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hp 48gII graphing calculator

user’s guide

Edition 4

HP part number F2226-90020

Notice

THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE

PROVIDED “AS IS” AND ARE SUBJECT TO CHANGE WITHOUT

NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY

OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT

NOT LIMITED TO, THE IMPLIED WARRANTIES OF

MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS FOR A

PARTICULAR PURPOSE.

HEWLETT-PACKARD CO. SHALL NOT BE LIABLE FOR ANY ERRORS

OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES IN

CONNECTION WITH THE FURNISHING, PERFORMANCE, OR USE

OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN.

Reproduction, adaptation, or translation of this manual is prohibited without

prior written permission of Hewlett-Packard Company, except as allowed

under the copyright laws.

Hewlett-Packard Company

4995 Murphy Canyon Rd,

Suite 301

San Diego,CA 92123

Printing History

Edition 4 April 2004

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 48gII

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

The hp 48gII can be operated in two different calculating modes, the Reverse

Polish Notation (RPN) mode and the Algebraic (ALG) mode (see page 1-11 in

user’s guide 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 in this user’s guide 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 are very convenient for producing complex figures in very little

time.

Thanks to the infrared port and the RS 232 cable available with your

calculator, you can connect your calculator with other calculators or

computers. The high-speed connection through infrared or RS 232 allows the

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.

