ <χ ) = 1-UTPC(ν,χ )

ꢀ H : σ > σ ,

P-value = P(χ >χ ) = UTPC(ν,χ )

ꢀ H : σ ≠ σ ,

P-value =2⋅min[P(χ <χ ), P(χ >χ )] =

2⋅min[1-UTPC(ν,χ ), UTPC(ν,χ )]

where the function min[x,y] produces the minimum value of x or y (similarly,

max[x,y] produces the maximum value of x or y). UTPC(ν,x) represents the

calculator’s upper-tail probabilities for ν = n - 1 degrees of freedom.

Page 18-47

The test criteria are the same as in hypothesis testing of means, namely,

ꢀ Reject H if P-value < α

ꢀ Do not reject H if P-value > α.

Please notice that this procedure is valid only if the population from which the

sample was taken is a Normal population.

Example 1 -- Consider the case in which σ = 25, α=0.05, n = 25, and s =

20, and the sample was drawn from a normal population. To test the

hypothesis, H : σ = σ , against H : σ < σ , we first calculate

With ν = n - 1 = 25 - 1 = 24 degrees of freedom, we calculate the P-value as,

P-value = P(χ <19.2) = 1-UTPC(24,19.2) = 0.2587…

Since, 0.2587… > 0.05, i.e., P-value > α, we cannot reject the null hypothesis,

H : σ =25(= σ ).

Inferences concerning two variances

The null hypothesis to be tested is , H : σ = σ , at a level of confidence (1-

α)100%, or significance level α, using two samples of sizes, n and n , and

variances s and s . The test statistic to be used is an F test statistic defined

s_N²

s_D²

F_o=

where s and s represent the numerator and denominator of the F statistic,

respectively. Selection of the numerator and denominator depends on the

alternative hypothesis being tested, as shown below. The corresponding F

distribution has degrees of freedom, ν = n -1, and ν = n -1, where n and

n , are the sample sizes corresponding to the variances s and s ,

respectively.

Page 18-48

The following table shows how to select the numerator and denominator for F

depending on the alternative hypothesis chosen:

____________________________________________________________________

Alternative

hypothesis

Test

statistic

Degrees

of freedom

____________________________________________________________________

H : σ < σ

(one-sided)

(two-sided)

F = s /s

ν = n -1, ν = n -1

H : σ > σ

F = s /s

ν = n -1, ν = n -1

H : σ ≠σ

F = s /s

ν = n -1,ν = n -1

=max(s ,s ), s =min(s ,s )

___________________________________________________________________

(*) n is the value of n corresponding to the s , and n is the value of n

corresponding to s .

____________________________________________________________________

The P-value is calculated, in all cases, as: P-value = P(F>F ) = UTPF(ν , ν ,F )

The test criteria are:

ꢀ Reject H if P-value < α

ꢀ Do not reject H if P-value > α.

Example1 -- Consider two samples drawn from normal populations such that n

= 21, n = 31, s = 0.36, and s

= 0.25. We test the null hypothesis, H :

= σ , at a significance level α = 0.05, against the alternative hypothesis,

H : σ ≠ σ . For a two-sided hypothesis, we need to identify s and s , as

follows:

=max(s ,s ) = max(0.36,0.25) = 0.36 = s

=min(s ,s ) = min (0.36,0.25) = 0.25 = s

Also,

n = n = 21,

n = n = 31,

ν = n - 1= 21-1=20,

ν = n -1 = 31-1 =30.

Page 18-49

Therefore, the F test statistics is F = s /s =0.36/0.25=1.44

The P-value is P-value = P(F>F ) = P(F>1.44) = UTPF(ν , ν ,F ) =

UTPF(20,30,1.44) = 0.1788…

Since 0.1788… > 0.05, i.e., P-value > α, therefore, we cannot reject the null

hypothesis that H : σ = σ .

Additional notes on linear regression

In this section we elaborate the ideas of linear regression presented earlier in

the chapter and present a procedure for hypothesis testing of regression

parameters.

The method of least squares

Let x = independent, non-random variable, and Y = dependent, random

variable. The regression curve of Y on x is defined as the relationship between

x and the mean of the corresponding distribution of the Y’s.

Assume that the regression curve of Y on x is linear, i.e., mean distribution of

Y’s is given by Α + Βx. Y differs from the mean (Α + Β⋅x) by a value ε, thus

Y = Α + Β⋅x + ε, where ε is a random variable.

To visually check whether the data follows a linear trend, draw a scattergram or

scatter plot.

Suppose that we have n paired observations (x, y); we predict y by means of

∧

y = a + b⋅x, where a and b are constant.

∧

Define the prediction error as, e = y - y = y - (a + b⋅x ).

The method of least squares requires us to choose a, b so as to minimize the

sum of squared errors (SSE)

SSE = e²= [y − (a + bx )]²

∑ ∑

i=1

the conditions

∂

∂a

∂

∂b

(SSE) = 0

Page 18-50

We get the, so-called, normal equations:

y = a ⋅ n + b ⋅

∑

i=1

x ⋅ y = a ⋅ x + b ⋅ x²

∑

i=1

This is a system of linear equations with a and b as the unknowns, which can

be solved using the linear equation features of the calculator. There is,

however, no need to bother with these calculations because you can use the 3.

Fit Data … option in the ‚Ù menu as presented earlier.

Notes:

ꢀ a,b are unbiased estimators of Α, Β.

ꢀ The Gauss-Markov theorem of probability indicates that among all

unbiased estimators for Α and Β, the least-square estimators (a,b) are the

most efficient.

Additional equations for linear regression

The summary statistics such as Σx, Σx , etc., can be used to define the following

quantities:

1 ⎛

⎞

⎟

⎠

S = (x − x)²= (n −1)⋅ s²= x ²−

⎜

∑

i=1

⎝

i=1

1 ⎛

⎞

S = (y − y)²= (n −1)⋅ s²=

y ²−

⎜

⎟

⎠

∑

i=1

⎝

i=1

1 ⎛

⎞⎛

∑ ∑

⎞

⎟

⎠

S_xy

(x − x)(y − y)²= (n −1) ⋅ s =

x y −

⎜

⎟⎜

⎠⎝

∑

i=1

⎝

Page 18-51

From which it follows that the standard deviations of x and y, and the

covariance of x,y are given, respectively, by

S_yy

S_yx

S_xx

s_x=

s_y=

s_xy=

n −1

, and

S_xy

S_xx⋅ S_yy

Also, the sample correlation coefficient is

r_xy=

In terms of ⎯x, ⎯y, S , S , and S , the solution to the normal equations is:

xx yy

S_xys_xy

b =

s_x²

S_xx

a = y − bx

Prediction error

The regression curve of Y on x is defined as Y = Α + Β⋅x + ε. If we have a set

of n data points (x , y ), then we can write Y = Α + Β⋅x + ε , (i = 1,2,…,n),

where Y = independent, normally distributed random variables with mean (Α +

Β⋅x ) and the common variance σ ; ε = independent, normally distributed

random variables with mean zero and the common variance σ .

Let y = actual data value, y = a + b⋅x = least-square prediction of the data.

Then, the prediction error is: e = y - y = y - (a + b⋅x ).

An estimate of σ is the, so-called, standard error of the estimate,

S_yy− (S_xy)²/S _xx

n −1

n − 2

s_e²=

[y − (a + bx )]²=

⋅ s_y²⋅(1− r_x²_y)

∑

n − 2

i=1

Confidence intervals and hypothesis testing in linear regression

Here are some concepts and equations related to statistical inference for linear

regression:

Page 18-52

ꢀ Confidence limits for regression coefficients:

For the slope (Β): b − (t

)⋅s /√S < Β < b + (t

)⋅s /√S ,

n-2,α/2 xx

n-2,α/2

For the intercept (Α):

1/2

a − (t

)⋅s ⋅[(1/n)+⎯x /S ]

< Α < a + (t

)⋅s ⋅[(1/n)+⎯x /

n-2,α/2

1/2

S ] , where t follows the Student’s t distribution with ν = n – 2, degrees

of freedom, and n represents the number of points in the sample.

ꢀ Hypothesis testing on the slope, Β:

Null hypothesis, H : Β = Β , tested against the alternative hypothesis, H :

Β ≠ Β . The test statistic is t = (b -Β )/(s /√S ), where t follows the

Student’s t distribution with ν = n – 2, degrees of freedom, and n represents

the number of points in the sample. The test is carried out as that of a

mean value hypothesis testing, i.e., given the level of significance, α,

determine the critical value of t, t , then, reject H if t > t

or if t < -

α/2

If you test for the value Β = 0, and it turns out that the test suggests that you

do not reject the null hypothesis, H : Β = 0, then, the validity of a linear

regression is in doubt. In other words, the sample data does not support

the assertion that Β ≠ 0. Therefore, this is a test of the significance of the

regression model.

ꢀ Hypothesis testing on the intercept , Α:

Null hypothesis, H : Α = Α , tested against the alternative hypothesis, H :

1/2

Α ≠ Α . The test statistic is t = (a-Α )/[(1/n)+⎯x /S ] , where t follows

the Student’s t distribution with ν = n – 2, degrees of freedom, and n

represents the number of points in the sample. The test is carried out as

that of a mean value hypothesis testing, i.e., given the level of significance,

α, determine the critical value of t, t , then, reject H if t > t

or if t <

α/2

- t

α/2

ꢀ Confidence interval for the mean value of Y at x = x , i.e., α+βx :

1/2

a+b⋅x−(t

)⋅s ⋅[(1/n)+(x -⎯x) /S ]

< α+βx <

n-2,α/2

1/2

a+b⋅x+(t

)⋅s ⋅[(1/n)+(x -⎯x) /S ]

n-2, α /2

ꢀ Limits of prediction: confidence interval for the predicted value Y =Y(x ):

1/2

a+b⋅x−(t

)⋅s ⋅[1+(1/n)+(x -⎯x) /S ]

< Y <

n-2,α/2

Page 18-53

1/2

a+b⋅x+(t

)⋅s ⋅[1+(1/n)+(x -⎯x) /S ]

n-2, α /2

Procedure for inference statistics for linear regression using the

calculator

1) Enter (x,y) as columns of data in the statistical matrix ΣDAT.

2) Produce a scatterplot for the appropriate columns of ΣDAT, and use

appropriate H- and V-VIEWS to check linear trend.

3) Use ‚Ù˜˜@@@OK@@@, to fit straight line, and get a, b, s

(Covariance), and r (Correlation).

4) Use ‚Ù˜@@@OK@@@, to obtain ⎯x, ⎯y, s , s . Column 1 will show the

statistics for x while column 2 will show the statistics for y.

5) Calculate

n −1

n − 2

s_e²=

⋅ s_y²⋅(1− r_x²_y)

S_xx= (n −1)⋅ s_x²

6) For either confidence intervals or two-tailed tests, obtain t , with (1-

α/2

α)100% confidence, from t-distribution with ν = n -2.

7) For one- or two-tailed tests, find the value of t using the appropriate

equation for either Α or Β. Reject the null hypothesis if P-value< α.

8) For confidence intervals use the appropriate formulas as shown above.

Example 1 -- For the following (x,y) data, determine the 95% confidence

interval for the slope B and the intercept A

2.0

5.5

2.5

7.2

3.0

9.4

3.5

10.0 12.2

4.0

Enter the (x,y) data in columns 1 and 2 of ΣDAT, respectively. A scatterplot of

the data shows a good linear trend:

Use the Fit Data.. option in the ‚Ù menu, to get:

3: '-.86 + 3.24*X'

2: Correlation: 0.989720229749

Page 18-54

1: Covariance: 2.025

These results are interpreted as a = -0.86, b = 3.24, r = 0.989720229749,

and s = 2.025. The correlation coefficient is close enough to 1.0 to confirm

the linear trend observed in the graph.

From the Single-var…option of the ‚Ù menu we find: ⎯x = 3, s =

0.790569415042,⎯y = 8.86, s = 2.58804945857.

Next, with n = 5, calculate

S_xx= (n −1) ⋅ s_x²= (5 −1)⋅0.790569415042²= 2.5

n −1

n − 2

s_e²=

⋅ s_y²⋅ (1− r_x²_y) =

5 −1

5 − 2

⋅ 2.5880...²⋅(1− 0.9897...²) = 0.1826...

Confidence intervals for the slope (Β) and intercept (A):

ꢀ First, we obtain t

= t ,

= 3.18244630528 (See chapter 17 for

n-2,α/2

3 0.025

a program to solve for t ):

ν,a

ꢀ Next, we calculate the terms

1/2

)⋅s /√S = 3.182…⋅(0.1826…/2.5)

= 0.8602…

n-2,α/2

1/2

)⋅s ⋅[(1/n)+⎯x /S ]

n-2,α/2

1/2

3.1824…⋅√0.1826…⋅[(1/5)+3 /2.5]

= 2.65

ꢀ Finally, for the slope B, the 95% confidence interval is

(-0.86-0.860242, -0.86+0.860242) = (-1.72, -0.00024217)

For the intercept A, the 95% confidence interval is (3.24-2.6514,

3.24+2.6514) = (0.58855,5.8914).

Page 18-55

Example 2 -- Suppose that the y-data used in Example 1 represent the

elongation (in hundredths of an inch) of a metal wire when subjected to a force

x (in tens of pounds). The physical phenomenon is such that we expect the

intercept, A, to be zero. To check if that should be the case, we test the null

hypothesis, H : Α = 0, against the alternative hypothesis, H : Α ≠ 0, at the level

of significance α = 0.05.

1/2

The test statistic is t = (a-0)/[(1/n)+⎯x /S ]

= (-0.86)/ [(1/5)+3 /2.5]

-0.44117. The critical value of t, for ν = n – 2 = 3, and α/2 = 0.025, can be

calculated using the numerical solver for the equation α = UTPT(γ,t) developed

in Chapter 17. In this program, γ represents the degrees of freedom (n-2), and

α represents the probability of exceeding a certain value of t, i.e., Pr[ t>t ] = 1

– α. For the present example, the value of the level of significance is α = 0.05,

g = 3, and t

= t

. Also, for γ = 3 and α = 0.025, t

= t

n-2,α/2

3,0.025

n-2,α/2 3,0.025

= 3.18244630528. Because t > - t

, we cannot reject the null

n-2,α/2

hypothesis, H : Α = 0, against the alternative hypothesis, H : Α ≠ 0, at the level

of significance α = 0.05.

This result suggests that taking A = 0 for this linear regression should be

acceptable. After all, the value we found for a, was –0.86, which is relatively

close to zero.

Example 3 – Test of significance for the linear regression. Test the null

hypothesis for the slope H : Β = 0, against the alternative hypothesis, H : Β ≠

0, at the level of significance α = 0.05, for the linear fitting of Example 1.

The test statistic is t = (b -Β )/(s /√S ) = (3.24-0)/(√0.18266666667/2.5) =

18.95. The critical value of t, for ν = n – 2 = 3, and α/2 = 0.025, was

obtained in Example 2, as t

= t

= 3.18244630528. Because, t >

n-2,α/2

3,0.025

, we must reject the null hypothesis H : Β ≠ 0, at the level of significance α

α/2

= 0.05, for the linear fitting of Example 1.

Page 18-56

Multiple linear fitting

Consider a data set of the form

…

1,m-1

2,m-1

2,m

3,m-1

3,m

n,m-1

n,m

m-1

1,m

Suppose that we search for a data fitting of the form y = b + b ⋅x + b ⋅x +

b ⋅x + … + b ⋅x . You can obtain the least-square approximation to the

values of the coefficients b = [b

b … b ], by putting together the

matrix X:

…

1,m

2,m

3,m

n,m

-1

Then, the vector of coefficients is obtained from b = (X ⋅X) ⋅X ⋅y, where y is

the vector y = [y y … y ] .

For example, use the following data to obtain the multiple linear fitting

y = b + b ⋅x + b ⋅x + b ⋅x

Page 18-57

1.20

2.50

3.50

4.00

6.00

3.10

4.50

5.00

2.00

2.50

3.00

3.50

5.70

8.20

5.00

8.20

9.50

With the calculator, in RPN mode, you can proceed as follows:

First, within your HOME directory, create a sub-directory to be called MPFIT

(Multiple linear and Polynomial data FITting) , and enter the MPFIT sub-

directory. Within the sub-directory, type this program:

« ꢀ X y « X TRAN X * INV X TRAN * y * » »

and store it in a variable called MTREG (MulTiple REGression).

Next, enter the matrices X and b into the stack:

[[1,1.2,3.1,2][1,2.5,3.1,2.5 ][1,3.5,4.5,2.5][1,4,4.5,3][1,6,5,3.5]]

`` (keep an extra copy)

[5.7,8.2,5.0,8.2,9.5] `

Press J@MTREG. The result is: [-2.1649…,–0.7144…,-1.7850…,7.0941…],

i.e.,

-2

y = -2.1649–0.7144⋅x -1.7850×10 ⋅x + 7.0941⋅x .

You should have in your calculator’s stack the value of the matrix X and the

vector b, the fitted values of y are obtained from y = X⋅b, thus, just press *

to obtain: [5.63.., 8.25.., 5.03.., 8.22.., 9.45..].

Page 18-58

Compare these fitted values with the original data as shown in the table below:

y-fitted

1.20

2.50

3.50

4.00

6.00

3.10

4.50

5.00

2.00

2.50

3.00

3.50

5.70

8.20

5.00

8.20

9.50

5.63

8.25

5.03

8.22

9.45

Polynomial fitting

Consider the x-y data set {(x ,y ), (x ,y ), …, (x ,y )}. Suppose that we want

to fit a polynomial or order p to this data set. In other words, we seek a fitting

of the form y = b + b ⋅x + b ⋅x + b ⋅x + … + b ⋅x . You can obtain the

least-square approximation to the values of the coefficients b = [b b b b

… b ], by putting together the matrix X

p-1

…

p-1

-1

Then, the vector of coefficients is obtained from b = (X ⋅X) ⋅X ⋅y, where y is

the vector y = [y y … y ] .

In Chapter 10, we defined the Vandermonde matrix corresponding to a vector

x = [x x … x ] . The Vandermonde matrix is similar to the matrix X of

interest to the polynomial fitting, but having only n, rather than (p+1) columns.

We can take advantage of the VANDERMONDE function to create the matrix X

if we observe the following rules:

If p = n-1, X = V .

If p < n-1, then remove columns p+2, …, n-1, n from V to form X.

Page 18-59

If p > n-1, then add columns n+1, …, p-1, p+1, to V to form matrix X.

In step 3 from this list, we have to be aware that column i (i= n+1, n+2, …,

p+1) is the vector [x x … x ]. If we were to use a list of data values for x

rather than a vector, i.e., x = { x x … x }, we can easily calculate the

sequence { x x … x }. Then, we can transform this list into a vector and use

the COL menu to add those columns to the matrix V until X is completed.

After X is ready, and having the vector y available, the calculation of the

coefficient vector b is the same as in multiple linear fitting (the previous matrix

application). Thus, we can write a program to calculate the polynomial fitting

that can take advantage of the program already developed for multiple linear

fitting. We need to add to this program the steps 1 through 3 listed above.

The algorithm for the program, therefore, can be written as follows:

Enter vectors x and y, of the same dimension, as lists. (Note: since the

function VANDERMONDE uses a list as input, it is more convenient to enter the

(x,y) data as a list.) Also, enter the value of p.

ꢀ Determine n = size of vector x.

ꢀ Use the function VANDERMONDE to generate the Vandermonde

matrix V for the list x entered.

ꢀ If p = n-1, then

X = V ,

Else If p < n-1

Remove columns p+2, …, n from V to form X

(Use a FOR loop and COL-)

Else

Add columns n+1, …, p+1 to V to form X

(FOR loop, calculate x , convert to vector, use COL+)

ꢀ Convert y to vector

ꢀ Calculate b using program MTREG (see example on multiple linear

fitting above)

Here is the translation of the algorithm to a program in User RPL language.

(See Chapter 21 for additional information on programming):

Page 18-60

Open program

Enter lists x and y, and p (levels 3,2,1)

Open subprogram 1

ꢀ x y p

x SIZE ꢀ n

Determine size of x list

Open subprogram 2

x VANDERMONDE

Place x in stack, obtain V

IF ‘p<n-1’ THEN

p 2 +

This IF implements step 3 in algorithm

Place n in stack

Calculate p+1

FOR j

Start loop j = n-1, n-2, …, p+1, step = -1

Remove column and drop it from stack

Close FOR-STEP loop

j COL− DROP

-1 STEP

ELSE

IF ‘p>n-1’ THEN

n 1 +

Calculate n+1

p 1 +

FOR j

Calculate p+1

Start a loop with j = n, n+1, …, p+1.

x j ^

OBJꢀ ꢀARRY

j COL+

Calculate x , as a list

Convert list to array

Add column to matrix

END

Close FOR-NEXT loop

Ends second IF clause.

Ends first IF clause. Its result is X

Convert list y to an array

X and y used by program MTREG

Convert to decimal format

Close sub-program 2

END

y OBJꢀ ꢀARRY

MTREG

ꢀNUM

Close sub-program 1

Close main program

Save it into a variable called POLY (POLYnomial fitting).

As an example, use the following data to obtain a polynomial fitting with p =

2, 3, 4, 5, 6.

Page 18-61

2.30

3.20

4.50

1.65

179.72

562.30

1969.11

65.87

9.32 31220.89

1.18

6.24

3.45

32.81

6731.48

737.41

9.89 39248.46

1.22 33.45

Because we will be using the same x-y data for fitting polynomials of different

orders, it is advisable to save the lists of data values x and y into variables xx

and yy, respectively. This way, we will not have to type them all over again in

each application of the program POLY. Thus, proceed as follows:

{ 2.3 3.2 4.5 1.65 9.32 1.18 6.24 3.45 9.89 1.22 } ` ‘xx’ K

{179.72 562.30 1969.11 65.87 31220.89 32.81 6731.48 737.41 39248.46

33.45} ` ‘yy’ K

To fit the data to polynomials use the following:

@@xx@@ @@yy@@ 2 @POLY, Result: [4527.73 -3958.52 742.23]

i.e.,

y = 4527.73-3958.52x+742.23x

@@xx@@ @@yy@@ 3 @POLY, Result: [ –998.05 1303.21 -505.27 79.23]

y = -998.05+1303.21x-505.27x +79.23x

@@xx@@ @@yy@@ 4 @POLY, Result: [20.92 –2.61 –1.52 6.05 3.51 ]

i.e.,

y = 20.92-2.61x-1.52x +6.05x +3.51x .

@@xx@@ @@yy@@ 5 @POLY, Result: [19.08 0.18 –2.94 6.36 3.48 0.00 ]

i.e.,

y = 19.08+0.18x-2.94x +6.36x +3.48x +0.0011x

@@xx@@ @@yy@@ 6 @POLY, Result: [-16.73 67.17 –48.69 21.11 1.07 0.19 0.00]

i.e.,

y = -16.72+67.17x-48.69x +21.11x +1.07x +0.19x -0.0058x

Selecting the best fitting

As you can see from the results above, you can fit any polynomial to a set of

data. The question arises, which is the best fitting for the data? To help one

decide on the best fitting we can use several criteria:

Page 18-62

ꢀ The correlation coefficient, r. This value is constrained to the range –1

< r < 1. The closer r is to +1 or –1, the better the data fitting.

ꢀ The sum of squared errors, SSE. This is the quantity that is to be

minimized by least-square approach.

ꢀ A plot of residuals. This is a plot of the error corresponding to each of

the original data points. If these errors are completely random, the

residuals plot should show no particular trend.

Before attempting to program these criteria, we present some definitions:

Given the vectors x and y of data to be fit to the polynomial equation, we form

the matrix X and use it to calculate a vector of polynomial coefficients b. We

can calculate a vector of fitted data, y’, by using y’ = X⋅b.

An error vector is calculated by e = y – y’.

The sum of square errors is equal to the square of the magnitude of the error

vector, i.e., SSE = |e| = e•e = Σ e = Σ (y -y’ ) .