Page TOC-1

Table of contents

A note about screenshots in this guide, Note-1

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, 1-3

Selecting SOFT menus or CHOOSE boxes, 1-4

The TOOL menu, 1-6

Setting time and date, 1-7

Introducing the calculator’s keyboard, 1-10

Setting calculator’s mode, 1-11

Operating mode, 1-12

Number format and decimal dot or comma, 1-16

Angle measure, 1-21

Coordinate system, 1-22

Beep, Key Click, and Last Stack, 1-23

Selecting CAS settings, 1-24

Selecting Display modes, 1-24

Selecting the display font, 1-25

Selecting properties of the line editor, 1-26

Selecting properties of the Stack, 1-26

Selecting properties of the Equation Writer (EQW), 1-27

Selecting the size of the header, 1-28

Selecting the clock display, 1-28

Chapter 2 - Introducing the calculator, 2-1

Calculator objects, 2-1

Editing expressions in the screen, 2-3

Creating arithmetic expressions, 2-3

Editing arithmetic expressions, 2-6

Page TOC-2

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, 2-16

Creating algebraic expressions, 2-19

Editing algebraic expressions, 2-20

Creating and editing summations, derivatives, and integrals, 2-28

Organizing data in the calculator, 2-32

Functions for manipulation of variables, 2-33

The HOME directory, 2-34

The CASDIR sub-directory, 2-35

Typing directory and variable names, 2-37

Creating sub-directories, 2-38

Moving among sub-directories, 2-42

Deleting sub-directories, 2-42

Variables, 2-46

Creating variables, 2-46

Checking variable contents, 2-51

Replacing the contents of variables, 2-53

Copying variables, 2-54

Reordering variables in a directory, 2-57

Moving variables using the FILES menu, 2-58

Deleting variables, 2-59

UNDO and CMD functions, 2-61

Flags, 2-62

Example of flag setting: general solution vs. principal value, 2-63

Other flags of interest, 2-64

CHOOSE boxes vs. SOFT menus, 2-65

Selected CHOOSE boxes, 2-66

Chapter 3 - Simple calculations with real numbers, 3-1

Checking calculator settings, 3-1

Checking calculator mode, 3-2

Real number calculations, 3-2

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

Page TOC-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, 3-4

Powers and roots, 3-5

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

Using powers of 10 in entering data, 3-5

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

Operation with units, 3-17

The UNITS menu, 3-17

Available units, 3-18

Converting to base units, 3-21

Attaching units to numbers, 3-22

Operations with units, 3-25

Units manipulation tools, 3-27

Physical constants in the calculator, 3-28

Special physical functions, 3-31

Function ZFACTOR, 3-32

Function F0λ, 3-32

Function SIDENS, 3-32

Function TDELTA, 3-33

Function TINC, 3-33

Defining and using functions, 3-33

Functions defined by more than one expression, 3-35

The IFTE function, 3-35

Combined IFTE functions, 3-36

Page TOC-4

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, 4-4

Entering the unit imaginary number, 4-5

The CMPLX menus, 4-5

CMPLX menu through the MTH menu, 4-5

CMPLX menu in keyboard, 4-7

Functions applied to complex numbers, 4-8

Functions from the MTH menu, 4-8

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, 5-2

Functions in the ALG menu, 5-3

COLLECT, 5-5

EXPAND, 5-5

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, 5-8

Expansion and factoring using trigonometric functions, 5-8

Functions in the ARITHMETIC menu, 5-9

DIVIS, 5-10

Page TOC-5

FACTORS, 5-10

LGCD, 5-10

PROPFRAC, 5-10

SIMP2, 5-10

INTEGER menu, 5-10

POLYNOMIAL menu, 5-11

MODULO menu, 5-12

Applications of the ARITHMETIC menu, 5-12

Modular arithmetic, 5-12

Finite arithmetic in the calculator, 5-15

Polynomials, 5-18

Modular arithmetic with polynomials, 5-19

The CHINREM function, 5-19

The EGCD function, 5-19

The GCD function, 5-19

The HERMITE function, 5-20

The HORNER function, 5-20

The variable VX, 5-20

The LAGRANGE function, 5-21

The LCM function, 5-21

The LEGENDRE function, 5-22

The PCOEF function, 5-22

The PROOT function, 5-22

The PTAYL function, 5-22

The QUOTIENT and REMAINDER functions, 5-23

The EPSX0 function and the CAS variable EPS, 5-23

The PEVAL function, 5-23

The TCHEBYCHEFF function, 5-24

Fractions, 5-24

The SIMP2 function, 5-24

The PROPFRAC function, 5-24

The PARTFRAC function, 5-25

The FCOEF function, 5-25

The FROOTS function, 5-26

Step-by-step operations with polynomial fractions, 5-26

The CONVERT menu and algebraic operations, 5-27

Page TOC-6

UNITS convert menu, 5-27

BASE convert menu, 5-28

TRIGONOMETRIC convert menu, 5-28

MATRICES convert menu, 5-28

REWRITE convert menu, 5-28

Chapter 6 - Solution to single equations

, 6-1

Symbolic solution of algebraic equations, 6-2

Function ISOL, 6-1

Function SOLVE, 6-2

Function SOLVEVX, 6-4

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, 6-14

The SOLVE soft menu, 6-27

The ROOT sub-menu, 6-27

Function ROOT, 6-27

Variable EQ, 6-28

The SOLVR sub-menu, 6-28

The DIFFE sub-menu, 6-30

The POLY sub-menu, 6-30

The SYS sub-menu, 6-31

The TVM sub-menu, 6-31

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, 7-6

Using the Multiple Equation Solver (MES), 7-10

Application 1 - Solution of triangles, 7-10

Page TOC-7

Application 2 - Velocity and acceleration in polar coordinates, 7-18

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, 8-3

Changing sign, 8-3

Addition, subtraction, multiplication, division, 8-3

Real number functions from the keyboard, 8-5

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

List size, 8-10

Extracting and inserting elements, 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, 8-14

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

Page TOC-8

Using the Matrix Writer (MTWR) 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 and division by a scalar, 9-10

Absolute value function, 9-10

The MTH/VECTOR menu, 9-10

Magnitude, 9-11

Dot product, 9-11

Cross product, 9-11

Decomposing a vector, 9-12

Building a two-dimensional vector, 9-12

Building a three-dimensional vector, 9-13

Changing coordinate system, 9-13

Application of vector operations, 9-16

Resultant of forces, 9-16

Angle between vectors, 9-16

Moment of a force, 9-17

Equation of a plane in space, 9-18

Row vectors, column vectors, and lists, 9-19

Function OBJ, 9-20

Function LIST, 9-20

Function DROP, 9-21

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

Transforming a column vector into a row vector, 9-22

Transforming a list into a vector, 9-24

Transforming a vector (or matrix) into a list, 9-25

Chapter 10 - Creating and manipulating matrices, 10-1

Definitions, 10-1

Entering matrices in the stack, 10-2

Using the Matrix Writer, 10-2

Typing the matrix directly into the stack, 10-3

Creating matrices with calculator functions, 10-3

Page TOC-9

Functions GET and PUT, 10-6

Functions GETI and PUTI, 10-6

Function SIZE, 10-7

Function TRN, 10-8

Function CON, 10-8

Function IDN, 10-9

Function RDM, 10-10

Function RANM, 10-11

Function SUB, 10-11

Function REPL, 10-12

Function DIAG, 10-12

Function DIAG, 10-13

Function VANDERMONDE, 10-14

Function HILBERT, 10-14

A program to build a matrix out of a number of lists, 10-15

Lists represent columns of the matrix, 10-15

Lists represent rows of the matrix, 10-17

Manipulating matrices by columns, 10-18

Function COL, 10-18

Function COL, 10-19

Function COL+, 10-20

Function COL-, 10-20

Function CSWP, 10-21

Manipulating matrices by rows, 10-21

Function ROW, 10-22

Function ROW , 10-23

Function ROW+, 10-24

Function ROW -, 10-24

Function RSWP, 10-25

Function RCI, 10-25

Function RCIJ, 10-26

Chapter 11 - Matrix Operations and Linear Algebra

, 11-1

Operations with matrices, 11-1

Addition and subtraction, 11-2

Multiplication, 11-2

Page TOC-10

Characterizing a matrix (the matrix NORM menu), 11-6

Function ABS, 11-7

Function SNRM, 11-7

Functions RNRM and CNRM, 11-8

Function SRAD, 11-9

Function COND, 11-9

Function RANK, 11-10

Function DET, 11-11

Function TRACE, 11-13

Function TRAN, 11-14

Additional matrix operations (the matrix OPER menu), 11-14

Function AXL, 11-15

Function AXM, 11-15

Function LCXM, 11-15

Solution of linear systems, 11-16

Using the numerical solver for linear systems, 11-17

Least-square solution (function LSQ), 11-23

Solution with the inverse matrix, 11-26

Solution by “division” of matrices, 11-26

Solving multiple set of equations with the same coefficient

matrix, 11-27

Gaussian and Gauss-Jordan elimination, 11-28

Step-by-step calculator procedure for solving linear systems, 11-37

Solution to linear systems using calculator functions, 11-46

Residual errors in linear system solutions (function RSD) , 11-43

Eigenvalues and eigenvectors, 11-44

Function PCAR, 11-44

Function EGVL, 11-45

Function EGV, 11-46

Function JORDAN, 11-47

Function MAD, 11-47

Matrix factorization, 11-48

Function LU, 11-49

Orthogonal matrices and singular value decomposition, 11-49

Function SCHUR, 11-50

Function LQ, 11-51

Page TOC-11

Function QR, 11-51

Matrix Quadratic Forms, 11-51

The QUADF menu, 11-52

Linear Applications, 11-54

Function IMAGE, 11-54

Function ISOM, 11-54

Function KER, 11-55

Function MKISOM, 11-55

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, 12-8

Graphics of transcendental functions, 12-10

Graph of ln(X) , 12-8

Graph of the exponential function, 12-10

The PPAR variable, 12-11

Inverse functions and their graphs, 12-12

Summary of FUNCTION plot operation, 12-13

Plots of trigonometric and hyperbolic functions and their inverses, 12-16

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

The TPAR variable, 12-18

Plots in polar coordinates, 12-19

Plotting conic curves, 12-21

Parametric plots, 12-23

Generating a table of parametric equations, 12-26

Plotting the solution to simple differential equations, 12-26

Truth plots, 12-29

Plotting histograms, bar plots, and scatterplots, 12-30

Bar plots, 12-30

Scatter plots, 12-32

Slope fields, 12-34

Fast 3D plots, 12-35

Wireframe plots, 12-37

Ps-Contour plots, 12-39

Page TOC-12

Y-Slice plots, 12-41

Gridmap plots, 12-42

Pr-Surface plots, 12-43

The VPAR variable, 12-44

Interactive drawing, 12-44

DOT+ and DOT-, 12-45

MARK, 12-46

LINE, 12-46

TLINE, 12-46

BOX, 12-47

CIRCL, 12-47

LABEL, 12-47

DEL, 12-47

ERASE, 12-48

MENU, 12-48

SUB, 12-48

REPL, 12-48

PICT, 12-48

X,Y, 12-48

Zooming in and out in the graphics display, 12-49

ZFACT, ZIN, ZOUT, and ZLAST, 12-49

BOXZ, 12-50

ZDFLT, ZAUTO, 12-50

HZIN, HZOUT, VZIN, and VZOUT, 12-50

CNTR, 12-50

ZDECI, 12-50

ZINTG, 12-51

ZSQR, 12-51

ZTRIG, 12-51

The SYMBOLIC menu and graphs, 12-51

The SYMB/GRAPH menu, 12-52

Function DRAW3DMATRIX, 12-54

Chapter 13 - Calculus Applications, 13-1

The CALC (Calculus) menu, 13-1

Limits and derivatives, 13-1

Page TOC-13

Function lim, 13-2

Derivatives, 13-3

Function DERIV and DERVX,13-3

The DERIV&INTEG menu, 13-3

Calculating derivatives with ∂,13-4

The chain rule,13-6

Derivatives of equations,13-6

Implicit derivatives,13-7

Application of derivatives,13-7

Analyzing graphics of functions,13-7

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, 13-18

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,13-21

Integration with units, 13-21

Infinite series,13-23

Taylor and Maclaurin’s series,13-23

Taylor polynomial and remainder,13-23

Functions TAYLR, TAYRL0, 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

Page TOC-14

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

Checking solutions in the calculator, 16-2

Slope field visualization of solutions, 16-3

The CALC/DIFF menu, 16-4

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 ODEs, 16-17

Page TOC-15

Fourier series, 16-27

Function FOURIER, 16-28

Fourier series for a quadratic function, 16-29

Fourier series for a triangular wave, 16-35

Fourier series for a square wave, 16-39

Fourier series applications in differential equations, 16-42

Fourier Transforms, 16-43

Definition of Fourier transforms, 16-46

Properties of the Fourier transform, 16-48

Fast Fourier Transform (FFT) , 16-49

Examples of FFT applications, 16-50

Solution to specific second-order differential equations, 16-53

The Cauchy or Euler equation , 16-53

Legendre’s equation, 16-54

Bessel’s equation, 16-55

Chebyshev or Tchebycheff polynomials, 16-57

Laguerre’s equation, 16-58

Weber’s equation and Hermite polynomials, 16-59

Numerical and graphical solutions to ODEs, 16-60

Numerical solution of first-order ODE, 16-60

Graphical solution of first-order ODE, 16-62

Numerical solution of second-order ODE, 16-64

Graphical solution for a second-order ODE, 16-66

Numerical solution for stiff first-order ODE, 16-68

Numerical solution to ODEs with the SOLVE/DIFF menu, 16-69

Function RKF, 16-70

Function RRK, 16-71

Function RKFSTEP, 16-72

Function RRKSTEP, 16-73

Function RKFST, 16-72

Function RKFERR, 16-72

Function RSBERR, 16-73

Chapter 17 - Probability Applications, 17-1

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

Factorials, combinations, and permutations, 17-1

Page TOC-16

Random numbers, 17-2

Discrete probability distributions, 17-4

Binomial distribution, 17-4

Poisson distribution, 17-5

Continuous probability distributions, 17-6

The gamma distribution, 17-6

The exponential distribution, 17-7

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

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, 18-15

The ΣPAR sub-menu, 18-16

The 1VAR sub-menu, 18-16

The PLOT sub-menu, 18-17

The FIT sub-menu, 18-18

Example of STAT menu operations, 18-18

Confidence intervals, 18-22

Estimation of Confidence Intervals, 18-23

Definitions, 18-23

Page TOC-17

Confidence intervals for the population mean when the

population variance is known, 18-23

Confidence intervals for the population mean when the

population variance is unknown, 18-24

Confidence interval for a proportion, 18-24

Sampling distributions 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, 18-34

Procedure for testing hypotheses, 18-35

Errors in hypothesis testing, 18-35

Inferences concerning one mean, 18-36

Inferences concerning two means, 18-39

Paired sample tests, 18-40

Inferences concerning one proportion, 18-41

Testing the difference between two proportions, 18-41

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, 18-49

The method of least squares, 18-49

Additional equations for linear regression, 18-51

Prediction error, 18-51

Confidence intervals and hypothesis testing in

linear regression, 18-52

Procedure for inference statistics for linear regression

using the calculator, 18-53

Multiple linear fitting, 18-56

Polynomial fitting, 18-58

Selecting the best fitting, 18-62

Chapter 19 - Numbers in Different Bases

, 19-1

Definitions, 19-1

The BASE menu, 19-1

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

Page TOC-18

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, 19-6

Hexadecimal numbers for pixel references, 19-7

Chapter 20 - Customizing menus and keyboard

, 20-1

Customizing menus, 20-1

The PRG/MODES/MENU, 20-1

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

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, 20-6

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

Assigning multiple user-defined key, 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, 21-6

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

Page TOC-19

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 number and tag, 21-33

“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, 21-44

Program branching, 21-46

Branching with IF, 21-46

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, 21-62

Errors and error trapping, 21-64

DOERR, 21-64

ERRN, 21-64

ERRM, 21-65

ERR0, 21-65

LASTARG, 21-65

Sub-menu IFERR, 21-65

User RPL programming in algebraic mode, 21-66

Chapter 22 - Programs for graphics manipulation, 22-1

The PLOT menu, 22-1

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

Page TOC-20

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

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, 22-22

PVIEW, 22-22

PXC, 22-22

CPX, 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), 22-30

The GROB menu, 22-31

A program with plotting and drawing functions, 22-33

Modular programming, 22-36

Running the program, 22-36

A program to calculate principal stresses, 22-38

Ordering the variables in the sub-directory, 22-39

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

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

Chapter 23 - Character strings, 23-1

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

String concatenation, 23-2

Page TOC-21

The CHARS menu, 23-2

The characters list, 23-3

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

Calculations 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, 26-2

Backup objects, 26-3

Backing up objects in port memory, 26-3

Backing up and restoring HOME, 26-4

Storing, deleting, and restoring backup objects, 26-5

Using data in backup objects, 26-5

Using libraries, 26-6

Installing and attaching a library, 26-6

Library numbers, 26-7

Deleting a library, 26-7

Page TOC-22

Creating libraries, 26-7

Backup battery, 26-7

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

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 - The MATHS menu, J-1

Appendix K - The MAIN menu, K-1

Appendix L - Line editor commands, L-1

Appendix M – Index, M-1

Limited Warranty –

Service, W-2

Regulatory information, W-4

Page Note-1

A note about screenshots in this guide

A screenshot is a representation of the calculator screen. For example, the

first time the calculator is turned on you get the following screen (calculator

screens are shown with a thick border in this section):

The top two lines represent the screen header and the remaining area in the

screen is used for calculator output.