To calculate the correlation coefficient we need to calculate first what is known

as the sum of squared totals, SST, defined as SST = Σ (y -⎯y) , where ⎯y is the

mean value of the original y values, i.e., ⎯y = (Σy )/n.

In terms of SSE and SST, the correlation coefficient is defined by

1/2

r = [1-(SSE/SST)]

Here is the new program including calculation of SSE and r (Once more,

consult the last page of this chapter to see how to produce the variable and

command names in the program):

Open program

Enter lists x and y, and number p

ꢀ x y p

Open subprogram1

Determine size of x list

Open subprogram 2

x SIZE ꢀ n

Page 18-63

x VANDERMONDE

Place x in stack, obtain V

IF ‘p<n-1’ THEN

p 2 +

This IF is step 3 in algorithm

Place n in stack

Calculate p+1

FOR j

j COL− DROP

-1 STEP

Start loop, j = n-1 to p+1, step = -1

Remove column, drop from stack

Close FOR-STEP loop

ELSE

IF ‘p>n-1’ THEN

n 1 +

Calculate n+1

p 1 +

FOR j

Calculate p+1

Start loop with j = n, n+1, …, p+1.

x j ^

OBJꢀ ꢀARRY

j COL+

Calculate x , as a list

Convert list to array

Add column to matrix

Close FOR-NEXT loop

Ends second IF clause.

Ends first IF clause. Produces X

Convert list y to an array

Enter matrix and array as X and y

Open subprogram 3

END

y OBJꢀ ꢀARRY

X yv

ꢀ

X yv MTREG

ꢀNUM

ꢀ b

X and y used by program MTREG

If needed, converts to floating point

Resulting vector passed as b

Open subprogram 4

b yv

X b *

Place b and yv in stack

Calculate X⋅b

Calculate e = y - X⋅b

Calculate SSE, make copy

Calculate ⎯y

Create vector of n values of ⎯y

Calculate SST

ABS SQ DUP

y ΣLIST n /

n 1 ꢀLIST SWAP CON

yv − ABS SQ

Calculate SSE/SST

1/2

NEG 1 + √

“r” ꢀTAG

SWAP

Calculate r = [1–SSE/SST ]

Tag result as “r”

Exchange stack levels 1 and 2

Page 18-64

“SSE” ꢀTAG

Tag result as SSE

Close sub-program 4

Close sub-program 3

Close sub-program 2

Close sub-program 1

Close main program

Save this program under the name POLYR, to emphasize calculation of the

correlation coefficient r.

Using the POLYR program for values of p between 2 and 6 produce the

following table of values of the correlation coefficient, r, and the sum of square

errors, SSE:

SSE

10731140.01

88619.36

7.48

0.9971908

0.9999768

0.9999999

0.9999998

8.92

432.60

While the correlation coefficient is very close to 1.0 for all values of p in the

table, the values of SSE vary widely. The smallest value of SSE corresponds to p

= 4. Thus, you could select the preferred polynomial data fitting for the

original x-y data as:

y = 20.92-2.61x-1.52x +6.05x +3.51x .

Page 18-65

Chapter 19

Numbers in Different Bases

In this Chapter we present examples of calculations of number in bases other

than the decimal basis.

Definitions

The number system used for everyday arithmetic is known as the decimal system

for it uses 10 (Latin, deca) digits, namely 0-9, to write out any real number.

Computers, on the other hand, use a system that is based on two possible

states, or binary system. These two states are represented by 0 and 1, ON

and OFF, or high-voltage and low-voltage. Computers also use number

systems based on eight digits (0-7) or octal system, and sixteen digits (0-9, A-F)

or hexadecimal. As in the decimal system, the relative position of digits

determines its value. In general, a number n in base b can be written as a

series of digits n = (a a …a .c c …c ) . The “point” separates n “integer”

1 2

m b

digits from m “decimal” digits. The value of the number, converted to our

-1

n-2

customary decimal system, is calculated by using n = a ⋅bn + a ⋅b + … +

-1

-2

-m

a b + c ⋅b + c ⋅b + … +c ⋅b . For example, (15.234) = 1⋅10 +

-1

-2

-3

5⋅10 + 2⋅10 + 3⋅10 + 4⋅10 , and (101.111) = 1⋅2 + 0⋅2 + 1⋅2 +

-1

-2

-3

1⋅2 + 1⋅2 + 1⋅2

The BASE menu

While the calculator would typically be operated using the decimal system, you

can produce calculations using the binary, octal, or hexadecimal system.

Many of the functions for manipulating number systems other than the decimal

system are available in the BASE menu, accessible through ‚ã(the 3

key). With system flag 117 set to CHOOSE boxes, the BASE menu shows the

following entries:

Page 19-1

With system flag 117 set to SOFT menus, the BASE menu shows the following:

With this format, it is evident that the LOGIC, BIT, and BYTE entries within the

BASE menu are themselves sub-menus. These menus are discussed later in this

Chapter.

Functions HEX, DEC, OCT, and BIN

Numbers in non-decimal systems are written preceded by the # symbol in the

calculator. The symbol # is readily available as „â(the 3 key). To select

which number system (current base) will be used for numbers preceded by #,

select one of the following functions in the first BASE menu, i.e., HEX(adecimal),

DEC(imal), OCT(al), or BIN(ary). For example, if @HEXꢀ! is selected, any number

written in the calculator that starts with # will be a hexadecimal number. Thus,

you can write numbers such as #53, #A5B, etc. in this system. As different

systems are selected, the numbers will be automatically converted to the new

current base.

The following examples show the same three numbers written with the # symbol

for different current bases:

HEX

DEC

OCT

BIN

Page 19-2

As the decimal (DEC) system has 10 digits (0,1,2,3,4,5,6,7,8,9), the

hexadecimal (HEX) system has 16 digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F), the

octal (OCT) system has 8 digits (0,1,2,3,4,5,6,7), and the binary (BIN) system

has only 2 digits (0,1).

Conversion between number systems

Whatever the number system selected, it is referred to as the binary system for

the purpose of using the functions RꢀB and BꢀR. For example, if @HEXꢀ! is

selected, the function BꢀR will convert any hexadecimal number (preceded by

#) into a decimal number, while the function RꢀB works in the opposite

direction. Try the following exercises, HEX is the current base:

The following examples show conversions when the base is the octal system:

We also present transformations using the binary system as the current base:

Notice that every time you enter a number starting with #, you get as the entry

the number you entered preceded by # and followed by the letter h, o, or b

(hexadecimal, octal, or binary). The type of letter used as suffix depends on

which non-decimal number system has been selected, i.e., HEX, OCT, or BIN.

To see what happens if you select the @DEC@ setting, try the following conversions:

Page 19-3

The only effect of selecting the DECimal system is that decimal numbers, when

started with the symbol #, are written with the suffix d.

Wordsize

The wordsize is the number of bits in a binary object. By default, the wordsize

is 64 bites. Function RCWS (ReCall WordSize) shows the current wordsize.

Function STWS (SeT the WordSize) allows the user to reset the wordsize to any

number between 0 and 64.

Changing the wordsize will affect the way that binary integer operations are

performed. For example, if a binary integer exceeds the current wordsize, the

leading bits will be dropped before any operation can be performed on such

number.

Operations with binary integers

The operations of addition, subtraction, change of sign, multiplication, and

division are defined for binary integers. Some examples, of addition and

subtraction, are shown below, for different current bases:

#A02h + #12Ah = #B2Ch

#2562d + #298d = #2860d

#5002o + #452o = #5454o

#101000000010b + #100101010b = #101100101100b

#A02h - #12Ah = #8D8h

#2562d - #298d = #2264d

#5002o - #452o = #4330o

#101000000010b - #100101010b = #100011011000b

Page 19-4

The LOGIC menu

The LOGIC menu, available through the BASE (‚ã) provides the following

functions:

The functions AND, OR, XOR (exclusive OR), and NOT are logical functions.

The input to these functions are two values or expressions (one in the case of

NOT) that can be expressed as binary logical results, i.e., 0 or 1.

Comparisons of numbers through the comparison operators =, ≠, >, <, ≤, and

≥, are logical statements that can be either true (1) or false (0). Some examples

of logical statements are shown below:

Functions AND, OR, XOR, and NOT can be applied to comparison statements

under the following rules:

1 AND 1 = 1 1 AND 0 = 0 0 AND 1 = 0 0 AND 0 = 0

1 OR 1 = 1

1 XOR 1 = 0 1 XOR 0 = 1 0 XOR 1 = 1 0 XOR 0 = 0

NOT(1) = 0 NOT(0) = 1

1 OR 0 = 1

0 OR 1 = 1

0 OR 0 = 0

These functions can be used to build logical statements for programming

purposes. In the context of this Chapter, they will by used to provide the result

of bit-by-bit operations along the lines of the rules provided above. In the

following examples, the base number system is indicated in parentheses:

Page 19-5

AND (BIN)

XOR (BIN)

OR (BIN)

NOT (HEX)

The BIT menu

The BIT menu, available through the BASE (‚ã) provides the following

functions:

Functions RL, SL, ASR, SR, RR, contained in the BIT menu, are used to

manipulate bits in a binary integer. The definition of these functions are shown

below:

RL: Rotate Left one bit, e.g., #1100b ꢀ #11000b

SL: Shift Left one bit, e.g., #1101b ꢀ #11010b

ASR: Arithmetic Shift Right one bit, e.g., #1100010b ꢀ #110001b

SR: Shift Right one bit, e.g., #11011b ꢀ#1101b

RR: Rotate Right one bit, e.g., #1101b ꢀ

#10000000000000000000000000000000000000000000000000

000000000001b

Page 19-6

The BYTE menu

The BYTE menu, available through the BASE (‚ã) provides the following

functions:

Functions RLB, SLB, SRB, RRB, contained in the BIT menu, are used to

manipulate bits in a binary integer. The definition of these functions are shown

below:

RLB: Rotate Left one byte, e.g., #1100b ꢀ #110000000000b

SLB: Shift Left one byte, e.g., #1101b ꢀ #110100000000b

SRB: Shift Right one byte, e.g., #11011b ꢀ#0b

RRB: Rotate Right one byte, e.g., #1101b ꢀ

#1101000000000000000000000000000000000000000000000

00000000000b

Hexadecimal numbers for pixel references

Many plot option specifications use pixel references as input, e.g., { #332h

#A23h } #Ah 0. 360. ARC, to draw an arc of a circle. We use functions

CꢀPX and PXꢀC to convert quickly between user-unit coordinates and pixel

references. These functions can be found through the command catalog

(‚N).

Some examples are shown below:

Page 19-7

Chapter 20

Customizing menus and keyboard

Through the use of the many calculator menus you have become familiar with

the operation of menus for a variety of applications. Also, you are familiar with

the many functions available by using the keys in the keyboard, whether

through their main function, or by combining them with the left-shift („), right-

shift (‚) or ALPHA (~) keys. In this Chapter we provide examples of

customized menus and keyboard keys that you may find useful in your own

applications.

Customizing menus

A custom menu is a menu created by the user. The specifications for the menu

are stored into the reserved variables CST. Thus, to create a menu you must put

together this variable with the features that you want to display in your menu

and the actions required by the soft menu keys. To show examples of

customizing menus we need to set system flag 117 to SOFT menu. Make sure

you do this before continuing (See Chapter 2 for instructions on setting system

flags).

The PRG/MODES/MENU menu

Commands useful in customizing menus are provided by the MENU menu,

accessible through the PRG menu („°). Setting system flag 117 to SOFT

menu, the sequence „°L@MODES @MENU produces the following MENU

soft menu:

The functions available are:

MENU:

CST:

Activates a menu given its number

Reference to the CST variable, e.g., ‚@@CST@@ shows CST contents.

TMENU: Use instead of MENU to create a temporary menu without

overwriting the contents of CST

RCLMENU:Returns menu number of current menu

Page 20-1

Menu numbers (RCLMENU and MENU functions)

Each pre-defined menu has a number attached to it. For example, suppose that

you activate the MTH menu („´). Then, using the function catalog

(‚N) find function RCLMENU and activate it. In ALG mode simple press

` after RCLMENU()shows up in the screen. The result is the number 3.01.

Thus, you can activate the MTH menu by using MENU(3.01), in ALG, or

3.01 MENU, in RPN.

Most menus can be activated without knowing their numbers by using the

keyboard. There are, however, some menus not accessible through the

keyboard. For example, the soft menu STATS is only accessible by using

function MENU. Its number is 96.01. Use MENU(96.01) in ALG mode, or

96.01 MENU in RPN mode to obtain the STAT soft menu.

Note: The number 96.01 in this example means the first (01) sub-menu of

menu 96.

Custom menus (MENU and TMENU functions)

Suppose that you need to activate four functions for a particular application.

Say, that you need to be able to quickly access the functions EXP, LN, GAMMA

and ! (~‚2) and you want to place them in a soft menu that you will

keep active for a while. You could do this by creating a temporary menu with

function TMENU, or a more permanent menu with function MENU. The main

difference is that function MENU creates variable CST, while TMENU does not.

With variable CST created permanently in your sub-directory you can always

reactivate the menu using the specifications in CST by pressing „£. With

TMENU the menu specifications are lost after you replace the temporary menu

with another one.

For example, in RPN mode, a menu is created by using:

{EXP LN GAMMA !} ` TMENU `

{EXP LN GAMMA !} ` MENU `

to produce the following menu:

Page 20-2

To activate any of those functions you simply need to enter the function

argument (a number), and then press the corresponding soft menu key.

In ALG mode, the list to be entered as argument of function TMENU or MENU is

more complicated:

{{“exp”,”EXP(“},{“ln”,”LN(“},{“Gamma”,”GAMMA(“},{“!”,”!(“}}

The reason for this is that, in RPN mode, the command names are both soft

menu labels and commands. In ALG mode, the command names will produce

no action since ALG functions must be followed by parentheses and arguments.

In the list shown above (for the ALG mode), within each sub-list you have a label

for the key, e.g., “exp”, followed by the way that the function will be entered in

the stack so that the argument to the function can be typed at the prompt, e.g.,

“EXP(“. We need not worry about the closing parenthesis, because the

calculator will complete the parentheses before executing the function. The

implementation of function TMENU in ALG mode with the argument list shown

above is as follows. First, we enter the list, then we produce the temporary

menu (see menu key labels) by using function TMENU(ANS(1)). We also

show, in the left-hand side, the result of pressing the @@exp! soft menu key, i.e., the

prompt EXP(. After typing 8` the result of the operation is shown in the

right-hand side:

A simpler version of the menu can be defined by using

MENU({{”EXP(“,“LN(“,“GAMMA(“,”!(“}).

Enhanced RPN menu

The list presented above for the ALG mode, can be modified slightly to use in

the RPN mode. The modified list will look like this:

{{“exp”,EXP},{“ln”,LN},{“Gamma”,GAMMA},{“!”,!}}

Page 20-3

You can try using this list with TMENU or MENU in RPN mode to verify that you

get the same menu as obtained earlier in ALG mode.

Menu specification and CST variable

From the two exercises shown above we notice that the most general menu

specification list include a number of sub-lists equal to the number of items to be

displayed in your custom menu. Each sub-list contains a label for the menu key

followed by a function, expression, label, or other object that constitutes the

effect of the menu key when pressed. Care must be exercised in specifying the

menu list in ALG mode versus RPN mode. In RPN mode, the menu key action

can be simply a calculator command (e.g., EXP, LN, etc., as shown above),

while in ALG mode it has to be a string with the command prompt whose

argument needs to be provided by the user before pressing ` and

completing the command. The examples above illustrate the difference.

The general form of the argument list for commands TMENU or MENU in ALG

mode is

{“label1”,”function1(“,”ls1(“,”rs1(“}, {“label2”, “function2(“,”ls2(“,”rs2(“},…}

While, in RPN mode, the argument list has this format

{“label1”, function1, ls1, rs1}, {“label2”, function2, ls2, rs2},…}

In these specifications, function1, function 2, etc., represent the main operation

of the key, while ls1, ls2, …, etc., represent the left-shift operation of the key.

Similarly, rs1, rs2, …, etc., represent the right-shift operation of the key. This list

will be stored in variable CST if command MENU is used. You can have a

different CST variable in each sub-directory, and you can always replace the

current contents of CST with those of other variables storing the properly

formatted list to produce another custom menu.

Note: You can use a 21x8 GROB (See Chapter 22) to produce an icon in the

soft menu keys. As an example, try, in RPN mode:

{{GROB 21 8 00000EF908FFF900FFF9B3FFF9A2FFF9A3FFF9A0FFF388FF

“hp” }}

` MENU

This will place the hp logo on key A. Pressing A places the text ‘hp’ in the

command line.

Page 20-4

Customizing the keyboard

Each key in the keyboard can be identified by two numbers representing their

row and column. For example, the VAR key (J) is located in row 3 of

column 1, and will be referred to as key 31. Now, since each key has up to ten

functions associated with it, each function is specified by decimal digits

between 0 and 1, according to the following specifications:

.0 or 1, unshifted key

0.01 or 0.11, not applicable

.2, key combined with „

.3, key combined with ‚

.4, key combined with ~

.5, key combined with ~„

.6, key combined with ~‚

.21, key simultaneous with „

.31, key simultaneous with ‚

.41, key combined with ~

.51, ~ key simultaneous with „

.61, ~ key simultaneous with ‚

Thus, the VAR function will be referred to as key 31.0 or 31.1, while the UPDIR

function will be key 31.2, the COPY function will be key 31.3, the upper-case J

is key 31.4, and lower case j is key 31.5. (Key 31.6 is not defined). In

general, a key will be described by the arrangement XY.Z, where X = row

number, Y = column number, Z = shifting.

We can combine a given key with the USER key (left-shift associated with the

~ key, or „Ì) to create a customized key action. In principle, the entire

keyboard can be re-defined to perform a number of customized operations.

The PRG/MODES/KEYS sub-menu

Commands useful in customizing the keyboard are provided by the KEYS menu

accessible through the PRG menu („°). Setting system flag 117 to SOFT

menu, the sequence „ °L @MODES @KEYS

produces the following KEYS soft menu:

Page 20-5

The functions available are:

ASN:

Assigns an object to a key specified by XY.Z

STOKEYS: Stores user-defined key list

RCLKEYS: Returns current user-defined key list

DELKEYS: Un-assigns one or more keys in the current user-defined key list, the

arguments are either 0, to un-assign all user-defined keys, or XY.Z,

to un-assign key XY.Z.

Recall current user-defined key list

Use command RCLKEYS to see the current user-defined key list. Before any

user-defined key assignments, the result should be a list containing the letter S,

i.e., {S}.

Assign an object to a user-defined key

Suppose that you want to have access to the old-fashioned PLOT command first

introduced with the HP 48G series calculator, but currently not directly available

from the keyboard. The menu number for this menu is 81.01. You can see this

menu active by using

ALG mode: MENU(81.01)

RPN mode: 81.01 ` MENU `

If you want to have a quick way to activate this menu from the keyboard, you

could assign this menu to the GRAPH key (C) whose reference number is

13.0, i.e., first row, third column, main function. To assign an object to a key

use function ASN, as follows:

ALG mode: ASN(<<MENU(81.01)>>,13.0)

RPN mode: << 18.01 MENU >> ` 13.0 ` ASN

Another useful menu is the original SOLVE menu (described at the end of

Chapter 6 in this Guide), which can be activated by using ‚(hold) 7.

Page 20-6

Operating user-defined keys

To operate this user-defined key, enter „Ì before pressing the C key.

Notice that after pressing „Ì the screen shows the specification 1USR in

the second display line. Pressing for „Ì C for this example, you

should recover the PLOT menu as follows:

If you have more than one user-defined key and want to operate more than one

of them at a time, you can lock the keyboard in USER mode by entering

„Ì„Ì before pressing the user-defined keys. With the keyboard

locked in USER mode, the specification USR will be shown in the second

display line. To unlock the keyboard press „Ì once more.

Un-assigning a user-defined key

To remove the assignment performed above, use function DELKEYS, as follows:

ALG mode:

RPN mode:

DELKEYS(13.0)

13.0 ` DELKEYS `

Assigning multiple user-defined keys

The simplest way to assign several user-defined is to provide a list of commands

and key specifications. For example, suppose that we assign the three

trigonometric functions (SIN, COS, TAN) and the three hyperbolic functions

(SINH, COSH, TANH) to keys A through F, respectively, as user-defined

keys. In RPN mode use:

{SIN[11.0[COS[12.0[T N[13.0[SINH[14.0[COSH[15.0[T NH[

16.0} ` STOKEYS `

In ALG mode use:

STOKEYS({"SIN(" , 11.0, "COS(", 12.0, "T N(", 13.0,

"SINH(", 14.0, "COSH(", 15.0, "T NH(", 16.0}) `

Operate these keys by using, for example, in RPN mode:

5„ÌA 4„ÌB 6„ÌC

2 „ÌD 1„ÌE 2„ÌF

Page 20-7

To un-assign all user-defined keys use:

ALG mode: DELKEYS(0)

RPN mode: 0 DELKEYS

Check that the user-key definitions were removed by using function RCLKEYS.

Page 20-8

Chapter 21

Programming in User RPL language

User RPL language is the programming language most commonly used to

program the calculator. The program components can be put together in the

line editor by including them between program containers « » in the

appropriate order. Because there is more experience among calculator users in

programming in the RPN mode, most of the examples in this Chapter will be

presented in the RPN mode. Also, to facilitate entering programming

commands, we suggest you set system flag 117 to SOFT menus. The programs

work equally well in ALG mode once they have been debugged and tested in

RPN mode. If you prefer to work in the ALG mode, simply learn how to do the

programming in RPN and then reset the operating mode to ALG to run the

programs. For a simple example of User RPL programming in ALG mode, refer

to the last page in this chapter.

An example of programming

Throughout the previous Chapters in this guide we have presented a number of

programs that can be used for a variety of applications (e.g., programs CRMC

and CRMT, used to create a matrix out of a number of lists, were presented in

Chapter 10). In this section we present a simple program to introduce concepts

related to programming the calculator. The program we will write will be used

to define the function f(x) = sinh(x)/(1+x ), which accepts lists as argument

(i.e., x can be a list of numbers, as described in Chapter 8). In Chapter 8 we

indicated that the plus sign, , acts as a concatenation operator for lists and not

to produce a term-by-term sum. Instead, you need to use the ADD operator to

achieve a term-by-term summation of lists. Thus, to define the function shown

above we will use the following program:

«'x' STO x SINH 1 x SQ ADD / 'x' PURGE »

To key in the program follow these instructions:

Keystroke sequence:

‚å

Produces:

Interpreted as:

Start an RPL program

Store level 1 into variable x

Place x in level 1

[']~„x™K

~„x

'x' STO

„´@HYP @SINH

SINH

Calculate sinh of level 1

1#~„x „º 1 x SQ

Enter 1 and calculate x

Page 21-1

„´@LIST @ADD@

ADD

Calculate (1+x ),

'x'

then divide

[']~„x™

„°@@MEM@@ @@DIR@@ @PURGE

PURGE

Purge variable x

Program in level 1

_____________________

_______________________

__________

To save the program use: [']~„gK

Press J to recover your variable menu, and evaluate g(3.5) by entering the

value of the argument in level 1 (3.5`) and then pressing @@@g@@@.

The result is 1.2485…, i.e., g(3.5) = 1.2485. Try also obtaining g({1 2 3}), by

entering the list in level 1 of the display: „ä1#2#3`

and pressing @@@g@@@. The result now is {SINH(1)/2 SINH(2)/5 SINH(3)/10}, if

your CAS is set to EXACTmode. If your CAS is set to APPROXIMATE mode, the

result will be {0.5876.. 0.7253… 1.0017…}.

Global and local variables and subprograms

The program @@@g@@@, defined above, can be displayed as

« 'x' STO x SINH 1 x SQ ADD / 'x' PURGE »

by using ‚@@@g@@@.