Most screenshots in this guide were generated using a computer-based

emulator (a program that simulates the operation of the calculator in a

computer), and are missing the screen header lines. Instead, they will show

additional screen output area in the location of the header lines, as shown

below:

This additional screen output area in many screen shots in this guide will not

show when you try those guide’s example in your calculator. Thus, while in

the guide you may see a screenshot as the following:

the calculator will actually show the following screen:

Page Note-2

Notice that the header lines cover the top first and a half lines of output in the

calculator’s screen. Nevertheless, the lines of output not visible are still

available for you to use. You can access those lines in your calculator by

pressing the up-arrow key (—), which will allow you to scroll down the

screen contents.

Also, as you perform the three operations listed in the screenshot, in the order

shown, your screen will show them occupying higher levels in the display as

shown next:

The keystrokes required to complete these exercises are the following:

S2.5`

R„Ü5.5+‚¹2.5`

The next operation,

2.3+5*„Ê\2.3`

will force the lines corresponding to the operation SIN(2.5) to move upwards

and be hidden by the header lines.

Many screenshots in this guide have also been modified to show only the

operation of interest. For example, the screenshot for the operation SIN(2.5),

shown above, may be simplified in this guide to look as:

Page Note-3

These simplifications of the screenshots are aimed at economizing output

space in the guide.

Be aware of the differences between the guide’s screenshots and the actual

screen display, and you should have no problem reproducing the exercises in

this guide.

Page 1-1

Chapter 1

Getting started

This chapter is aimed at providing basic information in the operation of your

calculator. The exercises are aimed at familiarizing yourself with the basic

operations and settings before actually performing a calculation.

Basic Operations

The following exercises are aimed at getting you acquainted with the

hardware of your calculator.

Batteries

The calculator uses 3 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 3 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 1-2

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 red 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 red 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 1-3

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:

RAD XYZ HEX R= 'X'

For details on the meaning of these specifications 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.

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

Page 1-4

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:

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

functions, from 1. HEX x to 6. BR. 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

Page 1-5

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 RB (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 soft menu key

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

normal calculator display.

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:

@CHK@

Page 1-6

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.

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

L for more information on editing)

Page 1-7

@VIEW B VIEW the contents of a variable

@@ RCL @@ C ReCaLl the contents of a variable

@@STO@ D STOre the contents of a variable

! PURGE E PURGE a variable

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 red right-shift button, ‚, with

the 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:

Page 1-8

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#@ F 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#@ F 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 1-9

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:

o AM : indicates that displayed time is AM time

o PM : indicates that displayed time is PM time

o 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#@ F soft menu key to

make the selection.

Setting the date

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

look as follows:

Page 1-10

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 ( B), to see the options for the date format:

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

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

Introducing the calculator’s keyboard

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

numbering of its rows and columns.

Page 1-11

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 green left-shift key, key (8,1),

the red right-shift key, key (9,1), and the blue ALPHA key, key (7,1), can be

Page 1-12

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:

P Main function, to activate the SYMBolic menu

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

… N Right-shift function, to activate the CATalog function

~p ALPHA function, to enter the upper-case letter P

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

~…p ALPHA-Right-Shift function, to enter the symbol 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 1-13

Press the !!@@OK#@ F 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 ( B). 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:

5.2

⋅

−⋅













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™˜—`

Page 1-14

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.

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#@ F soft menu key to complete the operation.

The display, for the RPN mode looks as follows:

Page 1-15

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.

Basically, what RPN means is that, instead of writing an operation such as 3

+ 2, in the calculator by using 3+2`, we write first the operands,

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 2`pushes

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. A simpler

way to calculate this operation is by using: 3`2+.

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

√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.

2X Completes the calculation.

Let's try now the expression proposed earlier:

5.2

⋅

−⋅













Page 1-16

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

Y 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

!¸ e

2.5

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

+ (3× (5 - 1/(3×3)))/23

2.5

= 12.18369, into lev. 1.

R √((3× (5 - 1/(3×3)))/23

2.5

) = 3.4905156, into 1.

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:

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

Page 1-17

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 —„—„—„ — @@CHK@@ @@OK@@ @@OK@@

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

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.

Page 1-18

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 ( B), and select the option Fixed with the arrow

down key ˜.

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.

Page 1-19

• 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 ( B), 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.

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

Page 1-20

• 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 ( B), 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 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 × 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 ( B), 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 (i.e., the B key). The input form will

look as follows:

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

number 123.456789012, entered earlier, now is shown as:

Page 1-22

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

key ( B). If using the latter approach, use up and down arrow

keys,— ˜, to select the preferred mode, and press the !!@@OK#@ F soft

menu key to complete the operation. For example, in the following

screen, the Radians mode is selected:

Coordinate System

Page 1-23

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:

)cos( yxrrx +=⋅= θ













=⋅=

−

tan)sin( θθ

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

and Spherical coordinate systems are related by the following quantities:













=⋅=













=⋅⋅=

++=⋅⋅=

−

zyxx

222

tan)cos(

tan)sin()sin(

)cos()sin(

φφρ

θθφρ

ρθφρ

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

Page 1-24

• 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 ( B). If using the latter approach, use up and down arrow

keys,— ˜, to select the preferred mode, and press the !!@@OK#@ F 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.

_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,

Page 1-25

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

Use the soft menu key (i.e., the B key) to change the selection.

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

soft menu key (i.e., the B key) to change the selection.

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

menu key (i.e., the B key) to change the selection.

Press the !!@@OK#@ F 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.

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 (D) to display the DISPLAY MODES input form.

@CHK@

Page 1-26

• 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

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 (B).

• 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

(D) 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. Pressing the @CHOOS soft menu key (B), 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

@CHK@

Page 1-27

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

(D) 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 Allows to place the cursor after the end of the line

_Indent 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

(D) to 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.

_Textbook Display mathematical expressions in graphical mathematical

notation

Page 1-28

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

(D) to display the DISPLAY MODES input form. Press the down arrow key,

˜, three 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.

Page 1-29

For the example of the integral

∫

∞

−

dXe

, presented above, selecting the

_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

(D) 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

(D) 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 right arrow key (™) to select the underline in front of

the options _Clock or _Analog. Toggle the 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.

@CHK@

Page 2-1

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..

Page 2-2

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

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.

Page 2-3

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

algebraic expression enclosed between apostrophes.

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 in 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:

0.20.3

5.7

0.1

0.5

−

⋅

To enter this expression use the following keystrokes:

Page 2-4

5.*„Ü1.+1./7.5™/

„ÜR3.-2.Q3

The resulting expression is: 5.*(1.+1./7.5)/(ƒ3.-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`

Page 2-5

The result will be shown as follows:

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. 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 2-6

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

µ 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:

rather than the intended expression:

5.7

−

⋅ . The incorrect expression

was entered by using:

³5*„Ü1+1/1.75™/„Ü

R5-2Q3`

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

Page 2-7

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:

Page 2-8

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

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

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

Page 2-9

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:

• 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:

Page 2-10

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

• 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.

Page 2-11

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:

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

Page 2-12

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

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+π

/2). First, we use the delete key (ƒ)

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

/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 π

expression. We do this by pressing the right arrow key (™)

once. At that point, we enter the following keystrokes: /2

Page 2-13

The expression now looks as follows:

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

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

)

5(25

+⋅+

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 π

/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:

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 C soft menu key.

For this case, the screen looks as follows:

Page 2-14

To recover the larger-font display, press the @BIG C 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 D

soft menu key.

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

the entire expression, by pressing ‚ ‘. Then, press the @EVAL D 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:

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

(i.e., …ï). The result is as follows:

Page 2-15

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 D soft menu key, resulting in:

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:

Page 2-16

Then, press the @EVAL D 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:

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:

Page 2-17

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:

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:

Page 2-18

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

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

2/3 by using: šƒƒ2/3

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

Page 2-19

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:













∆⋅+

⋅+

−

3/1

LNe

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, ∆y (~‚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.

Page 2-20

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 `.

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.

Page 2-21

• 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

2. θ

3. ∆y

4. µ

5. 2

6. x

7. µ in the exponential function

8. λ

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:

Page 2-22

™ ~‚2 Enters the factorial for the 3 in the square root

(entering the factorial changes the cursor to the selection cursor)

˜˜™™ Selects the µ in the exponential function

/3*~‚f Modifies exponential function argument

™™™™ Selects ∆y

R Places a square root symbol on ∆y

(this operation also changes the cursor to the selection cursor)

˜˜ ™—— S Select θ

1/3

and enter the SIN function

The resulting screen is the following:

Evaluating a sub-expression

Since we already have the sub-expression

(

)

3/1

θSIN highlighted, let’s press

the @EVAL D soft menu key to evaluate this sub-expression. The result is:

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:

Page 2-23

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 C 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

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 C soft menu key to get the screen to look as in the previous

figure (see above). Now, press the @SIMP C 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,

θ ,

transformed into

)(θLN

e . 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.

Page 2-24

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

Press ‚¯to recover the original expression. Next, enter the following

keystrokes: ˜˜˜™™™™™™™———‚™ to select

the last two terms in the expression, i.e.,

press the 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

@FACTO

Page 2-25

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:

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 (F), to get:

Page 2-26

Next, press the L key to recover the original Equation Writer menu, and

press the @EVAL@ soft menu key (D) 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 (F), to get

information on the command DERVX:

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):

Page 2-27

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⋅∆y 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:

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⋅∆y 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):

Page 2-28

To select the sub-expression of interest, use:

™™™™™™™™‚¢

™™™™™™™™™™‚¤

The screen shows the required sub-expression highlighted:

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 C 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:

∑

∞

Page 2-29

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:

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

∑

∞

You can use the Equation Writer to prove that

+∞=

∑

∞

This summation (representing an infinite series) is said to diverge.

Double summations are also possible, for example:

Page 2-30

Derivatives

We will use the Equation Writer to enter the following derivative:

)(

δβα +⋅+⋅ tt

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 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

Page 2-31

βαδβα +⋅=+⋅−⋅ ttt

2)(

Second order derivatives are possible, for example:

which evaluates to:

Note: The notation

()

x∂

∂

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:

∫

⋅⋅

)sin( dttt . Press ‚O to activate the Equation Writer.

Then 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:

Page 2-32

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

)cos()sin()sin(

τττ

⋅−=⋅⋅

∫

dttt

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:

Page 2-33

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:

@CHDIR (A): Change to selected directory

@CANCL (E): Cancel action

@@OK@@ (F): Approve a selection

For example, to change directory to the CASDIR, press the down-arrow key,

˜, and press @CHDIR (A). 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

Page 2-34

@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

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

Page 2-35

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:

This time the CASDIR is highlighted in the screen. To see the contents of the

directory press the @@OK@@ (F) 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

represents a graphic

variable.

• 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,

Page 2-36

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:

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 apostrophe 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

Page 2-37

variable, but one created by a previous exercise

CASINFO a graph that provides CAS information

MODULO Modulo for modular arithmetic (default = 13)

REALASSUME List of variable names assumed as real values

PERIOD Period for trigonometric functions (default = 2π)

VX Name of default independent variable (default = X)

EPS Value of small increment (epsilon), (default = 10

-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 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’

Page 2-38

³~~math`

³~~m„a„t„h`

³~~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.

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@@ (F) soft menu key. The screen may look like this:

Page 2-39

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@@ (C) . This will

produce the following input form:

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 (C) 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.