Notice that the program uses the variable name x to store the value placed in

level 1 of stack through the programming steps 'x' STO. The variable x, while

the program is executing, is stored in your variable menu as any other variable

you had previously stored. After calculating the function, the program purges

(erases) the variable x so it will not show in your variable menu after finishing

evaluating the program. If we were not to purge the variable x within the

program its value would be available to us after program execution. For that

reason, the variable x, as used in this program, is referred to as a global

variable. One implication of the use of x as a global variable is that, if we had

a previously defined a variable with the name x, its value would be replaced by

the value that the program uses and then completely removed from your

variable menu after program execution.

From the point of view of programming, therefore, a global variable is a

variable that is accessible to the user after program execution. It is possible to

Page 21-2

use a local variable within the program that is only defined for that program

and will not be available for use after program execution. The previous

program could be modified to read:

« → x « x SINH 1 x SQ ADD / » »

The arrow symbol (→) is obtained by combining the right-shift key ‚ with the

0 key, i.e., ‚é. Also, notice that there is an additional set of

programming symbols (« ») indicating the existence of a sub-program, namely «

x SINH 1 x SQ ADD / », within the main program. The main program starts

with the combination → x, which represents assigning the value in level 1 of

stack to a local variable x. Then, programming flow continues within the sub-

program by placing x in the stack, evaluating SINH(x), placing 1 in the stack,

placing x in the stack, squaring x, adding 1 to x, and dividing stack level 2

(SINH(x)) by stack level 1 (1+x ). The program control is then passed back to

the main program, but there are no more commands between the first set of

closing programming symbols (») and the second one, therefore, the program

terminates. The last value in the stack, i.e., SINH(x)/ (1+x ), is returned as the

program output.

The variable x in the last version of the program never occupies a place among

the variables in your variable menu. It is operated upon within the calculator

memory without affecting any similarly named variable in your variable menu.

For that reason, the variable x in this case is referred to as a variable local to

the program, i.e., a local variable.

Note: To modify program @@@g@@@, place the program name in the stack

(³@@@g@@@ `), then use „˜. Use the arrow keys (š™—˜) to

move about the program. Use the backspace/delete key, ƒ, to delete any

unwanted characters. To add program containers (i.e., « »), use ‚å,

since these symbols come in pairs you will have to enter them at the start and

end of the sub-program and delete one of its components with the delete key

ƒ to produce the required program, namely:

Page 21-3

« → x « x SINH 1 x SQ ADD / » ».

When done editing the program press ` . The modified program is stored

back into variable @@g@@.

Global Variable Scope

Any variable that you define in the HOME directory or any other directory or

sub-directory will be considered a global variable from the point of view of

program development. However, the scope of such variable, i.e., the location

in the directory tree where the variable is accessible, will depend on the

location of the variable within the tree (see Chapter 2).

The rule to determine a variable’s scope is the following: a global variable is

accessible to the directory where it is defined and to any sub-directory attached

to that directory, unless a variable with the same name exists in the sub-

directory under consideration. Consequences of this rule are the following:

ꢀ A global variable defined in the HOME directory will be accessible from

any directory within HOME, unless redefined within a directory or sub-

directory.

ꢀ If you re-define the variable within a directory or sub-directory this

definition takes precedence over any other definition in directories above

the current one.

ꢀ When running a program that references a given global variable, the

program will use the value of the global variable in the directory from

which the program is invoked. If no variable with that name exist in the

invoking directory, the program will search the directories above the current

one, up to the HOME directory, and use the value corresponding to the

variable name under consideration in the closest directory above the

current one.

A program defined in a given directory can be accessed from that directory or

any of its sub-directories.

Page 21-4

All these rule may sound confusing for a new calculator user. They all can be

simplified to the following suggestion: Create directories and sub-directories

with meaningful names to organize your data, and make sure you have all the

global variables you need within the proper sub-directory.

Local Variable Scope

Local variables are active only within a program or sub-program. Therefore,

their scope is limited to the program or sub-program where they’re defined. An

example of a local variable is the index in a FOR loop (described later in this

chapter), for example « → n x « 1 n FOR j x NEXT n ꢀLIST » »

The PRG menu

In this section we present the contents of the PRG (programming) menu with the

calculator’s system flag 117 set to SOFT menus. With this flag setting sub-

menus and commands in the PRG menu will be shown as soft menu labels. This

facilitates entering the programming commands in the line editor when you are

putting together a program.

To access the PRG menu use the keystroke combination „°. Within the

PRG menu we identify the following sub-menus (press L to move to the next

collection of sub-menus in the PRG menu):

Here is a brief description of the contents of these sub-menus, and their sub-

menus:

STACK: Functions for manipulating elements of the RPN stack

MEM:

DIR:

Functions related to memory manipulation

Functions related to manipulating directories

ARITH: Functions to manipulate indices stored in variables

BRCH: Collection of sub-menus with program branching and loop functions

IF: IF-THEN-ELSE-END construct for branching

CASE: CASE-THEN-END construct for branching

Page 21-5

START: START-NEXT-STEP construct for branching

FOR:

DO:

FOR-NEXT-STEP construct for loops

DO-UNTIL-END construct for loops

WHILE: WHILE-REPEAT-END construct for loops

TEST:

TYPE:

LIST:

Comparison operators, logical operators, flag testing functions

Functions for converting object types, splitting objects, etc.

Functions related to list manipulation

ELEM:

Functions for manipulating elements of a list

PROC: Functions for applying procedures to lists

GROB: Functions for the manipulation of graphic objects

PICT:

Functions for drawing pictures in the graphics screen

CHARS: Functions for character string manipulation

MODES: Functions for modifying calculator modes

FMT:

To change number formats, comma format

ANGLE: To change angle measure and coordinate systems

FLAG:

KEYS:

To set and un-set flags and check their status

To define and activate user-defined keys (Chapter 20)

MENU: To define and activate custom menus (Chapter 20)

MISC:

IN:

Miscellaneous mode changes (beep, clock, etc.)

Functions for program input

OUT:

TIME:

Functions for program output

Time-related functions

ALRM: Alarm manipulation

ERROR: Functions for error handling

IFERR:

RUN:

IFERR-THEN-ELSE-END construct for error handling

Functions for running and debugging programs

Navigating through RPN sub-menus

Start with the keystroke combination „°, then press the appropriate soft-

menu key (e.g., @@MEM@@ ). If you want to access a sub-menu within this sub-menu

(e.g., @@DIR@@ within the @@MEM@@ sub-menu), press the corresponding key. To move

up in a sub-menu, press the L key until you find either the reference to the

upper sub-menu (e.g., @@MEM@@ within the @@DIR@@ sub-menu) or to the PRG menu

(i.e., @@PRG@@ ).

Page 21-6

Functions listed by sub-menu

The following is a listing of the functions within the PRG sub-menus listed by sub-

menu.

STACK

DUP

SWAP

DROP

OVER

ROT

MEM/DIR

PURGE

RCL

STO

PATH

CRDIR

PGDIR

VARS

BRCH/IF

THEN

ELSE

BRCH/WHILE TYPE

WHILE

REPEAT

END

OBJꢀ

ꢀARRY

ꢀLIST

ꢀSTR

ꢀTAG

ꢀUNIT

CꢀR

END

TEST

BRCH/CASE

CASE

UNROT

ROLL

≠

ROLLD

PICK

TVARS

ORDER

THEN

END

≤

≥

AND

XOR

NOT

SAME

TYPE

RꢀC

NUM

CHR

DTAG

EQꢀ

TYPE

UNPICK

PICK3

DEPTH

DUP2

DUPN

DROP2

DROPN

DUPDU

NIP

MEM/ARITH BRCH/START

STO+

STO-

STOx

STO/

INCR

DECR

SINV

SNEG

SCONJ

START

STEP

VTYPE

BRCH/FOR

FOR

LIST

OBJꢀ

ꢀLIST

SUB

NDUPN

STEP

FS?

FC?

FS?C

FC?C

LININ

REPL

MEM

PURGE

MEM

BYTES

NEWOB

ARCHI

RESTO

BRCH/DO

UNTIL

END

BRCH

IFT

IFTE

Page 21-7

LIST/ELEM GROB

CHARS

MODES/FLAG MODES/MISC

GET

GETI

PUT

PUTI

SIZE

POS

HEAD

TAIL

ꢀGROB SUB

FS?

FC?

FS?C

FC?C

STOF

RCLF

RESET

BEEP

CLK

SYM

STK

ARG

CMD

INFO

BLANK

GOR

GXOR

SUB

REPL

POS

SIZE

NUM

CHR

REPL

ꢀLCD

LCDꢀ

SIZE

OBJꢀ

ꢀSTR

HEAD

LIST/PROC

DOLIST

DOSUB

NSUB

ANIMATE TAIL

SREPL

PICT

INFORM

NOVAL

CHOOSE

INPUT

KEY

MODES/KEYS

ASN

STOKEYS

RECLKEYS

DELKEYS

MODES/FMT

PICT

ENDSUB PDIM

STD

FIX

STREAM

REVLIST

SORT

LINE

WAIT

PROMPT

TLINE

BOX

ARC

PIXON

PIXOF

PIX?

PVIEW

PXꢀC

CꢀPX

SCI

ENG

FM,

MODES/MENU OUT

CST

TMENU

RCLMENU

SEQ

PVIEW

TEXT

MODES/ANGLE

DEG

RAD

CLLCD

DISP

FREEZE

MSGBOX

BEEP

GRAD

RECT

CYLIN

SPHERE

Page 21-8

TIME

DATE

ꢀDATE

ERROR

DOERR

ERRN

RUN

DBUG

SST

TIME

ꢀTIME

TICKS

ERRM

ERR0

LASTARG

SST↓

HALT

KILL

TIME/ALRM

ACK

ERROR/IFERR

IFERR

OFF

ACKALARM

STOALARM

RCLALARM

DELALARM

FINDALARM

THEN

ELSE

END

Shortcuts in the PRG menu

Many of the functions listed above for the PRG menu are readily available

through other means:

ꢀ Comparison operators (≠, ≤, <, ≥, >) are available in the keyboard.

ꢀ Many functions and settings in the MODES sub-menu can be activated

by using the input functions provided by the H key.

ꢀ Functions from the TIME sub-menu can be accessed through the

keystroke combination ‚Ó.

ꢀ Functions STO and RCL (in MEM/DIR sub-menu) are available in the

ꢀ Functions RCL and PURGE (in MEM/DIR sub-menu) are available

through the TOOL menu (I).

ꢀ Within the BRCH sub-menu, pressing the left-shift key („) or the right-

shift key (‚) before pressing any of the sub-menu keys, will create

constructs related to the sub-menu key chosen. This only works with the

calculator in RPN mode. Examples are shown below:

Page 21-9

„@@IF@@

„@START

„@@@DO@@

„@CASE@

„@@FOR@

„@WHILE

Notice that the insert prompt (ꢀ) is available after the key word for each

construct so you can start typing at the right location.

Keystroke sequence for commonly used commands

The following are keystroke sequences to access commonly used commands for

numerical programming within the PRG menu. The commands are first listed by

menu:

Page 21-10

@STACK

DUP

SWAP

DROP

„°@STACK @@DUP@@

„°@STACK @SWAP@

„°@STACK @DROP@

@@MEM@@ @@DIR@@

PURGE

ORDER

„°@@MEM@@ @@DIR@@ @PURGE

„°@@MEM@@ @@DIR@@ @ORDER

@@BRCH@ @@IF@@

„°@@BRCH@ @@IF@@ @@@IF@@@

„°@@BRCH@ @@IF@@ @THEN@

„°@@BRCH@ @@IF@@ @ELSE@

„°@@BRCH@ @@IF@@ @@@END@@

THEN

ELSE

END

@@BRCH@ @CASE@

CASE

THEN

END

„°@@BRCH@ @CASE@ @CASE@

„°@@BRCH@ @CASE@ @THEN@

„°@@BRCH@ @CASE@ @@END@

@@BRCH@ @START

START

STEP

„°@@BRCH@ @START @START

„°@@BRCH@ @START @NEXT

„°@@BRCH@ @START @STEP

@@BRCH@ @@FOR@

FOR

STEP

„°@@BRCH@ @@FOR@ @@FOR@@

„°@@BRCH@ @@FOR@ @@NEXT@

„°@@BRCH@ @@FOR@ @@STEP@

@@BRCH@ @@@DO@@

„°@@BRCH@ @@@DO@@ @@@DO@@

„°@@BRCH@ @@@DO@@ @UNTIL

„°@@BRCH@ @@@DO@@ @@END@@

UNTIL

END

Page 21-11

@@BRCH@ @WHILE@

WHILE

„°@@BRCH@ @WHILE@ @WHILE

„°@BRCH@ @WHILE@ @REPEA

„°@BRCH@ @WHILE@ @@END@

REPEAT

END

@TEST@

AND

XOR

NOT

SAME

FS?

FC?

FS?C

FC?C

„°@TEST@ @@@≠@@@

„°@TEST@ L @@AND@

„° @TEST@ L @@@OR@@

„°@TEST@ L @@XOR@

„° @TEST@ L @@NOT@

„°@TEST@ L @SAME

„°@TEST@ L L @@@SF@@

„°@TEST@ L L @@@CF@@

„°@TEST@ L L @@FS?@

„°@TEST@ L L @@FC?@

„°@TEST@ L L @FS?C

„°@TEST@ L L @FC?C

@TYPE@

OBJꢀ

ꢀARRY

ꢀLIST

ꢀSTR

ꢀTAG

NUM

CHR

„°@TYPE@ @OBJ ꢀ@

„°@TYPE@ @ ꢀARRY

„°@TYPE@ @ ꢀLIST

„°@TYPE@ @ ꢀSTR

„°@TYPE@ @ ꢀTAG

„°@TYPE@ L@NUM@

„°@TYPE@ L@CHR@

„°@TYPE@ L@TYPE@

TYPE

@LIST@ @ELEM@

GET

„°@LIST@ @ELEM@ @@GET@@

„°@LIST@ @ELEM@ @GETI@

„°@LIST@ @ELEM@ @@PUT@

„°@LIST@ @ELEM@ @PUTI@

„°@LIST@ @ELEM@ @SIZE@

„°@LIST@ @ELEM@ L @HEAD@

„°@LIST@ @ELEM@ L @TAIL@

GETI

PUT

PUTI

SIZE

HEAD

TAIL

Page 21-12

@LIST@ @PROC@

REVLIST

„°@LIST@ @PROC@ @REVLI@

SORT

SEQ

„°@LIST@ @PROC@ L @SORT@

„°@LIST@ @PROC@ L @@SEQ@@

@MODES @ANGLE@

DEG

„°L@MODES @ANGLE@ @@DEG@@

RAD

„°L@MODES @ANGLE@ @@RAD@@

@MODES @MENU@

CST

„°L@MODES @MENU@ @@CST@@

„°L@MODES @MENU@ @@MENU@

„°L@MODES @MISC@ @@BEEP@

BEEP

@@@IN@@

INFORM

INPUT

MSGBOX

PVIEW

„°L@@@IN@@ @INFOR@

„°L@@@IN@@ @INPUT@

„°L@@OUT@ @MSGBO@

„°L@@OUT@ @PVIEW@

@@RUN@

DBUG

SST

„°LL @@RUN@ @@DBG@

„°LL @@RUN@ @@SST@

„°LL @@RUN@ @SST↓@

„°LL @@RUN@ @HALT@

„°LL @@RUN@ @KILL

SST↓

HALT

KILL

Programs for generating lists of numbers

Please notice that the functions in the PRG menu are not the only functions that

can be used in programming. As a matter of fact, almost all functions in the

calculator can be included in a program. Thus, you can use, for example,

Page 21-13

functions from the MTH menu. Specifically, you can use functions for list

operations such as SORT, ΣLIST, etc., available through the MTH/LIST menu.

As additional programming exercises, and to try the keystroke sequences listed

above, we present herein three programs for creating or manipulating lists. The

program names and listings are as follows:

LISC:

« → n x « 1 n FOR j x NEXT n ꢀLIST » »

CRLST:

« → st en df « st en FOR j j df STEP en st - df / FLOOR 1 +

→LIST » »

CLIST:

« REVLIST DUP DUP SIZE 'n' STO ΣLIST SWAP TAIL DUP SIZE 1 - 1

SWAP FOR j DUP ΣLIST SWAP TAIL NEXT 1 GET n ꢀLIST REVLIST 'n'

PURGE »

The operation of these programs is as follows:

(1) LISC: creates a list of n elements all equals to a constant c.

Operation: enter n, enter c, press @LISC

Example: 5 ` 6.5 ` @LISC creates the list: {6.5 6.5 6.5 6.5 6.5}

(2) CRLST: creates a list of numbers from n to n with increment Δn, i.e.,

{n , n +Δn, n1+2⋅Δn, … n +N⋅Δn }, where N=floor((n -n )/Δn)+1.

2 1

Operation: enter n , enter n , enter Δn, press @CRLST

Example: .5 `3.5 `.5 ` @CRLST produces: {0.5 1 1.5 2 2.5 3 3.5}

(3) CLIST: creates a list with cumulative sums of the elements, i.e., if the original

list is {x x x … x }, then CLIST creates the list:

{x , x + x , x + x + x ,..., x }

∑

i=1

Operation: place the original list in level 1, press @CLIST.

Example: {1 2 3 4 5} `@CLIST produces {1 3 6 10 15}.

Page 21-14

Examples of sequential programming

In general, a program is any sequence of calculator instructions enclosed

between the program containers

and ». Subprograms can be included as

part of a program. The examples presented previously in this guide (e.g., in

Chapters 3 and 8) 6 can be classified basically into two types: (a) programs

generated by defining a function; and, (b) programs that simulate a sequence

of stack operations. These two types of programs are described next. The

general form of these programs is inputꢀprocessꢀoutput, therefore, we refer to

them as sequential programs.

Programs generated by defining a function

These are programs generated by using function DEFINE („à) with an

argument of the form:

'function_name(x , x , …) = expression containing variables x , x , …'

The program is stored in a variable called function_name. When the

program is recalled to the stack, by using ‚function_name. The program

shows up as follows:

« ꢀ x , x , … 'expression containing variables x , x , …'».

To evaluate the function for a set of input variables x , x , …, in RPN mode,

enter the variables into the stack in the appropriate order (i.e., x first, followed

by x , then x , etc.), and press the soft menu key labeled function_name. The

calculator will return the value of the function function_name(x , x , …).

Example: Manning’s equation for wide rectangular channel.

As an example, consider the following equation that calculates the unit

discharge (discharge per unit width), q, in a wide rectangular open channel

using Manning’s equation:

C_u

q =

y₀^{5/ 3}S₀

Page 21-15

where C is a constant that depends on the system of units used [C = 1.0 for

units of the International System (S.I.), and C = 1.486 for units of the English

System (E.S.)], n is the Manning’s resistance coefficient, which depends on the

type of channel lining and other factors, y is the flow depth, and S is the

channel bed slope given as a dimensionless fraction.

Note: Values of the Manning’s coefficient, n, are available in tables as

dimensionless numbers, typically between 0.001 to 0.5. The value of Cu is

also used without dimensions. However, care should be taken to ensure that the

value of y0 has the proper units, i.e., m in S.I. and ft in E.S. The result for q is

returned in the proper units of the corresponding system in use, i.e., m /s in S.I.

and ft /s in E.S. Manning’s equation is, therefore, not dimensionally consistent.

Suppose that we want to create a function q(Cu, n, y0, S0) to calculate the unit

discharge q for this case. Use the expression

‘q(Cu,n,y0,S0)=Cu/n*y0^(5./3.)*√S0’,

as the argument of function DEFINE. Notice that the exponent 5./3., in the

equation, represents a ratio of real numbers due to the decimal points. Press

J, if needed, to recover the variable list. At this point there will be a

variable called @@@q@@@ in your soft menu key labels. To see the contents of q, use

‚@@@q@@@. The program generated by defining the function q(Cu,n,y0,S0)is

shown as:

« → Cu n y0 S0 ‘Cu/n*y0^(5./3.)*√S0’ ».

This is to be interpreted as “enter Cu, n, y0, S0, in that order, then calculate the

expression between quotes.” For example, to calculate q for Cu = 1.0, n =

0.012, y0 = 2 m, and S0 = 0.0001, use, in RPN mode:

1 ` 0.012 ` 2 ` 0.0001 ` @@@q@@@

The result is 2.6456684 (or, q = 2.6456684 m /s).

Page 21-16

You can also separate the input data with spaces in a single stack line rather

than using `.

Programs that simulate a sequence of stack operations

In this case, the terms to be involved in the sequence of operations are assumed

to be present in the stack. The program is typed in by first opening the program

containers with ‚å. Next, the sequence of operations to be performed is

entered. When all the operations have been typed in, press ` to complete

the program. If this is to be a once-only program, you can at this point, press

μ to execute the program using the input data available. If it is to be a

permanent program, it needs to be stored in a variable name.

The best way to describe this type of programs is with an example:

Example: Velocity head for a rectangular channel.

Suppose that we want to calculate the velocity head, h , in a rectangular

channel of width b, with a flow depth y, that carries a discharge Q. The

specific energy is calculated as h = Q /(2g(by) ), where g is the acceleration

of gravity (g = 9.806 m/s in S.I. units or g = 32.2 ft/s in E.S. units). If we

were to calculate h for Q = 23 cfs (cubic feet per second = ft /s), b = 3 ft, and

y = 2 ft, we would use: h = 23 /(2⋅32.2⋅ (3⋅2) ). Using the RPN modethe

calculator, interactively, we can calculate this quantity as:

2`3*„º32.2*

2*23„º™/

Resulting in 0.228174, or h = 0.228174.

To put this calculation together as a program we need to have the input data

(Q, g, b, y) in the stack in the order in which they will be used in the

calculation. In terms of the variables Q, g, b, and y, the calculation just

performed is written as (do not type the following):

y ` b *„º g *2* Q „º™/

Page 21-17

As you can see, y is used first, then we use b, g, and Q, in that order.

Therefore, for the purpose of this calculation we need to enter the variables in

the inverse order, i.e., (do not type the following):

Q ` g `b `y `

For the specific values under consideration we use:

23 ` 32.2 ` 3 `2 `

The program itself will contain only those keystrokes (or commands) that result

from removing the input values from the interactive calculation shown earlier,

i.e., removing Q, g, b, and y from (do not type the following):

y ` b *„º g *2* Q „º™/

and keeping only the operations shown below (do not type the following):

` *„ *2* „º™/

Note: When entering the program do not use the keystroke ™, instead use

the keystroke sequence: „°@STACK @SWAP@.

Unlike the interactive use of the calculator performed earlier, we need to do

some swapping of stack levels 1 and 2 within the program. To write the

program, we use, therefore:

‚å

Opens program symbols

Multiply y with b

Square (b⋅y)

„º

Multiply (b⋅y) times g

Enter a 2 and multiply it with g⋅ (b⋅y)

„°@STACK @SWAP@

„º

Swap Q with 2⋅g⋅ (b⋅y)

Square Q

„°@STACK @SWAP@

Swap 2⋅g⋅ (b⋅y) with Q

Divide Q by 2⋅g⋅ (b⋅y)

Enter the program

The resulting program looks like this:

« * SQ * 2 * SWAP SQ SWAP / »

Page 21-18

Note: SQis the function that results from the keystroke sequence „º.

Save the program into a variable called hv:

³~„h~„v K

A new variable @@@hv@@@ should be available in your soft key menu. (Press J to

see your variable list.) The program left in the stack can be evaluated by using

function EVAL. The result should be 0.228174…, as before. Also, the program

is available for future use in variable @@@hv@@@. For example, for Q = 0.5 m /s, g

= 9.806 m/s , b = 1.5 m, and y = 0.5 m, use:

0.5 # 9.806 #1.5 # 0.5 @@@hv@@@

Note: #is used here as an alternative to ` for input data entry.

-2

The result now is 2.26618623518E-2, i.e., hv = 2.26618623518×10 m.

Note: Since the equation programmed in @@@hv@@@ is dimensionally consistent,

we can use units in the input.

As mentioned earlier, the two types of programs presented in this section are

sequential programs, in the sense that the program flow follows a single path,

i.e., INPUTꢀ OPERATION ꢀOUTPUT. Branching of the program flow is

possible by using the commands in the menu „°@@BRCH@ . More detail on

program branching is presented below.

Interactive input in programs

In the sequential program examples shown in the previous section it is not

always clear to the user the order in which the variables must be placed in the

stack before program execution. For the case of the program @@@q@@@, written as

« → Cu n y0 S0‘Cu/n*y0^(5/3)*√S0’ »,

Page 21-19

it is always possible to recall the program definition into the stack (‚@@@q@@@) to

see the order in which the variables must be entered, namely, → Cu n y0 S0.

However, for the case of the program @@hv@@, its definition

« * SQ * 2 * SWAP SQ SWAP / »

does not provide a clue of the order in which the data must be entered, unless,

of course, you are extremely experienced with RPN and the User RPL language.

One way to check the result of the program as a formula is to enter symbolic

variables, instead of numeric results, in the stack, and let the program operate

on those variables. For this approach to be effective the calculator’s CAS

(Calculator Algebraic System) must be set to symbolicand exactmodes. This

is accomplished by using H@CAS@, and ensuring that the check marks in the

options _Numericand _Approxare removed.

Press @@OK@@ @@OK@ to return to

normal calculator display. Press J to display your variables menu.

We will use this latter approach to check what formula results from using the

program @@hv@@ as follows: We know that there are four inputs to the program,

thus, we use the symbolic variables S4, S3, S2, and S1 to indicate the stack

levels at input:

~s4` ~s3` ~s2` ~s1`

Next, press @@hv@@. The resulting formula may look like this

‘SQ(S4)/(S3*SQ(S2*S1)*2)’,

if your display is not set to textbook style, or like this,

SQ(S4)

S3⋅ SQ(S2⋅ S1) ⋅ 2

if textbook style is selected. Since we know that the function SQ( ) stands for x ,

we interpret the latter result as

Page 21-20

S4²

2⋅ S3⋅(S2⋅ S1)²

which indicates the position of the different stack input levels in the formula. By

comparing this result with the original formula that we programmed, i.e.,

Q²

2g(by)²

h_v=

we find that we must enter y in stack level 1 (S1), b in stack level 2 (S2), g in

stack level 3 (S3), and Q in stack level 4 (S4).

Prompt with an input string

These two approaches for identifying the order of the input data are not very

efficient. You can, however, help the user identify the variables to be used by

prompting him or her with the name of the variables. From the various methods

provided by the User RPL language, the simplest is to use an input string and the

function INPUT („°L@@@IN@@ @INPUT@) to load your input data.

The following program prompts the user for the value of a variable a and places

the input in stack level 1:

« “Enter a:“ {“ꢀ:a:“ {2 0} V } INPUT OBJ→ »

This program includes the symbol :: (tag) and ꢀ (return), available through the

keystroke combinations „ê and ‚ë, both associated with the .

key. The tag symbol (::) is used to label strings for input and output. The return

symbol (ꢀ) is similar to a carriage return in a computer. The strings between

quotes (“ “) are typed directly from the alphanumeric keyboard.

Save the program in a variable called INPTa (for INPuT a).

Try running the program by pressing the soft menu key labeled @INPTa.

Page 21-21

The result is a stack prompting the user for the value of a and placing the cursor

right in front of the prompt :a: Enter a value for a, say 35, then press `.

The result is the input string :a:35 in stack level 1.

A function with an input string

If you were to use this piece of code to calculate the function, f(a) = 2*a^2+3,

you could modify the program to read as follows:

« “Enter a:“ {“ꢀ:a:“ {2 0} V }

INPUT OBJ→ → a « ‘2*a^2+3‘ » »

Save this new program under the name ‘FUNCa’ (FUNCtion of a):

Run the program by pressing @FUNCa. When prompted to enter the value of a

enter, for example, 2, and press `. The result is simply the algebraic 2a +3,

which is an incorrect result. The calculator provides functions for debugging

programs to identify logical errors during program execution as shown below.

Debugging the program

To figure out why it did not work we use the DBUG function in the calculator as

follows:

³@FUNCa `

„°LL @@RUN@ @@DBG@

Copies program name to stack level 1

Starts debugger

@SST↓@

Step-by-step debugging, result: “Enter a:”

Result: {“ ꢀ a:” {2 0} V}

@SST↓@

Result: user is prompted to enter value of a

Enter a value of 2 for a. Result: “ꢀ:a:2”

Result: a:2

@SST↓@

Page 21-22

@SST↓@

Result: empty stack, executing →a

Result: empty stack, entering subprogram «

Result: ‘2*a^2+3’

Result: ‘2*a^2+3’, leaving subprogram »

Result: ‘2*a^2+3’, leaving main program»

Further pressing the @SST↓@ soft menu key produces no more output since we

have gone through the entire program, step by step. This run through the

debugger did not provide any information on why the program is not

calculating the value of 2a +3for a = 2. To see what is the value of a in the

sub-program, we need to run the debugger again and evaluate a within the

sub-program. Try the following:

Recovers variables menu

Copies program name to stack level 1

Starts debugger

³@FUNCa `

„°LL @@RUN@ @@DBG@

@SST↓@

Step-by-step debugging, result: “Enter a:”

Result: {“ ꢀa:” {2 0} V}

Result: user is prompted to enter value of a

Enter a value of 2 for a. Result: “ꢀ:a:2”

Result: a:2

Result: empty stack, executing →a

Result: empty stack, entering subprogram «

At this point we are within the subprogram « ‘2*a^2+3’ » which uses the local

variable a. To see the value of a use:

~„aμ

This indeed shows that the local variable a = 2

Let’s kill the debugger at this point since we already know the result we will get.

To kill the debugger press @KILL. You receive an <!> Interruptedmessage

acknowledging killing the debugger. Press $to recover normal calculator

display.

Note: In debugging mode, every time we press @SST↓@ the top left corner of the

display shows the program step being executed. A soft key function called

@@SST@ is also available under the @RUN sub-menu within the PRG menu. This can

be used to execute at once any sub-program called from within a main

program. Examples of the application of @@SST@ will be shown later.

Page 21-23

Fixing the program

The only possible explanation for the failure of the program to produce a

numerical result seems to be the lack of the command ꢀNUM after the

algebraic expression ‘2*a^2+3’. Let’s edit the program by adding the missing

EVAL function. The program, after editing, should read as follows:

« “Enter a:“ {“ꢀ:a:“ {2 0} V } INPUT

OBJ→ → a « ‘2*a^2+3‘ ꢀNUM» »

Store it again in variable FUNCa, and run the program again with a = 2. This

time, the result is 11, i.e., 2*2 +3 = 11.

Input string for two or three input values

In this section we will create a sub-directory, within the directory HOME, to hold

examples of input strings for one, two, and three input data values. These will

be generic input strings that can be incorporated in any future program, taking

care of changing the variable names according to the needs of each program.

Let’s get started by creating a sub-directory called PTRICKS (Programming

TRICKS) to hold programming tidbits that we can later borrow from to use in

more complex programming exercises. To create the sub-directory, first make

sure that you move to the HOME directory. Within the HOME directory, use the

following keystrokes to create the sub-directory PTRICKS:

³~~ptricks`

„°@@MEM@@ @@DIR@@ @CRDIR

Enter directory name ‘PTRICKS’

Create directory

Recover variable listing

A program may have more than 3 input data values. When using input strings

we want to limit the number of input data values to 5 at a time for the simple

reason that, in general, we have visible only 7 stack levels. If we use stack level

7 to give a title to the input string, and leave stack level 6 empty to facilitate

reading the display, we have only stack levels 1 through 5 to define input

variables.

Page 21-24

Input string program for two input values

The input string program for two input values, say a and b, looks as follows:

« “Enter a and b:“ {“ꢀ:a:ꢀ:b:“ {2 0} V } INPUT OBJ→ »

This program can be easily created by modifying the contents of INPTa. Store

this program into variable INPT2.

Application: evaluating a function of two variables

Consider the ideal gas law, pV = nRT, where p = gas pressure (Pa), V = gas

volume(m ), n = number of moles (gmol), R = universal gas constant =

8.31451_J/(gmol*K), and T = absolute temperature (K).

We can define the pressure p as a function of two variables, V and T, as p(V,T)

= nRT/V for a given mass of gas since n will also remain constant. Assume

that n = 0.2 gmol, then the function to program is

p(V,T) = 8.31451⋅0.2⋅ = (1.662902 _ )⋅

We can define the function by typing the following program

« → V T ‘(1.662902_J/K)*(T/V)’ »

and storing it into variable @@@p@@@.

The next step is to add the input string that will prompt the user for the values of

V and T. To create this input stream, modify the program in @@@p@@@ to read:

« “Enter V and T:“ {“ꢀ :V:ꢀ :T:“ {2 0} V }

INPUT OBJ→ → V T ‘(1.662902_J/K)*(T/V)’

Store the new program back into variable @@@p@@@. Press @@@p@@@ to run the program.

Enter values of V = 0.01_m^3 and T = 300_K in the input string, then press

Page 21-25

`. The result is 49887.06_J/m^3. The units of J/m^3 are equivalent to

Pascals (Pa), the preferred pressure unit in the S.I. system.

Note: because we deliberately included units in the function definition, the

input values must have units attach to them in input to produce the proper result.

Input string program for three input values

The input string program for three input values, say a ,b, and c, looks as

follows:

« “Enter a, b and c:“ {“ꢀ :a:ꢀ :b:ꢀ :c:“ {2 0} V } INPUT

OBJ→ »

This program can be easily created by modifying the contents of INPT2 to make

it look like shown immediately above. The resulting program can then be stored

in a variable called INPT3. With this program we complete the collection of

input string programs that will allow us to enter one, two, or three data values.

Keep these programs as a reference and copy and modify them to fulfill the

requirements of new programs you write.

Application: evaluating a function of three variables

Suppose that we want to program the ideal gas law including the number of

moles, n, as an additional variable, i.e., we want to define the function

J n ⋅T

p(V,T,n) = (8.31451_

)

and modify it to include the three-variable input string. The procedure to put

together this function is very similar to that used earlier in defining the function

p(V,T). The resulting program will look like this:

« “Enter V, T, and n:“ {“ ꢀ :V:ꢀ :T:ꢀ :n:“ {2 0} V } INPUT

OBJ→ → V T n ‘(8.31451_J/(K*mol))*(n*T/V)‘ »

Store this result back into the variable @@@p@@@.To run the program, press @@@p@@@.

Page 21-26

Enter values of V = 0.01_m^3, T = 300_K, and n = 0.8_mol. Before pressing

`, the stack will look like this:

Press ` to get the result 199548.24_J/m^3, or 199548.24_Pa = 199.55

kPa.

Input through input forms

Function INFORM („°L@@@IN@@ @INFOR@.) can be used to create detailed

input forms for a program. Function INFORM requires five arguments, in this

order:

1. A title: a character string describing the input form

2. Field definitions: a list with one or more field definitions {s s … s },

where each field definition, s , can have one of two formats:

a. A simple field label: a character string

b. A list of specifications of the form {“label” “helpInfo” type

type … type }. The “label” is a field label. The “helpInfo” is

a character string describing the field label in detail, and the

type specifications is a list of types of variables allowed for the

field (see Chapter 24 for object types).

3. Field format information: a single number col or a list {col tabs}. In this

specification, col is the number of columns in the input box, and tabs

(optional) specifies the number of tab stops between the labels and the

fields in the form. The list could be an empty list. Default values are col

= 1 and tabs = 3.

4. List of reset values: a list containing the values to reset the different

fields if the option @RESET is selected while using the input form.

5. List of initial values: a list containing the initial values of the fields.

Page 21-27

The lists in items 4 and 5 can be empty lists. Also, if no value is to be

selected for these options you can use the NOVAL command

(„°L@@@IN@@ @NOVAL@).

After function INFORM is activated you will get as a result either a zero, in case

the @CANCEL option is entered, or a list with the values entered in the fields in the

order specified and the number 1, i.e., in the RPN stack:

2: {v v … v }

1 2

Thus, if the value in stack level 1 is zero, no input was performed, while it this

value is 1, the input values are available in stack level 2.

Example 1 - As an example, consider the following program, INFP1 (INput

Form Program 1) to calculate the discharge Q in an open channel through

1/2

Chezy’s formula: Q = C⋅(R⋅S) , where C is the Chezy coefficient, a function

of the channel surface’s roughness (typical values 80-150), R is the hydraulic

radius of the channel (a length), and S is the channel bed’s slope (a

dimensionless numbers, typically 0.01 to 0.000001). The following program

defines an input form through function INFORM:

« “ CHEZY’S EQN” { { “C:” “Chezy’s coefficient” 0} { “R:”

“Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { } { 120

1 .0001} { 110 1.5 .00001 } INFORM »

In the program we can identify the 5 components of the input as follows:

1. Title: “ CHEZY’S EQN”

2. Field definitions: there are three of them, with labels “C:”, “R:”, “S:”,

info strings “Chezy coefficient”, “Hydraulic radius”, “Channel bed

slope”, and accepting only data type 0 (real numbers) for all of the

three fields:

{ { “C:” “Chezy’s coefficient” 0} { “R:” “Hydraulic

radius” 0 } { “S:” “Channel bed slope” 0} }

Page 21-28

3. Field format information: { }(an empty list, thus, default values used)

4. List of reset values: { 120 1 .0001}

5. List of initial values: { 110 1.5 .00001}

Save the program into variable INFP1. Press @INFP1 to run the program. The

input form, with initial values loaded, is as follows:

To see the effect of resetting these values use L @RESET (select Reset all to reset

field values):

Now, enter different values for the three fields, say, C = 95, R = 2.5, and S =

0.003, pressing @@@OK@@@ after entering each of these new values. After these

substitutions the input form will look like this:

Now, to enter these values into the program press @@@OK@@@ once more. This

activates the function INFORM producing the following results in the stack:

Page 21-29

Thus, we demonstrated the use of function INFORM. To see how to use these

input values in a calculation modify the program as follows:

« “ CHEZY’S EQN” { { “C:” “Chezy’s coefficient” 0} { “R:”

“Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { } { 120

1 .0001} { 110 1.5 .00001 } INFORM IF THEN OBJꢀ DROP ꢀ C R S

‘C*(R*S)’ ꢀNUM “Q” ꢀTAG ELSE “Operation cancelled” MSGBOX

END »

The program steps shown above after the INFORM command include a

decision branching using the IF-THEN-ELSE-END construct (described in detail

elsewhere in this Chapter). The program control can be sent to one of two

possibilities depending on the value in stack level 1. If this value is 1 the

control is passed to the commands:

OBJꢀ DROP ꢀ C R S ‘C*√(R*S)’ ꢀNUM “Q” ꢀTAG

These commands will calculate the value of Q and put a tag (or label) to it. On

the other hand, if the value in stack level 1 is 0 (which happens when a @CANCEL

is entered while using the input box) , the program control is passed to the

commands:

“Operation cancelled” MSGBOX

These commands will produce a message box indicating that the operation was

cancelled.

Note: Function MSGBOX belongs to the collection of output functions under

the PRG/OUT sub-menu. Commands IF, THEN, ELSE, END are available

under the PRG/BRCH/IF sub-menu. Functions OBJꢀ, ꢀTAG are available

under the PRG/TYPE sub-menu. Function DROP is available under the PRG/

STACK menu. Functions ꢀ and ꢀNUM are available in the keyboard.

Example 2 – To illustrate the use of item 3 (Field format information) in the

arguments of function INFORM, change the empty list used in program INFP1

to { 2 1 }, meaning 2, rather than the default 3, columns, and only one tab stop

between labels and values. Store this new program in variable INFP2:

Page 21-30

« “ CHEZY’S EQN” { { “C:” “Chezy’s coefficient” 0} { “R:”

“Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } {

2 1 } { 120 1 .0001} { 110 1.5 .00001 } INFORM IF THEN

OBJꢀ DROP ꢀ C R S ‘C*(R*S)’ ꢀNUM “Q” ꢀTAG ELSE

“Operation cancelled” MSGBOX END »

Running program @INFP2 produces the following input form:

Example 3 – Change the field format information list to { 3 0 } and save the

modified program into variable INFP3. Run this program to see the new input

form:

Creating a choose box

Function CHOOSE („°L@@@IN@@ @CHOOS@) allows the user to create a

choose box in a program. This function requires three arguments:

1. A prompt (a character string describing the choose box)

2. A list of choice definitions {c c … c }. A choice definition c can have

any of two formats:

a. An object, e.g., a number, algebraic, etc., that will be

displayed in the choose box and will also be the result of the

choice.

b. A list {object_displayed object_result} so that object_displayed

is listed in the choose box, and object_result is selected as the

result if this choice is selected.

3. A number indicating the position in the list of choice definitions of the

default choice. If this number is 0, no default choice is highlighted.

Page 21-31

Activation of the CHOOSE function will return either a zero, if a @CANCEL action

is used, or, if a choice is made, the choice selected (e.g., v) and the number 1,

i.e., in the RPN stack:

Example 1 – Manning’s equation for calculating the velocity in an open

channel flow includes a coefficient, C , which depends on the system of units

used. If using the S.I. (Systeme International), C = 1.0, while if using the E.S.

(English System), C = 1.486. The following program uses a choose box to let

the user select the value of C by selecting the system of units. Save it into

variable CHP1 (CHoose Program 1):

« “Units coefficient” { { “S.I. units” 1}

{ “E.S. units” 1.486} } 1 CHOOSE »

Running this program (press @CHP1) shows the following choose box:

Depending on whether you select S.I. unitsor E.S. units, function

CHOOSE places either a value of 1 or a value of 1.486 in stack level 2 and a

1 in level 1. If you cancel the choose box, CHOICE returns a zero (0).

The values returned by function CHOOSE can be operated upon by other

program commands as shown in the modified program CHP2:

« “Units coefficient” { { “S.I. units” 1} { “E.S. units”

1.486} } 1 CHOOSE IF THEN “Cu” ꢀTAG ELSE “Operation

cancelled” MSGBOX END »

The commands after the CHOOSE function in this new program indicate a

decision based on the value of stack level 1 through the IF-THEN-ELSE-END

construct. If the value in stack level 1 is 1, the commands “Cu” ꢀTAGwill

produced a tagged result in the screen. If the value in stack level 1 is zero, the

Page 21-32

commands “Operation cancelled” MSGBOX will show a message box

indicating that the operation was cancelled.

Identifying output in programs

The simplest way to identify numerical program output is to “tag” the program

results. A tag is simply a string attached to a number, or to any object. The

string will be the name associated with the object. For example, earlier on,

when debugging programs INPTa (or INPT1) and INPT2, we obtained as

results tagged numerical output such as :a:35.

Tagging a numerical result

To tag a numerical result you need to place the number in stack level 2 and the

tagging string in stack level 2, then use the →TAG function („ ° @TYPE@

ꢀTAG) For example, to produce the tagged result B:5., use:

5`‚Õ~b„ ° @TYPE@ @ ꢀTAG

Decomposing a tagged numerical result into a number and a tag

To decompose a tagged result into its numerical value and its tag, simply use

function OBJꢀ („°@TYPE@ @OBJ ꢀ@). The result of decomposing a tagged

number with →OBJ is to place the numerical value in stack level 2 and the tag

in stack level 1. If you are interested in using the numerical value only, then

you will drop the tag by using the backspace key ƒ. For example,

decomposing the tagged quantity B:5(see above), will produce:

“De-tagging” a tagged quantity

“De-tagging” means to extract the object out of a tagged quantity. This

function is accessed through the keystroke combination: „ ° @TYPE@ L

@DTAG. For example, given the tagged quantity a:2, DTAG returns the

numerical value 2.

Page 21-33

Note: For mathematical operations with tagged quantities, the calculator will

"detag" the quantity automatically before the operation. For example, the left-

hand side figure below shows two tagged quantities before and after pressing

the * key in RPN mode:

Examples of tagged output

Example 1 – tagging output from function FUNCa

Let’s modify the function FUNCa, defined earlier, to produce a tagged output.

Use ‚ @FUNCa to recall the contents of FUNCa to the stack. The original

function program reads

« “Enter a:“ {“ꢀ:a:“ {2 0} V } INPUT OBJ→ → a« ‘2*a^2+3‘

ꢀNUM » »

Modify it to read:

« “Enter a:“ {“ꢀ:a:“ {2 0} V } INPUT OBJ→ → a« ‘2*a^2+3‘

ꢀNUM ”F” →TAG» »

Store the program back into FUNCa by using „ @FUNCa. Next, run the

program by pressing @FUNCa. Enter a value of 2 when prompted, and press

`. The result is now the tagged result F:11.

Example 2 – tagging input and output from function FUNCa

In this example we modify the program FUNCa so that the output includes not

only the evaluated function, but also a copy of the input with a tag.

Use ‚ @FUNCa to recall the contents of FUNCa to the stack:

« “Enter a:“ {“ꢀ:a:“ {2 0} V } INPUT OBJ→ → a « ‘2*a^2+3‘

ꢀNUM ”F” →TAG» »

Modify it to read:

Page 21-34

« “Enter a:“ {“ꢀ:a:“ {2 0} V } INPUT OBJ→ → a « ‘2*a^2+3‘

EVAL ”F” →TAG a SWAP» »

(Recall that the function SWAP is available by using „°@STACK @SWAP@).

Store the program back into FUNCa by using „ @FUNCa. Next, run the

program by pressing @FUNCa . Enter a value of 2 when prompted, and press

`. The result is now two tagged numbers a:2. in stack level 2, and F:11.

in stack level 1.

Note: Because we use an input string to get the input data value, the local

variable aactually stores a tagged value ( :a:2, in the example above).

Therefore, we do not need to tag it in the output. All what we need to do is

place an abefore the SWAP function in the subprogram above, and the

tagged input is placed in the stack. It should be pointed out that, in

performing the calculation of the function, the tag of the tagged input a is

dropped automatically, and only its numerical value is used in the calculation.

To see the operation of the function FUNCa, step by step, you could use the

DBUG function as follows:

³@FUNCa `

„°LL @@RUN@ @@DBG@

Copies program name to stack level 1

Starts debugger

@SST↓@

Step-by-step debugging, result: “Enter a:”

Result: {“ ꢀa:” {2 0} V}

Result: user is prompted to enter value of a

Enter a value of 2 for a. Result: “ꢀ:a:2”

Result: a:2

Result: empty stack, executing →a

Result: empty stack, entering subprogram «

Result: ‘2*a^2+3’

Result: empty stack, calculating

Result: 11.,

Result: “F”

Result: F: 11.

Result: a:2.

Result: swap levels 1 and 2

leaving subprogram »

leaving main program »

Page 21-35

Example 3 – tagging input and output from function p(V,T)

In this example we modify the program @@@p@@@ so that the output tagged input

values and tagged result. Use ‚@@@p@@@ to recall the contents of the program to

the stack:

« “Enter V, T, and n:“ {“ ꢀ :V:ꢀ :T:ꢀ :n:“ {2 0} V } INPUT

OBJ→ → V T n ‘(8.31451_J/(K*mol))*(n*T/V)‘ »

Modify it to read:

« “Enter V, T and n:“ {“ꢀ :V:ꢀ :T:ꢀ :n:“ {2 0} V } INPUT

OBJ→ → V T n « V T n‘(8.31451_J/(K*mol))*(n*T/V)‘ EVAL “p”

→TAG » »

Note: Notice that we have placed the calculation and tagging of the function

p(V,T,n), preceded by a recall of the input variables V T n, into a sub-program

[the sequence of instructions contained within the inner set of program symbols

« » ]. This is necessary because without the program symbol separating the

two listings of input variables (V T N « V T n), the program will assume that

the input command

→ V T N V T n

requires six input values, while only three are available. The result would have

been the generation of an error message and the interruption of the program

execution.