@CHK@

Page 2-40

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):

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 MANS}. 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.,

Page 2-42

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:

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

Page 2-43

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:

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@ (A) Proceed with deleting the sub-directory (or variable)

@ALL@ (B) Proceed with deleting all sub-directories (or variables)

!ABORT (E) Do not delete sub-directory (or variable) from a list

@@NO@@ (F) 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 (E).

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

Page 2-47

sub-directory {HOME MANS INTRO}, created in an earlier example, we

want to store the following variables with the values shown:

Name Contents Type

A 12.5 real

-0.25 real

A12 3×10

real

Q ‘r/(m+r)' algebraic

R [3,2,1] vector

z1 3+5i complex

p1 << → r 'π*r^2' >> 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 MANS 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 (A) soft menu key to see the contents in a graphical

format.

• Press the @TEXT (A) 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:

Using the STO command

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:

Page 2-54

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 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: „î K @@@z1@@ `. To check the new

contents of z1, use: ‚@@@z1@@

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

Page 2-55

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:

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:

Page 2-56

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:

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@@ `

Page 2-57

‚@@ @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.

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@@ Select MEMORY from the programming menu

˜˜˜˜ @@OK@@ Select DIRECTORY from the MEMORY menu

—— @@OK@@ Select ORDER from the DIRECTORY menu

The screen will show the following input line:

Page 2-58

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@@ `

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@@ Select MEMORY from the programming menu

˜˜˜˜ @@OK@@ Select DIRECTORY from the MEMORY menu

—— @@OK@@ 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 3-2

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)

R: real numbers

C: 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. Real number calculations will be demonstrated in both the

Algebraic (ALG) and Reverse Polish Notation (RPN) modes.

Page 3-3

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 /

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 -

Page 3-4

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

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`

Page 3-5

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

Using powers of 10 in entering data

Powers of ten, i.e., numbers of the form -4.5×10

-2

, etc., are entered by using

the V key. For example, in ALG mode:

\4.5V\2`

Page 3-6

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 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`

Page 3-7

„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 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:

Page 3-8

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

Page 3-9

example, to select option 4. HYPERBOLIC.. in the MTH menu, simply

press 4.

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:

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

-1

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

5 @@OK@@ Select the 5. TANH function

2.5` Evaluate tanh(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

5 @@OK@@ Select the 5. TANH function

Page 3-10

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@.

Note: To see additional options in these soft menus, press the L key or the

„«keystroke sequence.

Page 3-16

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: the base of natural logarithms.

• i: the imaginary unit, i

i 2

= -1.

• 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:

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.

Page 3-17

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:

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

Page 3-18

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:

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:

Page 3-19

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)

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)

Page 3-20

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 )

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)

Page 3-21

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: ‚N~„g

lbmol: ‚N~„l

rpm: ‚N~„r

dB: ‚N~„d

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

@@OK@@ Select the TOOLS menu

˜ @@OK@@ Select the UBASE function

1 ‚Ý Enter 1 and underline

‚Û Select the UNITS menu

— @@OK@@ Select the VISCOSITY option

@@OK@@ Select the UNITS menu

` Convert the units

Page 3-22

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:

1 Enter 1 (no underline)

‚Û Select the UNITS menu

— @@OK@@ Select the VISCOSITY option

@@OK@@ Select the unit P (poise)

‚Û Select the UNITS menu

@@OK@@ Select the TOOLS menu

˜ @@OK@@ Select the UBASE function

In ALG mode, system flag 117 set to SOFT menus:

‚Û Select the UNITS menu

)@TOOLS Select the TOOLS menu

@UBASE Select the UBASE function

1 ‚Ý Enter 1 and underline

‚Û Select the UNITS menu

„« @)VISC Select the VISCOSITY option

@@@P@@ Select the unit P (poise)

` Convert the units

In RPN mode, system flag 117 set to SOFT menus:

1 Enter 1 (no underline)

‚Û Select the UNITS menu

„« @)VISC Select the VISCOSITY option

@@@P@@ Select the unit P (poise)

‚Û Select the UNITS menu

)@TOOLS Select the TOOLS menu

@UBASE Select the UBASE function

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.

Page 3-23

Here is the sequence of steps to enter this number in ALG mode, system flag

117 set to CHOOSE boxes:

5‚Ý Enter number and underscore

‚Û Access the UNITS menu

8@@OK@@ Select units of force (8. Force..)

@@OK@@ 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:

5 Enter number (do not enter underscore)

‚Û Access the UNITS menu

8@@OK@@ Select units of force (8. Force..)

@@OK@@ Select Newtons (N)

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

Page 3-24

‚Û Access the UNITS menu

L @)@FORCE Select units of force

@ @@N@@ Select Newtons (N)

` Enter quantity with units in the stack

The same quantity, entered in RPN mode uses the following keystrokes:

5 Enter number (no underscore)

‚Û Access the UNITS menu

L @)@FORCE Select units of 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 Prefix Name x

____________________________________________________

Y yotta +24 d deci -1

Z zetta +21 c centi -2

E exa +18 m milli -3

P peta +15 µ micro -6

T tera +12 n nano -9

G giga +9 p pico -12

M mega +6 f femto -15

k,K kilo +3 a atto -18

h,H hecto +2 z zepto -21

D(*) deka +1 y yocto -24

_____________________________________________________

(*) In the SI system, this prefix is da rather than D. Use D for deka in the

calculator, however.

Page 3-25

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)

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).

Page 3-26

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) `:

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:

Page 3-27

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): convert unit object x to SI units

UVAL(x): extract the value from unit object x

UFACT(x,y): factors a unit y from unit object x

UNIT(x,y): 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, in your favorite calculator settings. The output

shown below was developed in ALG mode with system flat 117 set to SOFT

menu:

Examples of CONVERT

Page 4-3

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 result:

For this result 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

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 =

5.2e

1.5i

, 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…).

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:

Page 4-4

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

= -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

- y

) +

i (x

+ x

The division of two complex numbers is accomplished by multiplying both

numerator and denominator by the complex conjugate of the denominator,

i.e.,

2112

2121

yxyx

yyxx

iyx

−

⋅+

−

⋅

Thus, the inverse function INV (activated with the Y key) is defined as

2222

iyx

iyxiyx

⋅+

−

⋅

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

Page 4-5

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.