To include the subprogram mentioned above in the modified definition of

program @@@p@@@, will require you to use ‚å at the beginning and end of the

sub-program. Because the program symbols occur in pairs, whenever ‚å

is invoked, you will need to erase the closing program symbol (») at the

beginning, and the opening program symbol («) at the end, of the sub-program.

To erase any character while editing the program, place the cursor to the right

of the character to be erased and use the backspace key ƒ.

Page 21-36

Store the program back into variable p by using „@@@p@@@. Next, run the

program by pressing @@@p@@@. Enter values of V = 0.01_m^3, T = 300_K, and n =

0.8_mol, when prompted. Before pressing ` for input, the stack will look like

this:

After execution of the program, the stack will look like this:

In summary: The common thread in the three examples shown here is the

use of tags to identify input and output variables. If we use an input string to

get our input values, those values are already pre-tagged and can be easily

recall into the stack for output. Use of the →TAG command allows us to

identify the output from a program.

Using a message box

A message box is a fancier way to present output from a program. The

message box command in the calculator is obtained by using

„°L@@OUT@ @MSGBO@. The message box command requires that the output

string to be placed in the box be available in stack level 1. To see the

operation of the MSGBOX command try the following exercise:

‚Õ~‚t~„ê1.2

‚Ý ~„r~„a~„d

„°L@@OUT@ @MSGBO@

Page 21-37

The result is the following message box:

Press @@@OK@@@ to cancel the message box.

You could use a message box for output from a program by using a tagged

output, converted to a string, as the output string for MSGBOX. To convert any

tagged result, or any algebraic or non-tagged value, to a string, use the

function →STR available at „°@TYPE@ @ ꢀSTR.

Using a message box for program output

The function @@@p@@@ , from the last example, can be modified to read:

« “Enter V, T and n:“ {“ꢀ :V:ꢀ :T:ꢀ :n:“ {2 0} V } INPUT

OBJ→ → V T n « V T n‘(8.31451_J/(K*mol))*(n*T/V)‘ EVAL “p”

→TAG →STR MSGBOX » »

Store the program back into variable p by using „@@@p@@@. Run the program by

pressing @@@p@@@. Enter values of V = 0.01_m^3, T = 300_K, and n = 0.8_mol,

when prompted.

As in the earlier version of @@@p@@@, before pressing ` for input, the stack will

look like this:

The first program output is a message box containing the string:

Page 21-38

Press @@@OK@@@ to cancel message box output. The stack will now look like this:

Including input and output in a message box

We could modify the program so that not only the output, but also the input, is

included in a message box. For the case of program @@@p@@@, the modified

program will look like:

« “Enter V, T and n:“ {“ꢀ :V:ꢀ :T:ꢀ :n:“ {2 0} V } INPUT

OBJ→ → V T n « V →STR “ꢀ ” +T →STR “ꢀ ” + n→STR “ꢀ ” +

‘(8.31451_J/(K*mol))*(n*T/V)‘ EVAL “p” →TAG →STR + + +

MSGBOX » »

Notice that you need to add the following piece of code after each of the

variable names V, T, and n, within the sub-program:

→STR “ꢀ ” +

To get this piece of code typed in the first time use:

„°@TYPE@ @ ꢀSTR ‚Õ ‚ë ™+

Because the functions for the TYPE menu remain available in the soft menu keys,

for the second and third occurrences of the piece of code (→STR “ꢀ ” + )

within the sub-program (i.e., after variables T and n, respectively), all you need

to use is:

@ ꢀSTR ‚Õ ‚ë ™+

Page 21-39

You will notice that after typing the keystroke sequence ‚ë a new line is

generated in the stack.

The last modification that needs to be included is to type in the plus sign three

times after the call to the function at the very end of the sub-program.

Note: The plus sign (+) in this program is used to concatenate strings.

Concatenation is simply the operation of joining individual character strings.

To see the program operating:

ꢀ Store the program back into variable p by using „@@@p@@@.

ꢀ Run the program by pressing @@@p@@@.

ꢀ Enter values of V = 0.01_m^3, T = 300_K, and n = 0.8_mol, when

prompted.

As in the earlier version of [ p ], before pressing [ENTER] for input, the stack

will look like this:

The first program output is a message box containing the string:

Press @@@OK@@@ to cancel message box output.

Page 21-40

Incorporating units within a program

As you have been able to observe from all the examples for the different

versions of program @@@p@@@ presented in this chapter, attaching units to input

values may be a tedious process. You could have the program itself attach

those units to the input and output values. We will illustrate these options by

modifying yet once more the program @@@p@@@, as follows.

Recall the contents of program @@@p@@@ to the stack by using ‚@@@p@@@, and modify

them to look like this:

Note: I’ve separated the program arbitrarily into several lines for easy

reading. This is not necessarily the way that the program shows up in the

calculator’s stack. The sequence of commands is correct, however. Also,

recall that the character ꢀ does not show in the stack, instead it produces a

new line.

« “Enter V,T,n [S.I.]:“ {“ꢀ :V:ꢀ :T:ꢀ :n: “ {2 0} V }

INPUT OBJ→ → V T n

« V ‘1_m^3’ * T ‘1_K’ * n ‘1_mol’ * → V T n

« V “V” →TAG →STR “ꢀ ” +T “T” →TAG →STR “ꢀ ” + n “n” →TAG

→STR “ꢀ ” +

‘(8.31451_J/(K*mol))*(n*T/V)‘ EVAL “p” →TAG →STR + + +

MSGBOX » » »

This new version of the program includes an additional level of sub-

programming (i.e., a third level of program symbols « », and some steps using

lists, i.e.,

V ‘1_m^3’ * { } + T ‘1_K’ * + n ‘1_mol’ *+ EVAL → V T n

The interpretation of this piece of code is as follows. (We use input string values

of :V:0.01, :T:300, and :n:0.8):

1. V

: The value of V, as a tagged input (e.g., V:0.01) is

placed in the stack.

Page 21-41

2. ‘1_m^3’

: The S.I. units corresponding to V are then placed in

stack level 1, the tagged input for V is moved to

stack level 2.

3. *

: By multiplying the contents of stack levels 1 and 2,

we generate a number with units (e.g., 0.01_m^3),

but the tag is lost.

4. T ‘1_K’ *

5. n ‘1_mol’ *

6. → V T n

:Calculating value of T including S.I. units

: Calculating value of n including units

: The values of V, T, and n, located respectively in stack

levels 3, 2, and 1, are passed on to the next level of

sub-programming.

To see this version of the program in action do the following:

ꢀ Store the program back into variable p by using [ꢁ][ p ].

ꢀ Run the program by pressing [ p ].

ꢀ Enter values of V = 0.01, T = 300, and n = 0.8, when prompted (no units

required now).

Before pressing ` for input, the stack will look like this:

Press ` to run the program. The output is a message box containing the

string:

Page 21-42

Press @@@OK@@@ to cancel message box output.

Message box output without units

Let’s modify the program @@@p@@@ once more to eliminate the use of units

throughout it. The unit-less program will look like this:

« “Enter V,T,n [S.I.]:“ {“ꢀ :V:ꢀ :T:ꢀ :n: “ {2 0} V }

INPUT OBJ→ → V T n

« V DTAG T DTAG n DTAG → V T n

« “V=” V →STR + “ꢀ ”+ “T=” T →STR + “ꢀ ” + “n=” n →STR +

“ꢀ ” +

‘8.31451*n*T/V‘ EVAL →STR “p=” SWAP + + + + MSGBOX » » »

And when run with the input data V = 0.01, T = 300, and n = 0.8, produces

the message box output:

Press @@@OK@@@ to cancel the message box output.

Relational and logical operators

So far we have worked mainly with sequential programs. The User RPL

language provides statements that allow branching and looping of the program

flow. Many of these make decisions based on whether a logical statement is

true or not. In this section we present some of the elements used to construct

such logical statements, namely, relational and logical operators.

Relational operators

Relational operators are those operators used to compare the relative position

of two objects. For example, dealing with real numbers only, relational

Page 21-43

operators are used to make a statement regarding the relative position of two or

more real numbers. Depending on the actual numbers used, such a statement

can be true (represented by the numerical value of 1. in the calculator), or false

(represented by the numerical value of 0. in the calculator).

The relational operators available for programming the calculator are:

____________________________________________________

Operator

Meaning

Example

____________________________________________________

ꢁꢁ[

≠

“is equal to”

“is not equal to”

“is less than”

‘x==2’

‘3 ≠ 2’

‘m<n’

ꢀ

≥

≤

“is greater than”

“is greater than or equal to”

“is less than or equal to”

‘10>a’

‘p ≥ q’

‘7≤12’

_____________________________________________________

All of the operators, except == (which can be created by typing ‚Å

‚Å ), are available in the keyboard. They are also available in „°

@TEST@.

Two numbers, variables, or algebraics connected by a relational operator form

a logical expression that can take value of true (1.), false (0.), or could simply

not be evaluated. To determine whether a logical statement is true or not,

place the statement in stack level 1, and press EVAL (μ). Examples:

^‘2<10’μ, result: 1. (true)

^‘2>10’μ, result: 0. (false)

In the next example it is assumed that the variable m is not initialized (it has not

been given a numerical value):

^‘2==m’μ, result: ‘2==m’

The fact that the result from evaluating the statement is the same original

statement indicates that the statement cannot be evaluated uniquely.

Page 21-44

Logical operators

Logical operators are logical particles that are used to join or modify simple

logical statements. The logical operators available in the calculator can be

easily accessed through the keystroke sequence: „°@TEST@ L.

The available logical operators are: AND, OR, XOR (exclusive or), NOT, and

SAME. The operators will produce results that are true or false, depending on

the truth-value of the logical statements affected. The operator NOT (negation)

applies to a single logical statements. All of the others apply to two logical

statements.

Tabulating all possible combinations of one or two statements together with the

resulting value of applying a certain logical operator produces what is called

the truth table of the operator. The following are truth tables of each of the

standard logical operators available in the calculator:

NOT p

p AND q

p OR q

Page 21-45

p XOR q

The calculator includes also the logical operator SAME. This is a non-standard

logical operator used to determine if two objects are identical. If they are

identical, a value of 1 (true) is returned, if not, a value of 0 (false) is returned.

For example, the following exercise, in RPN mode, returns a value of 0:

‘SQ(2)’ ` 4 ` SAME

Please notice that the use of SAME implies a very strict interpretation of the

word “identical.” For that reason, SQ(2) is not identical to 4, although they

both evaluate, numerically, to 4.

Program branching

Branching of a program flow implies that the program makes a decision among

two or more possible flow paths. The User RPL language provides a number of

commands that can be used for program branching. The menus containing

these commands are accessed through the keystroke sequence:

„°@@BRCH@

This menu shows sub-menus for the program constructs

The program constructs IF…THEN..ELSE…END, and CASE…THEN…END will

be referred to as program branching constructs. The remaining constructs,

namely, START, FOR, DO, and WHILE, are appropriate for controlling repetitive

processing within a program and will be referred to as program loop

constructs. The latter types of program constructs are presented in more detail

in a later section.

Page 21-46

Branching with IF

In this section we presents examples using the constructs IF…THEN…END and

IF…THEN…ELSE…END.

The IF…THEN…END construct

The IF…THEN…END is the simplest of the IF program constructs. The general

format of this construct is:

IF logical_statement THEN program_statements END.

The operation of this construct is as follows:

1. Evaluate logical_statement.

2. If logical_statement is true, perform program _statements and continue

program flow after the END statement.

3. If logical_statement is false, skip program_statements and continue

program flow after the END statement.

To type in the particles IF, THEN, ELSE, and END, use:

„°@@BRCH@ @@IF@@

The functions @@@IF@@ @@THEN @@ELSE@ @@ END@@ are available in that menu to be typed

selectively by the user. Alternatively, to produce an IF…THEN…END construct

directly on the stack, use:

„°@@BRCH@ „ @@IF@@

This will create the following input in the stack:

Page 21-47

With the cursor ꢂ in front of the IF statement prompting the user for the logical

statement that will activate the IF construct when the program is executed.

Example: Type in the following program:

« → x« IF ‘x<3’ THEN ‘x^2‘ EVAL END ”Done” MSGBOX » »

and save it under the name ‘f1’. Press J and verify that variable @@@f1@@@ is

indeed available in your variable menu. Verify the following results:

@@@f1@@@ Result: 0

1.2 @@@f1@@@ Result: 1.44

3.5 @@@f1@@@ Result: no action

10 @@@f1@@@ Result: no action

These results confirm the correct operation of the IF…THEN…END construct.

The program, as written, calculates the function f (x) = x , if x < 3 (and not

output otherwise).

The IF…THEN…ELSE…END construct

The IF…THEN…ELSE…END construct permits two alternative program flow

paths based on the truth value of the logical_statement. The general format of

this construct is:

IF logical_statement THEN program_statements_if_true ELSE

program_statements_if_false END.

The operation of this construct is as follows:

1. Evaluate logical_statement.

2. If logical_statement is true, perform program statements_if_true and

continue program flow after the END statement.

3. If logical_statement is false, perform program statements_if_false and

continue program flow after the END statement.

To produce an IF…THEN…ELSE…END construct directly on the stack, use:

„°@@BRCH@ ‚ @@IF@@

This will create the following input in the stack:

Page 21-48

Example: Type in the following program:

« → x« IF ‘x<3’ THEN ‘x^2‘ ELSE ‘1-x’ END EVAL ”Done” MSGBOX » »

^{and save it under the name ‘f2’. Press}J and verify that variable @@@f2@@@ is

indeed available in your variable menu. Verify the following results:

@@@f2@@@ Result: 0

1.2 @@@f2@@@ Result: 1.44

3.5 @@@f2@@@ Result: -2.5

10 @@@f2@@@ Result: -9

These results confirm the correct operation of the IF…THEN…ELSE…END

construct. The program, as written, calculates the function

x², if x < 3

⎧

f (x) =

⎨

1− x, otherwise

⎩

Note: For this particular case, a valid alternative would have been to use an

IFTE function of the form: ‘f2(x) = IFTE(x<3,x^2,1-x)’

Nested IF…THEN…ELSE…END constructs

In most computer programming languages where the IF…THEN…ELSE…END

construct is available, the general format used for program presentation is the

following:

IF logical_statement THEN

program_statements_if_true

ELSE

program_statements_if_false

END

In designing a calculator program that includes IF constructs, you could start by

writing by hand the pseudo-code for the IF constructs as shown above. For

example, for program @@@f2@@@, you could write

Page 21-49

IF x<3 THEN

ELSE

1-x

END

While this simple construct works fine when your function has only two

branches, you may need to nest IF…THEN…ELSE…END constructs to deal with

function with three or more branches. For example, consider the function

x², if x < 3

1− x, if 3 ≤ x < 5

⎧

⎪

f (x) =

sin(x), if 5 ≤ x < 3π

exp(x),if 3π ≤ x < 15

− 2, elsewhere

⎨

⎪

⎩

Here is a possible way to evaluate this function using IF… THEN … ELSE …

END constructs:

IF x<3 THEN

ELSE

IF x<5 THEN

1-x

ELSE

IF x<3π

THEN

sin(x)

ELSE

IF x<15 THEN

exp(x)

ELSE

-2

END

Page 21-50

A complex IF construct like this is called a set of nested IF … THEN … ELSE …

END constructs.

A possible way to evaluate f3(x), based on the nested IF construct shown

above, is to write the program:

« → x« IF ‘x<3‘ THEN ‘x^2‘ ELSE IF ‘x<5‘ THEN ‘1-x‘ ELSE IF

‘x<3*π‘ THEN ‘SIN(x)‘ ELSE IF ‘x<15‘ THEN ‘EXP(x)‘ ELSE –2 END

END END END EVAL » »

Store the program in variable @@@f3@@@ and try the following evaluations:

1.5@@f3@@@

Result: 2.25 (i.e., x )

2.5 @@@f3@@@ Result: 6.25 (i.e., x )

4.2@@@f3@@@ Result: -3.2 (i.e., 1-x)

5.6 @@@f3@@@ Result -0.631266… (i.e., sin(x), with x in radians)

12 @@@f3@@@ Result: 162754.791419 (i.e., exp(x))

23 @@@f3@@@ Result: -2. (i.e., -2)

The CASE construct

The CASE construct can be used to code several possible program flux paths,

as in the case of the nested IF constructs presented earlier. The general format

of this construct is as follows:

CASE

Logical_statement THEN program_statements END

Default_program_statements (optional)

END

When evaluating this construct, the program tests each of the logical_statements

until it finds one that is true. The program executes the corresponding

Page 21-51

program_statements, and passes program flow to the statement following the

END statement.

The CASE, THEN, and END statements are available for selective typing by

using „°@@BRCH@ @CASE@ .

If you are in the BRCH menu, i.e., („°@@BRCH@ ) you can use the following

shortcuts to type in your CASE construct (The location of the cursor is indicated

by the symbol ꢂ ):

ꢀ „@CASE@: Starts the case construct providing the prompts: CASE ꢂ THEN

END END

ꢀ ‚@CASE@: Completes a CASE line by adding the particles THEN ꢂ END

Example – program f (x) using the CASE statement

The function is defined by the following 5 expressions:

x², if x < 3

1− x, if 3 ≤ x < 5

⎧

⎪

f (x) =

sin(x), if 5 ≤ x < 3π

exp(x),if 3π ≤ x < 15

− 2, elsewhere

⎨

⎪

⎩

Using the CASE statement in User RPL language we can code this function as:

« → x« CASE ‘x<3‘ THEN ‘x^2‘ END ‘x<5‘ THEN ‘1-x‘ END ‘x<3*π‘

THEN ‘SIN(x)‘ END ‘x<15‘ THEN ‘EXP(x)‘ END –2 END EVAL » »

Store the program into a variable called @@f3c@. Then, try the following exercises:

1.5 @@f3c@

Result: 2.25 (i.e., x )

2.5 @@f3c@

Result: 6.25 (i.e., x )

4.2 @@f3c@

Result: -3.2 (i.e., 1-x)

Page 21-52

5.6 @@f3c@

Result: -0.631266… (i.e., sin(x), with x in radians)

Result: 162754.791419 (i.e., exp(x))

Result -2. (i.e., -2)

@@f3c@

As you can see, f3c produces exactly the same results as f3. The only

difference in the programs is the branching constructs used. For the case of

function f (x), which requires five expressions for its definition, the CASE

construct may be easier to code than a number of nested IF … THEN … ELSE

… END constructs.

Program loops

Program loops are constructs that permit the program the execution of a number

of statements repeatedly. For example, suppose that you want to calculate the

summation of the square of the integer numbers from 0 to n, i.e.,

S = k ²

∑

k=0

To calculate this summation all that you have to do is use the ‚½ key

within the equation editor and load the limits and expression for the summation

(examples of summations are presented in Chapters 2 and 13). However, in

order to illustrate the use of programming loops, we will calculate this

summation with our own User RPL codes. There are four different commands

that can be used to code a program loop in User RPL, these are START, FOR,

DO, and WHILE. The commands START and FOR use an index or counter to

determine how many times the loop is executed. The commands DO and

WHILE rely on a logical statement to decide when to terminate a loop

execution. Operation of the loop commands is described in detail in the

following sections.

The START construct

The START construct uses two values of an index to execute a number of

statements repeatedly. There are two versions of the START construct:

START…NEXT and START … STEP. The START…NEXT version is used when the

index increment is equal to 1, and the START…STEP version is used when the

index increment is determined by the user.

Page 21-53

Commands involved in the START construct are available through:

„°@@BRCH@ @START @START

Within the BRCH menu („°@@BRCH@) the following keystrokes are available

to generate START constructs (the symbol indicates cursor position):

ꢀ „ @START: Starts the START…NEXT construct: START ꢂ NEXT

ꢀ ‚ @START: Starts the START…STEP construct: START ꢂ STEP

The START…NEXT construct

The general form of this statement is:

start_value end_value START program_statements NEXT

Because for this case the increment is 1, in order for the loop to end you should

ensure that start_value < end_value. Otherwise you will produce what

is called an infinite (never-ending) loop.

Example – calculating of the summation S defined above

The START…NEXT construct contains an index whose value is inaccessible to

the user. Since for the calculation of the sum the index itself (k, in this case) is

needed, we must create our own index, k, that we will increment within the loop

each time the loop is executed. A possible implementation for the calculation of

S is the program:

« 0. DUP → n S k « 0. n START k SQ S + 1. ‘k‘ STO+ ‘S‘ STO

NEXT S “S” →TAG » »

Type the program in, and save it in a variable called @@@S1@@@.

Here is a brief explanation of how the program works:

Page 21-54

1. This program needs an integer number as input. Thus, before execution,

that number (n) is in stack level 1. The program is then executed.

2. A zero is entered, moving n to stack level 2.

3. The command DUP, which can be typed in as ~~dup~, copies

the contents of stack level 1, moves all the stack levels upwards, and places

the copy just made in stack level 1. Thus, after DUP is executed, n is in

stack level 3, and zeroes fill stack levels 1 and 2.

4. The piece of code → n S k stores the values of n, 0, and 0, respectively

into local variables n, S, k. We say that the variables n, S, and k have

been initialized (S and k to zero, n to whatever value the user chooses).

5. The piece of code 0. n START identifies a START loop whose index will

take values of 0, 1, 2, …, n

6. The sum S is incremented by k in the piece of code that reads: k SQ S +

7. The index k is incremented by 1 in the piece of code that reads: 1. k +

8. At this point, the updated values of S and k are available in stack levels 2

and 1, respectively. The piece of code ‘k‘ STO stores the value from stack

level 1 into local variable k. The updated value of S now occupies stack

level 1.

9. The piece of code ‘S‘ STO stores the value from stack level 1 into local

variable k. The stack is now empty.

10. The particle NEXT increases the index by one and sends the control to the

beginning of the loop (step 6).

11. The loop is repeated until the loop index reaches the maximum value, n.

12. The last part of the program recalls the last value of S (the summation), tags

it, and places it in stack level 1 to be viewed by the user as the program

output.

To see the program in action, step by step, you can use the debugger as follows

(use n = 2). Let SL1 mean stack level 1:

J2[‘] @@@S1@@ `

Place a 2 in level 2, and the

program name, ‘S1’, in level 1

Page 21-55

„°LL @@RUN@ @@DBG@

Start the debugger. SL1 = 2.

SL1 = 0., SL2 = 2.