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:

Page 4-6

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).

This first screen shows functions RE, IM, and CR. 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 RC, ABS, and ARG. Notice that the

ABS function gets translated to |3.+5.i|, the notation of the absolute value.

Page 4-7

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:

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.

Page 4-8

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).

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:

Page 4-9

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

+iy

, and returns the equation of the straight line, say, y = a+bx, that

contains the points (x

) and (x

). For example, the line between points

A(5,-3) and B(6,2) can be found as follows (example in Algebraic mode):

Function DROITE is found in the command catalog (‚N).

Using EVAL(ANS(1)) simplifies the result to:

Page 5-1

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):

Page 5-4

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:

The help facility 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):

Thus, we leave for the user to explore the applications of the functions in the

ALG (or ALGB) menu. This is a list of the commands:

Page 5-29

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 5-30

LIN LNCOLLECT

POWEREXPAND SIMPLIFY

Page 6-1

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 6-2

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:

Page 6-3

The following examples show the use of function SOLVE in ALG and RPN

modes:

The screen shot shown above displays two solutions. In the first one, β

-5β

=125, SOLVE produces no solutions { }. In the second one, β

- 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:

Page 6-4

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:

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:

Page 6-5

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:

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.

Page 6-6

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.

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

+ a

n-1

+ …+ a

x +

= 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

- s + 1 = 0.

We want to place the coefficients of the equation in a vector [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@@ Enter vector of coefficients

@SOLVE@ Solve equation

The screen will show the solution as follows:

Page 6-7

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:

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@@ Select Solve poly…

˜„Ô1‚í5

‚í2\‚í 4@@OK@@ Enter vector of roots

@SOLVE@ Solve for coefficients

Page 6-8

Press ` to return to stack, the coefficients will be shown in the stack.

Press ˜ to trigger the line editor to see all the coefficients.

Note: If you want to get a polynomial with real coefficients, but having

complex 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 resulting 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 Enter vector of coefficients

‚í2\‚í 4@@OK@@

—@SYMB@ Generate symbolic expression

` Return to stack.

The expression thus generated is shown in the stack as:

'X^3+5*X^2+-2*X+4'.

Page 6-9

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…

˜„Ô1‚í3 Enter vector of roots

‚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)'.

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…

˜„Ô1‚í3 Enter vector of roots

‚í2\‚í 1@@OK@@

@SOLVE@ Solve for coefficients

˜@SYMB@ Generate symbolic expression

` Return to stack.

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.

Page 6-10

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.

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

60 @@OK@@ Enter n = 60

6.5 @@OK@@ Enter I%YR = 6.5 %

2000000 @@OK@@ Enter PV = 2,000,000 US$

˜ Skip PMT, since we will be solving for it

0 @@OK@@ Enter FV = 0, the option

End is highlighted

— š @@SOLVE! Highlight PMT and solve for it

The solution screen will look like this:

Page 6-11

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

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:

Page 6-12

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

60 @@OK@@ Enter n = 60

6.5 @@OK@@ Enter I%YR = 6.5 %

2000000 @@OK@@ Enter PV = 2,000,000 US$

˜ Skip PMT, since we will be solving for it

0 @@OK@@ Enter FV = 0, the option

End is highlighted

@@CHOOS !—@@OK@@ Change payment option to Begin

— š @@SOLVE! Highlight PMT and solve for it

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.

Page 6-13

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 Enter PURGE, prepare list of variables

³‚@@@n@@ Enter name of variable N

™ ‚í Enter a comma

³ ‚@I©YR@ Enter name of variable I%YR

™ ‚í Enter a comma

³ ‚@@PV@@ Enter name of variable PV

™ ‚í Enter a comma

³ ‚@@PMT@@ Enter name of variable PMT

™ ‚í Enter a comma

³ ‚@@PYR@@ Enter name of variable PYR

™ ‚í Enter a comma

³ ‚@@FV@@. 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:

Page 6-14

J „ä Prepare a list of variables to be purged

@@@n@@ Enter name of variable N

@I©YR@ Enter name of variable I%YR

@@PV@@ Enter name of variable PV

@@PMT@@ Enter name of variable PMT

@@PYR@@ Enter name of variable PYR

@@FV@@ Enter name of variable FV

` Enter list of variables in stack

I@PURGE Purge variables in list

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) = 0.

Simply enter the expression as an algebraic object and store it into variable

EQ. The required keystrokes in ALG mode are the following:

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.

Page 6-15

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:

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.

Page 6-16

• 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













zzzyzx

yzyyyx

xzxyxx

σσσ

The equation is

,)]([

zzyyxxxx

∆⋅++⋅−= ασσσ here e

is the unit

strain in the x-direction, σ

, σ

, and σ

, are the normal stresses on the

particle 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 ∆T 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, ∆T = 60

To calculate the strain e

use the following:

‚Ï@@OK@@ Access numerical solver to solve equations

‚O Access the equation writer to enter equation

Page 6-17

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.,

is translated as ex, etc. -- this is done to save typing time):

Use the following shortcuts for special characters:

σ: ~‚s α: ~‚a ∆: ~‚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:

With the ex: field highlighted, press @SOLVE@ to solve for ex:

Page 6-18

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 ∆T: field (with ∆T = 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, ∆T, α, σz, σy, n, σx, and E.

Example 2 – Specific energy in open channel flow

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

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.

yE +=

Page 6-19

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

• Solve for y.

Page 6-20

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 more:

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.

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

Page 6-21

written as

⋅⋅= . The quantity f is known as the friction factor of

the flow and 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 6-22

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:

Page 6-23

Thus, the equation we are solving, after combining the different variables in

the directory, is:













⋅=

DARCY

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

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 6-24

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

distance r is given by the equation

⋅

⋅=

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 6-25

Solve for F, and press to return to normal calculator display. The solution is F :

6.67259E-15_N, or F = 6.67259×10

-15

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:

Type an equation, say X^2 - 125 = 0, directly on the stack, and press @@@OK@@@ .

Page 6-26

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 6-27

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).

Page 6-28

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:

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 ].

Page 6-29

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

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:

Page 6-30

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.