SL1 = 0., SL2 = 0., SL3 = 2. (DUP)

Empty stack (-> n S k)

@SST↓@

Empty stack (« - start subprogram)

SL1 = 0., (start value of loop index)

SL1 = 2.(n), SL2 = 0. (end value of loop

index)

@SST↓@

Empty stack (START – beginning of loop)

--- loop execution number 1 for k = 0

@SST↓@

SL1 = 0. (k)

SL1 = 0. (SQ(k) = k )

SL1 = 0.(S), SL2 = 0. (k )

SL1 = 0. (S + k )

SL1 = 1., SL2 = 0. (S + k )

SL1 = 0.(k), SL2 = 1., SL3 = 0. (S + k )

SL1 = 1.(k+1), SL2 = 0. (S + k )

SL1 = ‘k’, SL2 = 1., SL3 = 0. (S + k )

SL1 = 0. (S + k ) [Stores value of SL2 = 1, into

SL1 = ‘k’]

SL1 = ‘S’, SL2 = 0. (S + k )

@SST↓@

Empty stack [Stores value of SL2 = 0, into SL1

= ‘S’]

Empty stack (NEXT – end of loop)

@SST↓@

--- loop execution number 2 for k = 1

@SST↓@

SL1 = 1. (k)

SL1 = 1. (SQ(k) = k )

SL1 = 0.(S), SL2 = 1. (k )

SL1 = 1. (S + k )

SL1 = 1., SL2 = 1. (S + k )

SL1 = 1.(k), SL2 = 1., SL3 = 1. (S + k )

SL1 = 2.(k+1), SL2 = 1. (S + k )

SL1 = ‘k’, SL2 = 2., SL3 = 1. (S + k )

Page 21-56

@SST↓@

SL1 = 1. (S + k ) [Stores value of SL2 = 2, into

SL1 = ‘k’]

@SST↓@

SL1 = ‘S’, SL2 = 1. (S + k )

Empty stack [Stores value of SL2 = 1, into SL1

= ‘S’]

@SST↓@

Empty stack (NEXT – end of loop)

--- loop execution number 3 for k = 2

@SST↓@

SL1 = 2. (k)

SL1 = 4. (SQ(k) = k )

SL1 = 1.(S), SL2 = 4. (k )

SL1 = 5. (S + k )

SL1 = 1., SL2 = 5. (S + k )

SL1 = 2.(k), SL2 = 1., SL3 = 5. (S + k )

SL1 = 3.(k+1), SL2 = 5. (S + k )

SL1 = ‘k’, SL2 = 3., SL3 = 5. (S + k )

SL1 = 5. (S + k ) [Stores value of SL2 = 3, into

SL1 = ‘k’]

SL1 = ‘S’, SL2 = 5. (S + k )

@SST↓@

Empty stack [Stores value of SL2 = 0, into SL1

= ‘S’]

Empty stack (NEXT – end of loop)

@SST↓@

--- for n = 2, the loop index is exhausted and control is passed to the statement

following NEXT

@SST↓@

SL1 = 5 (S is recalled to the stack)

SL1 = “S”, SL2 = 5 (“S” is placed in the stack)

SL1 = S:5 (tagging output value)

SL1 = S:5 (leaving sub-program »)

SL1 = S:5 (leaving main program »)

The step-by-step listing is finished. The result of running program @@@S1@@ with n =

2, is S:5.

Check also the following results: J

Page 21-57

@@@S1@@

10 @@@S1@@

Result: S:14

Result: S:55

Result: S:385

Result: S:9455

@@@S1@@

20 @@@S1@@

Result: S:30

Result: S:204

Result: S:2870

Result: S:338350

30 @@@S1@@

100 @@@S1@@

The START…STEP construct

The general form of this statement is:

start_value end_value START program_statements increment

The start_value, end_value, and incrementof the loop index can be positive

or negative quantities. For increment > 0, execution occurs as long as the

index is less than or equal to end_value. For increment < 0, execution

occurs as long as the index is greater than or equal to end_value.

Example – generating a list of values

Suppose that you want to generate a list of values of x from x = 0.5 to x = 6.5

in increments of 0.5. You can write the following program:

« → xs xe dx « xs DUP xe START DUP dx + dx STEP DROP xe xs

– dx / ABS 1 + →LIST » »

and store it in variable @GLIST.

In this program , xs = starting value of the loop, xe = ending value of the loop,

dx = increment value for loop. The program places values of xs, xs+dx,

xs+2⋅dx, xs+3⋅dx, … in the stack. Then, it calculates the number of elements

generated using the piece of code: xe xs – dx / ABS 1. +

Finally, the program puts together a list with the elements placed in the stack.

ꢀ Check out that the program call 0.5 ` 2.5 ` 0.5 ` @GLIST

produces the list {0.5 1. 1.5 2. 2.5}.

ꢀ To see step-by-step operation use the program DBUG for a short list, for

example:

Page 21-58

J1 # 1.5 # 0.5 `

[ ‘ ] @GLIST `

„°LL @@RUN@ @@DBG@

Enter parameters 1 1.5 0.5

Enter the program name in level 1

Start the debugger.

Use @SST↓@ to step into the program and see the detailed operation of each

command.

The FOR construct

As in the case of the START command, the FOR command has two variations:

the FOR…NEXT construct, for loop index increments of 1, and the FOR…STEP

construct, for loop index increments selected by the user. Unlike the START

command, however, the FOR command does require that we provide a name

for the loop index (e.g., j, k, n). We need not to worry about incrementing the

index ourselves, as done in the examples using START. The value

corresponding to the index is available for calculations.

Commands involved in the FOR construct are available through:

„°@@BRCH@ @@FOR

Within the BRCH menu („°@@BRCH@) the following keystrokes are available

to generate FOR constructs (the symbol ꢂ indicates cursor position):

ꢀ „ @@FOR: Starts the FOR…NEXT construct: FOR ꢂ NEXT

ꢀ ‚ @@FOR: Starts the FOR…STEP construct: FOR ꢂ STEP

The FOR…NEXT construct

The general form of this statement is:

start_value end_value FOR loop_index program_statements

Page 21-59

To avoid an infinite loop, make sure that start_value < end_value.

Example – calculate the summation S using a FOR…NEXT construct

The following program calculates the summation

S = k ²

∑

k=0

Using a FOR…NEXT loop:

« 0 → n S « 0 n FOR k k SQ S + ‘S‘ STO NEXT S “S” →TAG » »

Store this program in a variable @@S2@@. Verify the following exercises: J

@@@S2@@

10 @@@S2@@

Result: S:14

Result: S:55

Result: S:385

Result: S:9455

@@@S2@@ Result: S:30

@@@S2@@ Result: S:204

20 @@@S2@@ Result: S:2870

30 @@@S2@@

100 @@@S2@@ Result: S:338350

You may have noticed that the program is much simpler than the one stored in

@@@S1@@. There is no need to initialize k, or to increment k within the program. The

program itself takes care of producing such increments.

The FOR…STEP construct

The general form of this statement is:

start_value end_value FOR loop_index program_statements

increment STEP

The start_value, end_value, and incrementof the loop index can be positive

or negative quantities. For increment > 0, execution occurs as long as the

index is less than or equal to end_value. For increment < 0, execution

occurs as long as the index is greater than or equal to end_value. Program

statements are executed at least once (e.g., 1 0 START 1 1 STEPreturns 1)

Page 21-60

Example – generate a list of numbers using a FOR…STEP construct

Type in the program:

« → xs xe dx « xe xs – dx / ABS 1. + → n « xs xe FOR x x

dx STEP n →LIST » » »

and store it in variable @GLIS2.

ꢀ Check out that the program call 0.5 ` 2.5 ` 0.5 ` @GLIS2

produces the list {0.5 1. 1.5 2. 2.5}.

ꢀ To see step-by-step operation use the program DBUG for a short list, for

example:

J1 # 1.5 # 0.5 `

[‘] @GLIS2 `

Enter parameters 1 1.5 0.5

Enter the program name in level 1

Start the debugger.

„°LL @@RUN@ @@DBG@

Use @SST↓@ to step into the program and see the detailed operation of each

command.

The DO construct

The general structure of this command is:

DO program_statements UNTIL logical_statement END

The DO command starts an indefinite loop executing the program_statements

until the logical_statement returns FALSE (0). The logical_statementmust

contain the value of an index whose value is changed in the

program_statements.

Example 1 - This program produces a counter in the upper left corner of the

screen that adds 1 in an indefinite loop until a keystroke (press any key) stops

the counter: « 0 DO DUP 1 DISP 1 + UNTIL KEY END DROP »

Command KEY evaluates to TRUE when a keystroke occurs.

Example 2 – calculate the summation S using a DO…UNTIL…END construct

Page 21-61

The following program calculates the summation

S = k ²

∑

k=0

Using a DO…UNTIL…END loop:

« 0. → n S « DO n SQ S + ‘S‘ STO n 1 – ‘n‘ STO UNTIL ‘n<0‘ END

S “S” →TAG » »

Store this program in a variable @@S3@@. Verify the following exercises: J

@@@S3@@

10 @@@S3@@

Result: S:14

Result: S:55

Result: S:385

Result: S:9455

@@@S3@@ Result: S:30

@@@S3@@ Result: S:204

20 @@@S3@@ Result: S:2870

30 @@@S3@@

100 @@@S3@@ Result: S:338350

Example 3 – generate a list using a DO…UNTIL…END construct

Type in the following program

« → xs xe dx « xe xs – dx / ABS 1. + xs → n x « xs DO

‘x+dx’ EVAL DUP ‘x’ STO UNTIL ‘x≥xe’ END n →LIST » » »

and store it in variable @GLIS3.

ꢀ Check out that the program call 0.5 ` 2.5 ` 0.5 ` @GLIS3

produces the list {0.5 1. 1.5 2. 2.5}.

ꢀ To see step-by-step operation use the program DBUG for a short list, for

example:

J1 # 1.5 # 0.5 `

[‘] @GLIS3 `

Enter parameters 1 1.5 0.5

Enter the program name in level 1

Start the debugger.

„°LL @@RUN@ @@DBG@

Use @SST↓@ to step into the program and see the detailed operation of each

command.

Page 21-62

The WHILE construct

The general structure of this command is:

WHILE logical_statement REPEAT program_statements END

The WHILE statement will repeat the program_statementswhile

logical_statementis true (non zero). If not, program control is passed to

the statement right after END. The program_statementsmust include a

loop index that gets modified before the logical_statementis checked at

the beginning of the next repetition. Unlike the DO command, if the first

evaluation of logical_statement is false, the loop is never executed.

Example 1 – calculate the summation S using a WHILE…REPEAT…END

construct

The following program calculates the summation

S = k ²

∑

k=0

Using a WHILE…REPEAT…END loop:

« 0. → n S « WHILE ‘n≥0‘ REPEAT n SQ S + ‘S‘ STO n 1 – ‘n‘ STO

END S “S” →TAG » »

Store this program in a variable @@S4@@. Verify the following exercises: J

@@@S4@@

10 @@@S4@@

Result: S:14

Result: S:55

Result: S:385

Result: S:9455

@@@S4@@

20 @@@S4@@

Result: S:30

Result: S:204

Result: S:2870

Result: S:338350

30 @@@S4@@

100 @@@S4@@

Example 2 – generate a list using a WHILE…REPEAT…END construct

Type in the following program

« → xs xe dx « xe xs – dx / ABS 1. + xs → n x « xs WHILE

‘x<xe‘ REPEAT ‘x+dx‘ EVAL DUP ‘x‘ STO END n →LIST » » »

Page 21-63

and store it in variable @GLIS4.

ꢀ Check out that the program call 0.5 ` 2.5 ` 0.5 ` @GLIS4

produces the list {0.5 1. 1.5 2. 2.5}.

ꢀ To see step-by-step operation use the program DBUG for a short list, for

example:

J1 # 1.5 # 0.5 `

[‘] @GLIS4 `

Enter parameters 1 1.5 0.5

Enter the program name in level 1

Start the debugger.

„°LL @@RUN@ @@DBG@

Use @SST↓@ to step into the program and see the detailed operation of each

command.

Errors and error trapping

The functions of the PRG/ERROR sub-menu provide ways to manipulate errors in

the calculator, and trap errors in programs. The PRG/ERROR sub-menu,

available through „°LL@ERROR@ , contains the following functions

and sub-menus:

DOERR

This function executes an user-define error, thus causing the calculator to behave

as if that particular error has occurred. The function can take as argument

either an integer number, a binary integer number, an error message, or the

number zero (0). For example, in RPN mode, entering 5` @DOERR,

produces the following error message: Error: Memory Clear

If you enter #11h ` @DOERR, produces the following message: Error:

Undefined FPTR Name

Page 21-64

If you enter “TRY AGAIN” ` @DOERR, produces the following message: TRY

AGAIN

Finally, 0` @DOERR, produces the message: Interrupted

ERRN

This function returns a number representing the most recent error. For example,

if you try 0Y$@ERRN, you get the number #305h. This is the binary

integer representing the error: Infinite Result

ERRM

This function returns a character string representing the error message of the

most recent error. For example, in Approx mode, if you try 0Y$@ERRM,

you get the following string: “Infinite Result”

ERR0

This function clears the last error number, so that, executing ERRN afterwards, in

Approx mode, will return # 0h. For example, if you try 0Y$@ERR0 @ERRN,

you get # 0h. Also, if you try 0Y$@ERR0 @ERRM, you get the empty string

“ “.

LASTARG

This function returns copies of the arguments of the command or function

executed most recently. For example, in RPN mode, if you use: 3/

2`, and then use function LASTARG (@LASTA), you will get the values 3 and

2 listed in the stack. Another example, in RPN mode, is the following:

5U`. Using LASTARG after these entries produces a 5.

Sub-menu IFERR

The @IFERR sub-menu provides the following functions:

Page 21-65

These are the components of the IFERR … THEN … END construct or of the

IFERR … THEN … ELSE … END construct. Both logical constructs are used for

trapping errors during program execution. Within the @ERROR sub-menu,

entering „@IFERR, or ‚@IFERR, will place the IFERR structure components in

the stack, ready for the user to fill the missing terms, i.e.,

The general form of the two error-trapping constructs is as follows:

IF trap-clause THEN error-clause END

IF trap-clause THEN error-clause ELSE normal-clause END

The operation of these logical constructs is similar to that of the IF … THEN …

END and of the IF … THEN … ELSE … END constructs. If an error is detected

during the execution of the trap-clause, then the error-clause is executed.

Otherwise, the normal-clause is executed.

As an example, consider the following program (@ERR1) that takes as input two

matrices, A and b, and checks if there is an error in the trap clause: A b / (RPN

mode, i.e., A/b). If there is an error, then the program calls function LSQ (Least

SQuares, see Chapter 11) to solve the system of equations:

« ꢀ A b « IFERR A b / THEN LSQ END » »

Try it with the arguments A = [ [ 2, 3, 5 ] , [1, 2, 1 ] ] and b = [ [ 5 ] , [ 6 ] ].

A simple division of these two arguments produces an error: /Error: Invalid

Dimension.

However, with the error-trapping construct of the program, @ERR1, with the same

arguments produces: [0.262295…, 0.442622…].

Page 21-66

User RPL programming in algebraic mode

While all the programs presented earlier are produced and run in RPN mode,

you can always type a program in User RPL when in algebraic mode by using

function RPL>. This function is available through the command catalog. As an

example, try creating the following program in algebraic mode, and store it into

variable P2:

« → X ‘2.5-3*X^2’ »

First, activate the RPL> function from the command catalog (‚N). All

functions activated in ALG mode have a pair of parentheses attached to their

name. The RPL> function is not exception, except that the parentheses must be

removed before we type a program in the screen. Use the arrow keys (š™)

and the delete key (ƒ) to eliminate the parentheses from the RPL>() statement.

At this point you will be ready to type the RPL program. The following figures

show the RPL> command with the program before and after pressing the `

key.

To store the program use the STO command as follows:

„îK~p2`

An evaluation of program P2 for the argument X = 5 is shown in the next

screen:

While you can write programs in algebraic mode, without using the function

RPL>, some of the RPL constructs will produce an error message when you press

`, for example:

Page 21-67

Whereas, using RPL, there is no problem when loading this program in

algebraic mode:

Page 21-68

Chapter 22

Programs for graphics manipulation

This chapter includes a number of examples showing how to use the calculator’s

functions for manipulating graphics interactively or through the use of programs.

As in Chapter 21 we recommend using RPN mode and setting system flag 117

to SOFT menu labels. « »

We introduce a variety of calculator graphic applications in Chapter 12. The

examples of Chapter 12 represent interactive production of graphics using the

calculator’s pre-programmed input forms. It is also possible to use graphs in

your programs, for example, to complement numerical results with graphics. To

accomplish such tasks, we first introduce function in the PLOT menu.

The PLOT menu

Commands for setting up and producing plots are available through the PLOT

menu. You can access the PLOT menu by using: 81.01

„°L@MODES @MENU@ @@MENU@.

The menu thus produced provides the user access to a variety of graphics

functions. For application in subsequent examples, let’s user-define the C

(GRAPH) key to provide access to this menu as described below.

User-defined key for the PLOT menu

Enter the following keystrokes to determine whether you have any user-defined

keys already stored in your calculator: „°L@MODES @@KEYS@ @@RCLKE@.

Unless you have user-defined some keys, you should get in return a list

containing an S, i.e., {S}. This indicates that the Standard keyboard is the only

key definition stored in your calculator.

Page 22-1

To user-define a key you need to add to this list a command or program

followed by a reference to the key (see details in Chapter 20). Type the list

{ S << 81.01 MENU >> 13.0 } in the stack and use function STOKEYS

(„°L@MODES @@KEYS@ @@STOK@) to user-define key C as the access to the

PLOT menu. Verify that such list was stored in the calculator by using

„°L@MODES @@KEYS@ @@RCLK@.

Note: We will not work any exercise while presenting the PLOT menu, its

functions or sub-menus. This section will be more like a tour of the contents of

PLOT as they relate to the different type of graphs available in the calculator.

To activate a user defined key you need to press „Ì (same as the ~

key) before pressing the key or keystroke combination of interest. To activate the

PLOT menu, with the key definition used above, press: „Ì C. You will

get the following menu (press L to move to second menu)

Description of the PLOT menu

The following diagram shows the menus in PLOT. The number accompanying

the different menus and functions in the diagram are used as reference in the

subsequent description of those objects.

The soft menu key labeled 3D, STAT, FLAG, PTYPE, and PPAR, produce

additional menus, which will be presented in more detail later. At this point we

describe the functions directly accessible through soft menu keys for menu

number 81.02. These are:

Page 22-2

LABEL (10)

The function LABEL is used to label the axes in a plot including the variable

names and minimum and maximum values of the axes. The variable names are

selected from information contained in the variable PPAR.

AUTO (11)

The function AUTO (AUTOscale) calculates a display range for the y-axis or for

both the x- and y-axes in two-dimensional plots according to the type of plot

defined in PPAR. For any of the three-dimensional graphs the function AUTO

produces no action. For two-dimensional plots, the following actions are

performed by AUTO:

ꢀ FUNCTION: based on the plotting range of x, it samples the function in EQ

and determines the minimum and maximum values of y.

ꢀ CONIC: sets the y-axis scale equal to the x-axis scale

ꢀ POLAR: based on the values of the independent variable (typically θ), it

samples the function in EQ and determines minimum and maximum values

of both x and y.

ꢀ PARAMETRIC: produces a similar result as POLAR based on the values of

the parameter defining the equations for x and y.

ꢀ TRUTH: produces no action.

ꢀ BAR: the x-axis range is set from 0 to n+1 where n is the number of

elements in ΣDAT. The range of values of y is based on the contents of

ΣDAT. The minimum and maximum values of y are determined so that the

x-axis is always included in the graph.

ꢀ HISTOGRAM: similar to BAR.

ꢀ SCATTER: sets x- and y-axis range based on the contents of the

independent and dependent variables from ΣDAT.

INFO (12)

The function INFO is interactive only (i.e., it cannot be programmed). When

the corresponding soft menu key is pressed it provides information about the

current plot parameters.

Page 22-3

EQ (3)

The variable name EQ is reserved by the calculator to store the current equation

in plots or solution to equations (see chapter …). The soft menu key labeled EQ

in this menu can be used as it would be if you have your variable menu

available, e.g., if you press [ EQ ] it will list the current contents of that variable.

ERASE (4)

The function ERASE erases the current contents of the graphics window. In

programming, it can be used to ensure that the graphics window is cleared

before plotting a new graph.

DRAX (5)

The function DRAX draws the axes in the current plot, if any is visible.

DRAW (6)

The function DRAW draws the plot defined in PPAR.

The PTYPE menu under PLOT (1)

The PTYPE menu lists the name of all two-dimensional plot types pre-

programmed in the calculator. The menu contains the following menu keys:

These keys correspond to the plot types Function, Conic, Polar, Parametric, Truth,

and Diff Eq, presented earlier. Pressing one of these soft menu keys, while

typing a program, will place the corresponding function call in the program.

Press L @PLOT to get back to the main PLOT menu.

The PPAR menu (2)

The PPAR menu lists the different options for the PPAR variable as given by the

following soft menu key labels. Press L to move to next menus:

Page 22-4

Note: the SCALE commands shown here actually represent SCALE, SCALEW,

SCALEH, in that order.

The following diagram illustrates the functions available in the PPAR menu. The

letters attached to each function in the diagram are used for reference purposes

in the description of the functions shown below.

INFO (n) and PPAR (m)

If you press @INFO, or enter ‚ @PPAR, while in this menu, you will get a listing of

the current PPAR settings, for example:

This information indicates that X is the independent variable (Indep), Y is the

dependent variable (Depnd), the x-axis range goes from –6.5 to 6.5 (Xrng), the

y-axis range goes from –3.1 to 3.2 (Yrng). The last piece of information in the

screen, the value of Res (resolution) determines the interval of the independent

variable used for generating the plot.

The soft menu key labels included in the PPAR(2) menu represent commands

that can be used in programs. These commands include:

Page 22-5

INDEP (a)

The command INDEP specifies the independent variable and its plotting range.

These specifications are stored as the third parameter in the variable PPAR. The

default value is 'X'. The values that can be assigned to the independent

variable specification are:

ꢀ A variable name, e.g., 'Vel'

ꢀ A variable name in a list, e.g., { Vel}

ꢀ A variable name and a range in a list, e.g., { Vel 0 20}

ꢀ A range without a variable name, e.g., { 0 20}

ꢀ Two values representing a range, e.g., 0 20

In a program, any of these specifications will be followed by the command

INDEP.

DEPND (b)

The command DEPND specifies the name of the dependent variable. For the

case of TRUTH plots it also specifies the plotting range. The default value is Y.

The type of specifications for the DEPND variable are the same as those for the

INDEP variable.

XRNG (c) and YRNG (d)

The command XRNG specifies the plotting range for the x-axis, while the

command YRNG specifies the plotting range for the y-axis. The input for any of

these commands is two numbers representing the minimum and maximum

values of x or y. The values of the x- and y-axis ranges are stored as the

ordered pairs (x , y ) and (x , y ) in the two first elements of the

min min

max max

variable PPAR. Default values for x

and x

are -6.5 and 6.5, respectively.

min

max

Default values for x

and x

are –3.1 and 3.2, respectively.

min

max

RES (e)

The RES (RESolution) command specifies the interval between values of the

independent variable when producing a specific plot. The resolution can be

expressed in terms of user units as a real number, or in terms of pixels as a

binary integer (numbers starting with #, e.g., #10). The resolution is stored as

the fourth item in the PPAR variable.

Page 22-6

CENTR (g)

The command CENTR takes as argument an ordered pair (x,y) or a value x,

and adjusts the first two elements in the variable PPAR, i.e., (x , y ) and

min min

(x , y ), so that the center of the plot is (x,y) or (x,0), respectively.

max max

SCALE (h)

The SCALE command determines the plotting scale represented by the number

of user units per tick mark. The default scale is 1 user-unit per tick mark. When

the command SCALE is used, it takes as arguments two numbers, x

and

scale

, representing the new horizontal and vertical scales. The effect of the

scale

SCALE command is to adjust the parameters (x , y ) and (x , y ) in

min min

max max

PPAR to accommodate the desired scale. The center of the plot is preserved.

SCALEW (i)

Given a factor x

, the command SCALEW multiplies the horizontal scale by

factor

that factor. The W in SCALEW stands for 'width.' The execution of SCALEW

changes the values of x

and x

in PPAR.

min

max

SCALEH (j)

Given a factor y

, the command SCALEH multiplies the vertical scale by that

factor

factor. The H in SCALEH stands for 'height.' The execution of SCALEW

changes the values of y and y in PPAR.

min

max

Note: Changes introduced by using SCALE, SCALEW, or SCALEH, can be

used to zoom in or zoom out in a plot.

ATICK (l)

The command ATICK (Axes TICK mark) is used to set the tick-mark annotations

for the axes. The input value for the ATICK command can be one of the

following:

ꢀ A real value x : sets both the x- and y-axis tick annotations to x units

ꢀ A list of two real values { x y }: sets the tick annotations in the x- and y-axes

to x and y units, respectively.

ꢀ A binary integer #n: sets both the x- and y-axis tick annotations to #n pixels

Page 22-7

A list of two binary integers {#n #m}: sets the tick annotations in the x- and y-

axes to #n and #m pixels, respectively.

AXES (k)

The input value for the axes command consists of either an ordered pair (x,y) or

a list {(x,y) atick "x-axis label" "y-axis label"}. The parameter atick stands for

the specification of the tick marking annotations as described above for the

command ATICK. The ordered pair represents the center of the plot. If only an

ordered pair is given as input to AXES, only the axes origin is altered. The

argument to the command AXES, whether an ordered pair or a list of values, is

stored as the fifth parameter in PPAR.

To return to the PLOT menu, press @PLOT.

Press L to reach the second menu of the PLOT menu set.

RESET (f)

This button will reset the plot parameters to default values.

The 3D menu within PLOT (7)

The 3D menu contains two sub-menus, PTYPE and VPAR, and one variable, EQ.

We are familiar already with the meaning of EQ, therefore, we will concentrate

on the contents of the PTYPE and VPAR menus. The diagram below shows the

branching of the 3D menu.

Page 22-8

The PTYPE menu within 3D (IV)

The PTYPE menu under 3D contains the following functions:

These functions correspond to the graphics options Slopefield, Wireframe, Y-

Slice, Ps-Contour, Gridmap and Pr-Surface presented earlier in this chapter.

Pressing one of these soft menu keys, while typing a program, will place the

corresponding function call in the program. Press L@@3D@@ to get back to the

main 3D menu.

The VPAR menu within 3D (V)

The variable VPAR stands for Volume PARameters, referring to a parallelepiped

in space within which the three-dimensional graph of interest is constructed.

When press [VPAR] in the 3D menu, you will get the following functions. Press

L to move to the next menu:

Next, we describe the meaning of these functions:

INFO (S) and VPAR (W)

When you press @INFO (S) you get the information shown in the left-hand side

screen shot above. The ranges in Xvol, Yvol, and Zvol describe the extent of the

parallelepiped in space where the graph will be generated. Xrng and Yrng

describe the range of values of x and y, respectively, as independent variables

in the x-y plane that will be used to generate functions of the form z = f(x,y).

Press L and @INFO (Y) to obtain the information in the right-hand side screen

shot above. These are the value of the location of the viewpoint for the three-

dimensional graph (Xeye, Yeye, Zeye), and of the number of steps in x and y to

generate a grid for surface plots.

Page 22-9

XVOL (N), YVOL (O), and ZVOL (P)

These functions take as input a minimum and maximum value and are used to

specify the extent of the parallelepiped where the graph will be generated (the

viewing parallelepiped). These values are stored in the variable VPAR. The

default values for the ranges XVOL, YVOL, and ZVOL are –1 to 1.

XXRNG (Q) and YYRNG (R)

These functions take as input a minimum and maximum value and are used to

specify the ranges of the variables x and y to generate functions z = f(x,y). The

default value of the ranges XXRNG and YYRNG will be the same as those of

XVOL and YVOL.

EYEPT (T)

The function EYEPT takes as input real values x, y, and z representing the

location of the viewpoint for a three-dimensional graph. The viewpoint is a

point in space from which the three-dimensional graph is observed. Changing

the viewpoint will produce different views of the graph. The figure below

illustrates the idea of the viewpoint with respect to the actual graphic space and

its projection in the plane of the screen.

NUMX(U) and NUMY (V)

The functions NUMX and NUMY are used to specify the number of points or

steps along each direction to be used in the generation of the base grid from

which to obtain values of z = f(x,y).

VPAR (W)

This is just a reference to the variable VPAR.

RESET (X)

Resets parameters in screen to their default values.

Press L@@3D@@ to return to the 3D menu.

Press @PLOT to return to the PLOT menu.

Page 22-10

The STAT menu within PLOT

The STAT menu provides access to plots related to statistical analysis. Within

this menu we find the following menus:

The diagram below shows the branching of the STAT menu within PLOT. The

numbers and letters accompanying each function or menu are used for

reference in the descriptions that follow the figure.

Page 22-11

The PTYPE menu within STAT (I)

The PTYPE menu provides the following functions:

These keys correspond to the plot types Bar (A), Histogram (B), and Scatter(C),

presented earlier. Pressing one of these soft menu keys, while typing a

program, will place the corresponding function call in the program. Press @STAT

to get back to the STAT menu.

The DATA menu within STAT (II)

The DATA menu provides the following functions:

The functions listed in this menu are used to manipulate the ΣDAT statistical

matrix. The functions Σ+ (D) and Σ- (E), add or remove data rows from the

matrix ΣDAT. CLΣ (F) clears the ΣDAT (G) matrix, and the soft menu key

labeled ΣDAT is just used as a reference for interactive applications. More

details on the use of these functions are presented in a later chapter on

statistical applications. Press @STAT to return to the STAT menu.

The ΣPAR menu within STAT (III)

The ΣPAR menu provides the following functions:

INFO (M) and ΣPAR (K)

The key INFO in ΣPAR provides the information shown in the screen shot above.

The information listed in the screen is contained in the variable ΣPAR. The

values shown are the default values for the x-column, y-column, intercept and

slope of a data fitting model, and the type of model to be fit to the data in

ΣDAT.

Page 22-12

XCOL (H)

The command XCOL is used to indicate which of the columns of ΣDAT, if more

than one, will be the x- column or independent variable column.

YCOL (I)

The command YCOL is used to indicate which of the columns of ΣDAT, if more

than one, will be the y- column or dependent variable column.

MODL (J)

The command MODL refers to the model to be selected to fit the data in ΣDAT,

if a data fitting is implemented. To see which options are available, press @!MODL.

You will get the following menu:

These functions correspond to Linear Fit, Logarithmic Fit, Exponential Fit, Power

Fit, or Best Fit. Data fitting is described in more detail in a later chapter. Press

£@PAR to return to the ΣPAR menu.

ΣPAR (K)

ΣPAR is just a reference to the variable ΣPAR for interactive use.

RESET (L)

This function resets the contents of ΣPAR to its default values.

Press L@STAT to return to the STAT menu. Press [PLOT] to return to the main

PLOT menu.

The FLAG menu within PLOT

The FLAG menu is actually interactive, so that you can select any of the

following options:

ꢀ AXES: when selected, axes are shown if visible within the plot area or

volume.

ꢀ CNCT: when selected the plot is produced so that individual points are

connected.

Page 22-13

ꢀ SIMU: when selected, and if more than one graph is to be plotted in the

same set of axes, plots all the graphs simultaneously.

Press @PLOT to return to the PLOT menu.

Generating plots with programs

Depending on whether we are dealing with a two-dimensional graph defined

by a function, by data from ΣDAT, or by a three-dimensional function, you need

to set up the variables PPAR, ΣPAR, and /or VPAR before generating a plot in a

program. The commands shown in the previous section help you in setting up

such variables.

Following we describe the general format for the variables necessary to

produce the different types of plots available in the calculator.

Two-dimensional graphics

The two-dimensional graphics generated by functions, namely, Function, Conic,

Parametric, Polar, Truth and Differential Equation, use PPAR with the format:

{ (x , y ) (x , y ) indep res axes ptype depend }

min

max

The two-dimensional graphics generated from data in the statistical matrix

ΣDAT, namely, Bar, Histogram, and Scatter, use the ΣPAR variable with the

following format:

{ x-column y-column slope intercept model}

while at the same time using PPAR with the format shown above.

The meaning of the different parameters in PPAR and ΣPAR were presented in

the previous section.

Page 22-14

Three-dimensional graphics

The three-dimensional graphics available, namely, options Slopefield,

Wireframe, Y-Slice, Ps-Contour, Gridmap and Pr-Surface, use the VPAR variable

with the following format:

{x , x

, y , y , z , z , x , x , y , y ,

near far low high min max min max

left

right

x , y , z , x , y

}

eye

step

These pairs of values of x, y, and z, represent the following:

ꢀ Dimensions of the view parallelepiped (x , x

, y , y ,

near far

left

right

z , z

)

low

high

ꢀ Range of x and y independent variables (x , x , y , y )

max

min

max

min

ꢀ Location of viewpoint (x , y , z )

eye

ꢀ Number of steps in the x- and y-directions (x , y

)

step

Three-dimensional graphics also require the PPAR variable with the parameters

shown above.

The variable EQ

All plots, except those based on ΣDAT, also require that you define the function

or functions to be plotted by storing the expressions or references to those

functions in the variable EQ.

In summary, to produce a plot in a program you need to load EQ, if required.

Then load PPAR, PPAR and ΣPAR, or PPAR and VPAR. Finally, use the name of

the proper plot type: FUNCTION, CONIC, POLAR, PARAMETRIC, TRUTH,

DIFFEQ, BAR, HISTOGRAM, SCATTER, SLOPE, WIREFRAME, YSLICE,

PCONTOUR, GRIDMAP, or PARSURFACE, to produce your plot.

Examples of interactive plots using the PLOT menu

To better understand the way a program works with the PLOT commands and

variables, try the following examples of interactive plots using the PLOT menu.

Example 1 – A function plot:

„ÌC

Get PLOT menu (*)

@PTYPE @FUNCT

Select FUNCTION as the plot type

Store function ‘√r’ into EQ

‘√r’ `„ @@EQ@@

Page 22-15

@PPAR

Show plot parameters

~„r` @INDEP

~„s` @DEPND

1 \# 10 @XRNG

1 \# 5 @YRNG L

{ (0,0) {.4 .2} “Rs” “Sr”} `

@AXES

Define ‘r’ as the indep. variable

Define ‘s’ as the dependent variable

Define (-1, 10) as the x-range

Define (-1, 5) as the y-range

Axes definition list

Define axes center, ticks, labels

Return to PLOT menu

L @PLOT

@ERASE @DRAX L@LABEL

L @DRAW

@EDIT L@MENU

Erase picture, draw axes, labels

Draw function and show picture

Removes menu labels

LL@PICT @CANCL

Returns to normal calculator display

(*) PLOT menu available through user-defined key C as shown earlier in this

Chapter.

Example 2 – A parametric plot (Use RAD as angles):

„ÌC

Get PLOT menu

@PTYPE @PARAM

Select PARAMETRIC as the plot type

Define complex function X+iY

Store complex function into EQ

Show plot parameters

{ ‘SIN(t)+i*SIN(2*t)’ } `

„ @@EQ@@

@PPAR

{t 0 6.29} ` @INDEP

~y` @DEPND

2.2 \# 2.2 @XRNG

1.1 \# 1.1 @YRNG L

{ (0,0) {.4 .2} “X(t)” “Y(t)”} `

@AXES

Define ‘t’ as the indep.variable

Define ‘Y’ as the dependent variable

Define (-2.2,2.2) as the x-range

Define (-1.1,1.1) as the y-range

Axes definition list

Define axes center, ticks, labels

Return to PLOT menu

L @PLOT

@ERASE @DRAX L@LABEL

L @DRAW

@EDIT L@MENU LL@PICT @CANCL

Erase picture, draw axes, labels

Draw function and show picture

Finish plot

Example 3 – A polar plot:

„ÌC

Get PLOT menu

Select POLAR as the plot type

@PTYPE @POLAR

‘1+SIN(θ)’ ` „ @@EQ@@

Store complex funct. r = f(θ) into EQ

Page 22-16

@PPAR

Show plot parameters

{ θ 0 6.29} ` @INDEP

~y` @DEPND

3 \# 3 @XRNG

0.5 \# 2.5 @YRNG L

{ (0,0) {.5 .5} “x” “y”} `

@AXES

Define ‘θ’ as the indep. Variable

Define ‘Y’ as the dependent variable

Define (-3,3) as the x-range

Define (-0.5,2.5) as the y-range

Axes definition list

Define axes center, ticks, labels

Return to PLOT menu

L @PLOT

@ERASE @DRAX L@LABEL

L @DRAW

@EDIT L@MENU

Erase picture, draw axes, labels

Draw function and show picture

Remove menu labels

LL@PICT @CANCL

Return to normal calculator display

From these examples we see a pattern for the interactive generation of a two-

dimensional graph through the PLOT menu:

1 – Select PTYPE.

2 – Store function to plot in variable EQ (using the proper format, e.g.,

‘X(t)+iY(t)’ for PARAMETRIC).

3 – Enter name (and range, if necessary) of independent and dependent

variables

4 – Enter axes specifications as a list { center atick x-label y-label }

5 – Use ERASE, DRAX, LABEL, DRAW to produce a fully labeled graph with

axes

This same approach can be used to produce plots with a program, except that

in a program you need to add the command PICTURE after the DRAW function

is called to recall the graphics screen to the stack.

Examples of program-generated plots

In this section we show how to implement with programs the generation of the

last three examples. Activate the PLOT menu before you start typing the

program to facilitate entering graphing commands („ÌC, see above).

Example 1 – A function plot. Enter the following program:

Page 22-17

Start program

Purge current PPAR and EQ

Store ‘√r’ into EQ

Set independent variable to ‘r’

Set dependent variable to ‘s’

Select FUNCTION as the plot type

{PPAR EQ} PURGE

‘√r’ STEQ

‘r’ INDEP

‘s’ DEPND

FUNCTION

{ (0.,0.) {.4 .2}

“Rs” “Sr” } AXES

–1. 5. XRNG

–1. 5. YRNG

ERASE DRAW DRAX LABEL

PICTURE »

Set axes information

Set x range

Set y range

Erase & draw plot, axes, and labels

Recall graphics screen to stack

Store the program in variable PLOT1. To run it, press J, if needed, then

press @PLOT1.

Example 2 – A parametric plot. Enter the following program:

Start program

Change to radians, purge vars.

Store ‘X(t)+iY(t)’ into EQ

Set indep. variable to ‘r’, with range

Set dependent variable to ‘Y’

Select PARAMETRIC as the plot type

RAD {PPAR EQ} PURGE

‘SIN(t)+i*SIN(2*t)’ STEQ

{ t 0. 6.29} INDEP

‘Y’ DEPND

PARAMETRIC

{ (0.,0.) {.5 .5} “X(t)”

“Y(t)” } AXES

–2.2 2.2 XRNG

–1.1 1.1 YRNG

ERASE DRAW DRAX LABEL

PICTURE

Set axes information

Set x range

Set y range

Erase & draw plot, axes, and labels

Recall graphics screen to stack

End program

Store the program in variable PLOT2. To run it, press J, if needed, then

press @PLOT2.

Page 22-18

Example 3 – A polar plot. Enter the following program:

Start program

Change to radians, purge vars.

Store ‘f(θ)’ into EQ

Set indep. variable to ‘θ’, with range

Set dependent variable to ‘Y’

Select POLAR as the plot type

RAD {PPAR EQ} PURGE

‘1+SIN(θ)’ STEQ

{ θ 0. 6.29} INDEP

‘Y’ DEPND

POLAR

{ (0.,0.) {.5 .5}

“x” “y”} AXES

–3. 3. XRNG

–.5 2.5 YRNG

ERASE DRAW DRAX LABEL

PICTURE

Set axes information

Set x range

Set y range

Erase & draw plot, axes, and labels

Recall graphics screen to stack

End program

Store the program in variable PLOT3. To run it, press J, if needed, then

press @PLOT3.

These exercises illustrate the use of PLOT commands in programs. They just

scratch the surface of programming applications of plots. I invite the reader to

try their own exercises on programming plots.

Drawing commands for use in programming

You can draw figures in the graphics window directly from a program by using

commands such as those contained in the PICT menu, accessible by

„°L@PICT@. The functions available in this menu are the following.

Press L to move to next menu:

Obviously, the commands LINE, TLINE, and BOX, perform the same operations

as their interactive counterpart, given the appropriate input. These and the

other functions in the PICT menu refer to the graphics windows whose x- and y-

ranges are determined in the variable PPAR, as demonstrated above for

different graph types. The functions in the PICT command are described next:

Page 22-19

PICT

This soft key refers to a variable called PICT that stores the current contents of

the graphics window. This variable name, however, cannot be placed within

quotes, and can only store graphics objects. In that sense, PICT is like no other

calculator variables.

PDIM

The function PDIM takes as input either two ordered pairs (x ,y ) (x ,y

)

min min

max max

or two binary integers #w and #h. The effect of PDIM is to replace the current

contents of PICT with an empty screen. When the argument is (x ,y

)

min min

(x ,y ), these values become the range of the user-defined coordinates in

max max

PPAR. When the argument is #w and #h, the ranges of the user-defined

coordinates in PPAR remain unchanged, but the size of the graph changes to #h

× #v pixels.

PICT and the graphics screen

PICT, the storage area for the current graph, can be thought of as a two

dimensional graph with a minimum size of 131 pixels wide by 64 pixels high.

The maximum width of PICT is 2048 pixels, with no restriction on the maximum

height. A pixel is each one of the dots in the calculator’s screen that can be

turned on (dark) or off (clear) to produce text or graphs. The calculator screen

has 131 pixels by 64 pixels, i.e., the minimum size for PICT. If your PICT is

larger than the screen, then the PICT graph can be thought of as a two

dimensional domain that can be scrolled through the calculator’s screen, as

illustrated in the diagram shown next.

LINE

This command takes as input two ordered pairs (x ,y ) (x , y ), or two pairs of

pixel coordinates {#n #m } {#n #m }. It draws the line between those

coordinates.

TLINE

This command (Toggle LINE) takes as input two ordered pairs (x ,y ) (x , y ),

or two pairs of pixel coordinates {#n #m } {#n #m }. It draws the line

between those coordinates, turning off pixels that are on in the line path and

vice versa.

Page 22-20

BOX

This command takes as input two ordered pairs (x ,y ) (x , y ), or two pairs of

pixel coordinates {#n #m } {#n #m }. It draws the box whose diagonals are

represented by the two pairs of coordinates in the input.

ARC

This command is used to draw an arc. ARC takes as input the following

objects:

ꢀ Coordinates of the center of the arc as (x,y) in user coordinates or {#n, #m}

in pixels.

ꢀ Radius of arc as r (user coordinates) or #k (pixels).

ꢀ Initial angle θ and final angle θ .

PIX?, PIXON, and PIXOFF

These functions take as input the coordinates of point in user coordinates, (x,y),

or in pixels {#n, #m}.

Page 22-21

ꢀ PIX? Checks if pixel at location (x,y) or {#n, #m} is on.

ꢀ PIXOFF turns off pixel at location (x,y) or {#n, #m}.

ꢀ PIXON turns on pixel at location (x,y) or {#n, #m}.

PVIEW

This command takes as input the coordinates of a point as user coordinates

(x,y) or pixels {#n, #m}, and places the contents of PICT with the upper left

corner at the location of the point specified. You can also use an empty list as

argument, in which case the picture is centered in the screen. PVIEW does not

activate the graphics cursor or the picture menu. To activate any of those

features use PICTURE.

PXꢀC

The function PXꢀC converts pixel coordinates {#n #m} to user-unit coordinates

(x,y).

CꢀPX

The function CꢀPX converts user-unit coordinates (x,y) to pixel coordinates {#n

#m}.

Programming examples using drawing functions

In this section we use the commands described above to produce graphics with

programs. Program listing are provided in the attached diskette or CD ROM.

Example 1 - A program that uses drawing commands

The following program produces a drawing in the graphics screen. (This

program has no other purpose than to show how to use calculator commands to

produce drawings in the display.)

Start program

Select degrees for angular measures

Set x range

Set y range

Erase picture

Draw box from (5,5) to (95,95)

DEG

0. 100. XRNG

0. 50. YRNG

ERASE

(5., 2.5) (95., 47.5) BOX

(50., 50.) 10. 0. 360. ARC

Draw a circle center (50,50), r =10.

Page 22-22

(50., 50.) 12. –180. 180. ARC

1 8 FOR j

Draw a circle center (50,50), r= 12.

Draw 8 lines within the circle

Lines are centered as (50,50)

Calculate x, other end at 50 + x

Calculates y, other end at 50 + y

Convert x y to (x,y), complex num.

Add (50,50) to (x,y)

(50., 50.) DUP

‘12*COS(45*(j-1))’ ꢀNUM

‘12*SIN(45*(j-1))’ ꢀNUM

R ꢀ C

LINE

Draw the line

{ } PVIEW

End of FOR loop

Show picture

Example 2 - A program to plot a natural river cross-section

This application may be useful for determining area and wetted perimeters of

natural river cross-sections. Typically, a natural river cross section is surveyed

and a series of points, representing coordinates x and y with respect to an

arbitrary set of coordinates axes. These points can be plotted and a sketch of

the cross section produced for a given water surface elevation. The figure

below illustrate the terms presented in this paragraph.

The program, available in the diskette or CD ROM that comes with your

calculator, utilizes four sub-programs FRAME, DXBED, GTIFS, and INTRP. The

main program, called XSECT, takes as input a matrix of values of x and y, and

the elevation of the water surface Y (see figure above), in that order. The

program produces a graph of the cross section indicating the input data with

points in the graph, and shows the free surface in the cross-section.

Page 22-23

It is suggested that you create a separate sub-directory to store the programs.

You could call the sub-directory RIVER, since we are dealing with irregular open

channel cross-sections, typical of rivers.

To see the program XSECT in action, use the following data sets. Enter them as

matrices of two columns, the first column being x and the second one y. Store

the matrices in variables with names such as XYD1 (X-Y Data set 1) and XYD2

(X-Y Data set 2). To run the program place one of the data sets in the stack,

e.g., J@XYD1!, then type in a water surface elevation, say 4.0, and press

@XSECT. The calculator will show an sketch of the cross-section with the

corresponding water surface. To exit the graph display, press $.

Try the following examples:

@XYD1! 2 @XSECT

@XYD1! 3 @XSECT

@XYD1! 4 @XSECT

@XYD1! 6 @XSECT

Please be patient when running program XSECT. Due to the relatively large

number of graphics functions used, not counting the numerical iterations, it may

take some time to produce the graph (about 1 minute).

Page 22-24

Data set 1

Data set 2

0.4

1.0

2.0

3.4

4.0

5.8

7.2

7.8

9.0

6.3

4.9

4.3

3.0

1.2

2.0

3.8

5.3

7.2

0.7

1.0

1.5

2.2

3.5

4.5

5.0

6.0

7.1

8.0

9.0

10.0

10.5

11.0

4.8

3.0

2.0

0.9

0.4

1.0

2.0

2.5

2.0

0.7

0.0

1.5

3.4

5.0

Note: The program FRAME, as originally programmed (see diskette or CD

ROM), does not maintain the proper scaling of the graph. If you want to

maintain proper scaling, replace FRAME with the following program:

« STOΣ MINΣ MAXΣ 2 COLꢀ DUP ꢀCOL DROP – AXL ABS AXL 20 /

DUP NEG SWAP 2 COLꢀ + ꢀROW DROP SWAP ꢀ yR xR « 131 DUP

RꢀB SWAP yR OBJꢀ DROP – xR OBJꢀ DROP - / * FLOOR RꢀB

PDIM yR OBJꢀ DROP YRNG xR OBJꢀ DROP XRNG ERASE » »

This program keeps the width of the PICT variable at 131 pixels – the minimum

pixel size for the horizontal axis – and adjusts the number of pixels in the

vertical axes so that a 1:1 scale is maintained between the vertical and

horizontal axes.

Pixel coordinates

The figure below shows the graphic coordinates for the typical (minimum)

screen of 131×64 pixels. Pixels coordinates are measured from the top left

corner of the screen {# 0h # 0h}, which corresponds to user-defined coordinates

Page 22-25

(x , y ). The maximum coordinates in terms of pixels correspond to the

min max

lower right corner of the screen {# 82h #3Fh}, which in user-coordinates is the

point (x , y ). The coordinates of the two other corners both in pixel as well

max min

as in user-defined coordinates are shown in the figure.

Animating graphics

Herein we present a way to produce animation by using the Y-Slice plot type.

Suppose that you want to animate the traveling wave, f(X,Y) = 2.5 sin(X-Y). We

can treat the X as time in the animation producing plots of f(X,Y) vs. Y for

different values of X. To produce this graph use the following:

ꢀ „ô simultaneously. Select Y-Slice for TYPE. ‘2.5*SIN(X-Y)’ for

EQ. ‘X’ for INDEP. Press L@@@OK@@@.

ꢀ „ò, simultaneously (in RPN mode). Use the following values:

ꢀ Press @ERASE @DRAW. Allow some time for the calculator to generate all the

needed graphics. When ready, it will show a traveling sinusoidal

wave in your screen.

Page 22-26

Animating a collection of graphics

The calculator provides the function ANIMATE to animate a number of graphics

that have been placed in the stack. You can generate a graph in the graphics

screen by using the commands in the PLOT and PICT menus. To place the

generated graph in the stack, use PICT RCL. When you have n graphs in levels

n through 1 of the stack, you can simply use the command n ANIMATE to

produce an animation made of the graphs you placed in the stack.

Example 1 – Animating a ripple in a water surface

As an example, type in the following program that generates 11 graphics

showing a circle centered in the middle of the graphics screen and whose

radius increase by a constant value in each subsequent graph.

Begin program

RAD

Set angle units to radians

Set PICT to 131×64 pixels

Set x- and y-ranges to 0-100

Start loop with j = 1 .. 11

Erase current PICT

131 RꢀB 64 RꢀB PDIM

0 100 XRNG 0 100 YRNG

1 11 FOR j

ERASE

(50., 50.) ‘5*(j-1)’ ꢀNUM

0 ‘2*π’ ꢀNUM ARC

PICT RCL

11 ANIMATE

Centers of circles (50,50)

Draw circle center r = 5(j-1)

Place current PICT on stack

End FOR-NEXT loop

Animate

End program

Store this program in a variable called PANIM (Plot ANIMation). To run the

program press J (if needed) @PANIM. It takes the calculator more than one

minute to generate the graphs and get the animation going. Therefore, be

really patient here. You will see the hourglass symbol up in the screen for what

seems a long time before the animation, resembling the ripples produced by a

pebble dropped on the surface of a body of quiescent water, appears in the

screen. To stop the animation, press $.

The 11 graphics generated by the program are still available in the stack. If

you want to re-start the animation, simply use: 11 ANIMATE. (Function

Page 22-27

ANIMATE is available by using „°L@GROB L @ANIMA). The animation

will be re-started. Press $ to stop the animation once more. Notice that the

number 11 will still be listed in stack level 1. Press ƒ to drop it from the

stack.

Suppose that you want to keep the figures that compose this animation in a

variable. You can create a list of these figures, let’s call it WLIST, by using:

11 „°@TYPE@ @ ꢀLIST ³ ~~wlist~ K

Press J to recover your list of variables. The variable @WLIST should now be

listed in your soft-menu keys. To re-animate this list of variables you could use

the following program:

Start program

WLIST

OBJꢀ

ANIMATE

Place list WLIST in stack

Decompose list, stack level 1 = 11

Start animation

End program

Save this program in a variable called RANIM (Re-ANIMate). To run it, press

@RANIM.

The following program will animate the graphics in WLIST forward and

backwards:

Start program

WLIST DUP

REVLIST +

OBJꢀ

ANIMATE

Place list WLIST in stack, make extra copy

Reverse order, concatenate 2 lists

Decompose list in elements, level 1 = 22

Start animation

End program

Save this program in a variable called RANI2 (Re-ANImate version 2). To run

it, press @RANI2. The animation now simulates a ripple in the surface of

otherwise quiescent water that gets reflected from the walls of a circular tank

back towards the center. Press $ to stop the animation.

Page 22-28

Example 2 - Animating the plotting of different power functions

Suppose that you want to animate the plotting of the functions f(x) = x , n = 0,

1, 2, 3, 4, in the same set of axes. You could use the following program:

Begin program

RAD

Set angle units to radians

Set PICT screen to 131×64 pixels

Set x- and y-ranges

131 RꢀB 64 RꢀB PDIM

0 2 XRNG 0 20 YRNG

0 4 FOR j

Start loop with j = 0,1,…,4

Store ‘X^j’ in variable EQ

Erase current PICT

‘X^j’ STEQ

ERASE

DRAX LABEL DRAW

PICT RCL

Draw axes, labels, function

Place current PICT on stack

End FOR-NEXT loop

5 ANIMATE

Animate

Store this program in a variable called PWAN (PoWer function ANimation).

To run the program press J (if needed) @PWAN. You will see the calculator

drawing each individual power function before starting the animation in which

the five functions will be plotted quickly one after the other. To stop the

animation, press $.

More information on the ANIMATE function

The ANIMATE function as used in the two previous examples utilized as input

the graphics to be animated and their number. You can use additional

information to produce the animation, such as the time interval between

graphics and the number of repetitions of the graphics. The general format of

the ANIMATE function in such cases is the following:

n-graphs { n {#X #Y} delay rep } ANIMATE

nrepresents the number of graphics, {#X #Y}stand for the pixel coordinates

of the lower right corner of the area to be plotted (see figure below), delayis

the number of seconds allowed between consecutive graphics in the animation,

and repis the number of repetitions of the animation.

Graphic objects (GROBs)

The word GROB stands for GRaphics OBjects and is used in the calculator’s

environment to represent a pixel-by-pixel description of an image that has been

Page 22-29

produced in the calculator’s screen. Therefore, when an image is converted

into a GROB, it becomes a sequence of binary digits (binary digits = bits), i.e.,

0’s and 1’s. To illustrate GROBs and conversion of images to GROBS consider

the following exercise.

When we produce a graph in the calculator, the graph become the contents of

a special variable called PICT. Thus, to see the last contents of PICT, you could

The display shows in stack level 1 the line Graphic 131×80(if using the

standard screen size) followed by a sketch of the top part of the graph. For

example,

If you press ˜ then the graph contained in level 1 is shown in the calculator’s

graphics display. Press @CANCL to return to normal calculator display.

The graph in level 1 is still not in GROB format, although it is, by definition, a

graphics object. To convert a graph in the stack into a GROB, use: 3`

„°L@GROB @ꢀGROB . Now we have the following information in level 1:

The first part of the description is similar to what we had originally, namely,

Graphic 131×64, but now it is expressed as Graphic 13128 × 8.

However, the graphic display is now replaced by a sequence of zeroes and

ones representing the pixels of the original graph. Thus, the original graph as

now been converted to its equivalent representation in bits.

You can also convert equations into GROBs. For example, using the equation

writer type in the equation ‘X^2+3’ into stack level 1, and then press

Page 22-30

1` „°L@GROB @ꢀGROB . You will now have in level 1 the GROB

described as:

As a graphic object this equation can now be placed in the graphics display.

To recover the graphics display press š. Then, move the cursor to an empty

sector in the graph, and press @EDIT LL@REPL. The equation ‘X^2-5’ is

placed in the graph, for example:

Thus, GROBs can be used to document graphics by placing equations, or text,

in the graphics display.

The GROB menu

The GROB menu, accessible through „°L@GROB @ꢀGROB, contains the

following functions. Press L to move to the next menu:

ꢀGROB

Of these functions we have already used SUB, REPL, (from the graphics EDIT

menu), ANIMATE [ANIMA], and ꢀGROB. ([ PRG ] is simply a way to return

to the programming menu.) While using ꢀGROB in the two previous examples

you may have noticed that I used a 3 while converting the graph into a GROB,

while I used a 1 when I converted the equation into a GROB. This parameter of

the function ꢀGROB indicates the size of the object that is being converted into

a GROB as 0 or 1 – for a small object, 2 – medium, and 3 – large. The other

functions in the GROB menu are described following.

Page 22-31

BLANK

The function BLANK, with arguments #n and #m, creates a blank graphics

object of width and height specified by the values #n and #m, respectively. This

is similar to the function PDIM in the GRAPH menu.

GOR

The function GOR (Graphics OR) takes as input grob (a target GROB), a set of

coordinates, and grob , and produces the superposition of grob onto grob

(or PICT) starting at the specified coordinates. The coordinates can be specified

as user-defined coordinates (x,y), or pixels {#n #m}. GOR uses the OR function

to determine the status of each pixel (i.e., on or off) in the overlapping region

between grob and grob .

GXOR

The function GXOR (Graphics XOR) performs the same operation as GOR, but

using XOR to determine the final status of pixels in the overlapping area

between graphic objects grob and grob .

Note: In both GOR and GXOR, when grob2 is replaced by PICT, these

functions produce no output. To see the output you need to recall PICT to the

stack by using either PICT RCL or PICTURE.

ꢀLCD

Takes a specified GROB and displays it in the calculator's display starting at the

upper left corner.

LCDꢀ

Copies the contents of the stack and menu display into a 131 x 64 pixels

GROB.

SIZE

The function SIZE, when applied to a GROB, shows the GROB’s size in the form

of two numbers. The first number, shown in stack level 2, represents the width of

the graphics object, and the second one, in stack level 1, shows its height.

Page 22-32

An example of a program using GROB

The following program produces the graph of the sine function including a

frame – drawn with the function BOX – and a GROB to label the graph. Here

is the listing of the program:

RAD

Begin program

Set angle units to radians

131 RꢀB 64 RꢀB PDIM

-6.28 6.28 XRNG –2. 2. YRNG

FUNCTION

‘SIN(X)’ STEQ

ERASE DRAX LABEL DRAW

(-6.28,-2.) (6.28,2.) BOX

PICT RCL

Set PICT screen to 131×64 pixels

Set x- and y-ranges

Select FUNCTION type for graphs

Store the function sine into EQ

Clear, draw axes, labels, graph

Draw a frame around the graph

Place contents of PICT on stack

Place graph label string in stack

Convert string into a small GROB

Coordinates to place label GROB

Combine PICT with the label GROB

Save combined GROB into PICT

Bring PICT to the stack

“SINE FUNCTION”

1 ꢀGROB

(-6., 1.5) SWAP

GOR

PICT STO

{ } PVIEW

End program

Save the program under the name GRPR (GROB PRogram). Press @GRPR to run

the program. The output will look like this:

A program with plotting and drawing functions

In this section we develop a program to produce, draw and label Mohr’s circle

for a given condition of two-dimensional stress. The left-hand side figure below

shows the given state of stress in two-dimensions, with σ and σ being

normal stresses, and τ = τ being shear stresses. The right-hand side figure

Page 22-33

shows the state of stresses when the element is rotated by an angle φ. In this

case, the normal stresses are σ’ and σ’ , while the shear stresses are τ’ and

τ’ .

The relationship between the original state of stresses (σ , σ , τ , τ ) and the

xx yy xy yx

state of stress when the axes are rotated counterclockwise by f (σ’ , σ’ , τ’ ,

τ’ ), can be represented graphically by the construct shown in the figure below.

To construct Mohr’s circle we use a Cartesian coordinate system with the x-axis

corresponding to the normal stresses (σ), and the y-axis corresponding to the

shear stresses (τ). Locate the points A(σ ,τ ) and B (σ , τ ), and draw the

xx xy

yy xy

segment AB. The point C where the segment AB crosses the σ axis will be the

center of the circle. Notice that the coordinates of point C are (½⋅(σ + σ ),

0). When constructing the circle by hand, you can use a compass to trace the

circle since you know the location of the center C and of two points, A and B.

Let the segment AC represent the x-axis in the original state of stress. If you

want to determine the state of stress for a set of axes x’-y’, rotated

counterclockwise by an angle φ with respect to the original set of axes x-y, draw

segment A’B’, centered at C and rotated clockwise by and angle 2φ with respect

to segment AB. The coordinates of point A’ will give the values (σ’ ,τ’ ), while

xx xy

those of B’ will give the values (σ’ ,τ’ ).

yy xy

Page 22-34

The stress condition for which the shear stress, τ’ , is zero, indicated by

segment D’E’, produces the so-called principal stresses, σ (at point D’) and

(at point E’). To obtain the principal stresses you need to rotate the

coordinate system x’-y’ by an angle φ , counterclockwise, with respect to the

system x-y. In Mohr’s circle, the angle between segments AC and D’C

measures 2φ .

The stress condition for which the shear stress, τ’ , is a maximum, is given by

segment F’G’. Under such conditions both normal stresses, σ’ = σ’ , are

equal. The angle corresponding to this rotation is φ . The angle between

segment AC and segment F’C in the figure represents 2φ .

Modular programming

To develop the program that will plot Mohr’s circle given a state of stress, we

will use modular programming. Basically, this approach consists in

decomposing the program into a number of sub-programs that are created as

Page 22-35

separate variables in the calculator. These sub-programs are then linked by a

main program, that we will call MOHRCIRCL. We will first create a sub-

directory called MOHRC within the HOME directory, and move into that

directory to type the programs.

The next step is to create the main program and sub-programs within the sub-

directory.

The main program MOHRCIRCL uses the following sub-programs:

ꢀ INDAT: Requests input of σx, σy, τxy from user, produces a list σL =

{σx, σy, τxy} as output.

ꢀ CC&r: Uses σL as input, produces σc = ½(σx+σy), r = radius of

Mohr’s circle, φn = angle for principal stresses, as output.

ꢀ DAXES: Uses σc and r as input, determines axes ranges and draws

axes for the Mohr’s circle construct

ꢀ PCIRC: Uses σc, r, and φn as input, draw’s Mohr’s circle by producing

a PARAMETRIC plot

ꢀ DDIAM: Uses σL as input, draws the segment AB (see Mohr’s circle

figure above), joining the input data points in the Mohr’s circle

ꢀ

σLBL: Uses σL as input, places labels to identify points A and B with

labels “σx” and “σy”.

ꢀ

σAXS: Places the labels “σ” and “τ” in the x- and y-axes, respectively.

ꢀ PTTL: Places the title “Mohr’s circle” in the figure.

Running the program

If you typed the programs in the order shown above, you will have in your sub-

directory MOHRC the following variables: PTTL, σAXS, PLPNT, σLBL, PPTS,

DDIAM. Pressing L you find also: PCIRC, DAXES, ATN2, CC&r, INDAT,

MOHRC. Before re-ordering the variables, run the program once by pressing

the soft-key labeled @MOHRC. Use the following:

@MOHRC

Launches the main program MOHRCIRCL

Enter σx = 25

Enter σy = 75

25˜

75˜

50`

Enter τxy = 50, and finish data entry.

Page 22-36

At this point the program MOHRCIRCL starts calling the sub-programs to

produce the figure. Be patient. The resulting Mohr’s circle will look as in the

picture to the left.

Because this view of PICT is invoked through the function PVIEW, we cannot get

any other information out of the plot besides the figure itself. To obtain

additional information out of the Mohr’s circle, end the program by pressing

$. Then, press š to recover the contents of PICT in the graphics

environment. The Mohr’s circle now looks like the picture to the right (see

above).

Press the soft-menu keys @TRACE and @(x,y)@. At the bottom of the screen you will

find the value of φ corresponding to the point A(σx, τxy), i.e., φ = 0,

(2.50E1, 5.00E1).

Press the right-arrow key (™) to increment the value of φ and see the

corresponding value of (σ’ , τ’ ). For example, for φ = 45 , we have the

values (σ’ , τ’ ) = (1.00E2, 2.50E1) = (100, 25). The value of σ’ will be

found at an angle 90 ahead, i.e., where φ = 45 + 90 = 135 . Press the ™

key until reaching that value of φ, we find (σ’ , τ’ ) = (-1.00E-10,-2.5E1) = (0,

25).

To find the principal normal values press š until the cursor returns to the

intersection of the circle with the positive section of the σ-axis. The values found

at that point are φ = 59 , and (σ’ , τ’ ) = (1.06E2,-1.40E0) = (106, -1.40).

Now, we expected the value of τ’ = 0 at the location of the principal axes.

What happens is that, because we have limited the resolution on the

independent variable to be Δφ = 1 , we miss the actual point where the shear

stresses become zero. If you press š once more, you find values of are φ =

58 , and (σ’ , τ’ ) = (1.06E2,5.51E-1) = (106, 0.551). What this

Page 22-37

information tell us is that somewhere between φ = 58 and φ = 59 , the shear

stress, τ’ , becomes zero.

To find the actual value of φn, press $. Then type the list corresponding to

the values {σx σy τxy}, for this case, it will be { 25 75 50 } [ENTER]

Then, press @CC&r. The last result in the output, 58.2825255885 , is the actual

value of φn.

A program to calculate principal stresses

The procedure followed above to calculate φn, can be programmed as follows:

Program PRNST:

Start program PRNST (PRiNcipal STresses)

Enter data as in program MOHRCIRC

Calculate σc, r, and fn, as in MOHRCIRC

Tag angle for principal stresses

INDAT

CC&r

“φn” ꢀTAG

3 ROLLD

Move tagged angle to level 3

RꢀC DUP

Convert σc and r to (σc, r), duplicate

Calculate principal stress σPx, tag it

Swap,calculate stress σPy, tag it.

End program PRNST

CꢀR + “σPx” ꢀTAG

SWAP CꢀR - “σPy” ꢀTAG

To run the program use:

J@PRNST

25˜

Start program PRNST

Enter σx = 25

75˜

Enter σy = 75

50`

Enter τxy = 50, and finish data entry.

The result is:

Ordering the variables in the sub-directory

Running the program MOHRCIRCL for the first time produced a couple of new

variables, PPAR and EQ. These are the Plot PARameter and EQuation variables

Page 22-38

necessary to plot the circle. It is suggest that we re-order the variables in the

sub-directory, so that the programs @MOHRC and @PRNST are the two first variables

in the soft-menu key labels. This can be accomplished by creating the list {

MOHRCIRCL PRNST } using: J„ä@MOHRC @PRNST `

And then, ordering the list by using:

„°@@MEM@@ @@DIR@@ @ORDER.

After this call to the function ORDER is performed, press J. You will now see

that we have the programs MOHRCIRCL and PRNST being the first two

variables in the menu, as we expected.

A second example of Mohr’s circle calculations

Determine the principal stresses for the stress state defined by σ = 12.5 kPa,

= -6.25 kPa, and τ = - 5.0 kPa. Draw Mohr’s circle, and determine from

the figure the values of σ’ , σ’ , and τ’ if the angle φ = 35 .

To determine the principal stresses use the program @PRNST, as follows:

J@PRNST

Start program PRNST

Enter σx = 12.5

Enter σy = -6.25

12.5˜

6.25\˜

5\`

Enter τxy = -5, and finish data entry.

The result is:

To draw Mohr’s circle, use the program @MOHRC, as follows:

J@MOHRC

Start program PRNST

Enter σx = 12.5

Enter σy = -6.25

12.5˜

6.25\˜

5\`

Enter τxy = -5, and finish data entry.

The result is:

Page 22-39

To find the values of the stresses corresponding to a rotation of 35 in the angle

of the stressed particle, we use:

$š

Clear screen, show PICT in graphics screen

@TRACE @(x,y)@.

To move cursor over the circle showing φ and (x,y)

Next, press ™ until you read φ = 35. The corresponding coordinates are

(1.63E0, -1.05E1), i.e., at φ = 35 , σ’ = 1.63 kPa, and σ’ = -10.5kPa.

An input form for the Mohr’s circle program

For a fancier way to input data, we can replace sub-program INDAT, with the

following program that activates an input form:

« “MOHR’S CIRCLE” { { “σx:” “Normal stress in x” 0 } {

“σy:” “Normal stress in y” 0 } { “τxy:” “Shear stress” 0}

} { } { 1 1 1 } { 1 1 1 } INFORM DROP »

With this program substitution, running @MOHRC will produce an input form as

shown next:

Press @@@OK@@@ to continue program execution. The result is the following figure:

Page 22-40

Since program INDAT is used also for program @PRNST (PRiNcipal STresses),

running that particular program will now use an input form, for example,

The result, after pressing @@@OK@@@, is the following:

Page 22-41

Chapter 23

Character strings

Character strings are calculator objects enclosed between double quotes. They

are treated as text by the calculator. For example, the string “SINE

FUNCTION”, can be transformed into a GROB (Graphics Object), to label a

graph, or can be used as output in a program. Sets of characters typed by the

user as input to a program are treated as strings. Also, many objects in

program output are also strings.

String-related functions in the TYPE sub-menu

The TYPE sub-menu is accessible through the PRG (programming) menu, i.e.,

„°. The functions provided in the TYPE sub-menu are also shown below.

Among the functions in the TYPE menu that are useful for manipulating strings

we have:

OBJꢀ: Converts string to the object it represents

ꢀSTR: Converts an object to its string representation

ꢀTAG: Tags a quantity

DTAG: Removes the tag from a tagged quantity (de-tags)

CHR: Creates a one-character string corresponding to the number used as

argument

NUM: Returns the code for first character in a string

Examples of application of these functions to strings are shown next:

Page 23-1

String concatenation

Strings can be concatenated (joined together) by using the plus sign +, for

example:

Concatenating strings is a practical way to create output in programs. For

example, concatenating "YOU ARE " AGE + " YEAR OLD" creates the string

"YOU ARE 25 YEAR OLD", where 25 is stored in the variable called AGE.

The CHARS menu

The CHARS sub-menu is accessible through the PRG (programming) menu, i.e.,

„°.

The functions provided by the CHARS sub-menu are the following:

Page 23-2

The operation of NUM, CHR, OBJꢀ, and ꢀSTR was presented earlier in this

Chapter. We have also seen the functions SUB and REPL in relation to graphics

earlier in this chapter. Functions SUB, REPL, POS, SIZE, HEAD, and TAIL have

similar effects as in lists, namely:

SIZE: number of a sub-string in a string (including spaces)

POS: position of first occurrence of a character in a string

HEAD: extracts first character in a string

TAIL: removes first character in a string

SUB: extract sub-string given starting and ending position

REPL: replace characters in a string with a sub-string starting at given position

SREPL: replaces a sub-string by another sub-string in a string

To see those effects on action try the following exercises: Store the string “MY

NAME IS CYRILLE” into variable S1. We’ll use this string to show examples of

the functions in the CHARS menu:

The characters list

The entire collection of characters available in the calculator is accessible

through the keystroke sequence ‚

When you highlight any character,

say they line feed character ꢀ , you will see at the left side of the bottom of the

Page 23-3

screen the keystroke sequence to get such character (ꢁ. for this case) and the

numerical code corresponding to the character (10 in this case).

Characters that are not defined appear as a dark square in the characters list

(ꢀ) and show (None) at the bottom of the display, even though a numerical code

exists for all of them. Numerical characters show the corresponding number at

the bottom of the display.

Letters show the code α (i.e., ~) followed by the corresponding letter, for

example, when you highlight M, you will see αM displayed at the lower left side

of the screen, indicating the use of ~m. On the other hand, m shows the

keystroke combination αꢀM, or ~„m.

Greek characters, such as σ, will show the code αꢁS, or ~‚s. Some

characters, like ρ, do not have a keystroke sequence associated with them.

Therefore, the only way to obtain such characters is through the character list by

highlighting the desired character and pressing @ECHO1@ or @ECHO@.

Use @ECHO1@ to copy one character to the stack and return immediately to normal

calculator display. Use @ECHO@ to copy a series of characters to the stack. To

return to normal calculator display use $.

See Appendix D for more details on the use of special characters. Also,

Appendix G shows shortcuts for producing special characters.

Page 23-4

Chapter 24

Calculator objects and flags

Numbers, lists, vectors, matrices, algebraics, etc., are calculator objects. They

are classified according to its nature into 30 different types, which are

described below. Flags are variables that can be used to control the calculator

properties. Flags were introduced in Chapter 2.

Description of calculator objects

The calculator recognizes the following object types:

_________________________________________________________________

Number

Type

Example

_________________________________________________________________

Real Number

Complex Number

String

-1.23E-5

(-1.2,2.3)

"Hello, world "

[[1 2][3 4]]

[[(1 2) (3 4)]

[(5 6) (7 8)]

{3 1 'PI'}

Real Array

Complex Array

List

Global Name

Local Name

Program

<< ꢀ a 'a^2' >>

'a^2+b^2'

Algebraic object

Binary Integer

Graphic Object

Tagged Object

Unit Object

# 2F1E h

Graphic 131×64

R: 43.5

3_m^2/s

XLIB 342 8

DIR É END

Library 1230"...

Backup MYDIR

COS

XLIB Name