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:

Page 6-31

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 independent variable. In other words, if the polynomial is a

+ a

n-1

+ … + a

+ a

x + a

, the vector of coefficients should be entered as [a

, a

, …

, a

]. For example, the roots of the polynomial whose

coefficients are [1, -5, 6] are [2, 3].

Function PCOEF

This function produces the coefficients [a

, a

n-1

, …

, a

] of a

polynomial a

+ a

n-1

+ … + a

+ a

x + a

, given a vector of its roots

, 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

, …

, a

], and a value x

, i.e., PEVAL calculates a

+ a

n-1

+ …

+ a

. For example, for coefficients [2, 3, -1, 2] and a value of

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:

Page 6-32

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:

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:

Page 6-33

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 7-1

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.

Page 7-2

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@@@ .

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 OBJ, 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

θθ

, respectively, are given by

)(

222

abr

PPba

PbPa

oioi

−⋅

−⋅⋅

−

⋅−⋅

θθ

)(

222

abr

PPba

PbPa

oioi

−⋅

−⋅⋅

−

⋅−⋅

=σ

Page 8-7

%({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.,

%({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. The screen also

shows that the resulting complex-number list is stored into variable L5:

Functions such as LN, EXP, SQ, etc., can also be applied to a list of complex

numbers, e.g.,

Page 8-8

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:

Page 8-9

Next, with system flag 117 set to SOFT menus:

This menu contains the following functions:

∆LIST : Calculate increment among consecutive elements in list

ΣLIST : Calculate summation of elements in the list

ΠLIST : Calculate product of elements in the list

SORT : 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:

Page 8-10

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 8-11

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 8-12

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 function f

or the 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}:

The following call to function MAP uses a program instead of a function as

second argument:

Page 8-13

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@@@,

to replace the plus sign (+) with ADD:

Page 8-14

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 ADD 3)*Y') :

You can also define the function as G(X,Y) = (X--3)*Y.

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

}. Suppose that the sample of interest is the list

{1, 5, 3, 1, 2, 1, 3, 4, 2, 1}

Page 8-15

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













+++

∑

sssn

1111

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 8-16

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

xxxxx L

⋅==

∏

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

Thus, the geometric mean of list S is s

= 1.003203…

Page 8-18

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

Class Frequency

Class mark count

boundaries s

0 - 2 1 5

2 - 4 3 12

4 - 6 5 18

6 - 8 7 1

8 -10 9 3

The class mark data can be stored in variable S, while the frequency count

can be stored in variable W, as follows:

Page 8-19

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:

∑

⋅

where

∑

represents the total frequency count.

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

ssw

∑

−⋅

)()(

Page 8-20

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 9-1

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:

]2,5,3,1[,

−=













−

= uv

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

, as A = A

i + A

j + A

Alternative notation for this vector are: A = [A

, A

], A = (A

, A

), or 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

] or A = [A

, A

], to refer to two- and three-dimensional

vectors from now on. The magnitude of a vector A is defined as |A| =

222

zyx

AAA ++ . 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

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

]. 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

Page 9-2

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),

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

, and A×B = [A

-A

]. The angle between two vectors can be found from the definition of the

dot product as cos(θ) = A•B/|A||B|= e

•e

. Thus, if two vectors A and B are

perpendicular (θ = 90

= π/2

rad

), 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] A general row vector

[1.5,-2.2] 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:

Page 9-3

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.

Storing vectors into variables

Vectors can be stored into variables. The screen shots below show the vectors

= [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! @VEC

 ←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 9-14

„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

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.

Page 9-15

The conversion from Cartesian to cylindrical coordinates is such that r =

)

1/2

, θ = tan

-1

(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) 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:

Page 9-16

Notice that the vectors that were written in cylindrical polar coordinates have

now been changed to the spherical coordinate system. The transformation is

such that ρ = (r

)

1/2

, θ = θ, and φ = tan

-1

(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,

= -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

= (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

-1

(A•B/|A||B|)

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):

Page 9-17

Thus, the result is θ = 122.891

. In RPN mode use the following:

[3,-5,6] ` [2,1,-3] ` DOT

[3,-5,6] ` ABS [2,1,-3] ` ABS *

/ ACOS 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:

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 angle as, θ = sin

-1

(|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:

Page 9-18

Thus the angle between vectors r and F is θ = 41.038

. RPN mode, we can

use: [3,-5,4] ` [2,5,-6] ` CROSS ABS [3,-5,4] `

ABS [2,5,-6] ` ABS * / ASIN NUM

Equation of a plane in space

Given a point in space P

) 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:

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:

Page 9-19

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 EXPAND

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:

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 OBJ, 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 OBJ,

ARRY, and LIST will be accessible by using „° @)TYPE!. Functions

Page 9-20

OBJ, 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 OBJ, LIST, ARRY, and

DROP with some examples.

Function OBJ

This function decomposes an object into its components. If the argument is a

list, function OBJ will list the list elements in the stack, with the number of

elements in stack level 1, for example: {1,2,3} ` „°@)TYPE! @OBJ@

results in:

When function OBJ 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! @OBJ@ results in:

If we now apply function OBJ 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:, …,

Page 9-21

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 OBJ

2 - Press 1+ to transform the list in stack level 1: from {3} to {3,1}

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! @OBJ@ 1 + !ARRY@

`³~~rxc` K

Page 9-22

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:

<< OBJ 1 + ARRY >>

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 OBJ to decompose the column vector

2 - Use function OBJ to decompose the list in stack level 1:

Page 9-23

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! @OBJ@ @OBJ@

„°@)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:

<< OBJ OBJ DROP ARRY >>

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

Page 9-24

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 OBJ 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! @OBJ@ 1 !LIST@ !ARRY@ `

³~~lxv ` K

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:

<< OBJ 1 LIST ARRY >>

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@@.

Page 9-25

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] ` AXL. The following screen shot shows the application

of function AXL to the same vector in ALG mode.

Page 10-1

Chapter 10

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

shown next:

.][

22221

11211













nmnn

mnij

aaa

OMM

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

]

. The main diagonal of a square matrix is the collection

of elements a

. An identity matrix, I

, is a square matrix whose 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













100

010

001

An identity matrix can be written as I

= [δ

], where δ

is a function known

as Kronecker’s delta

, and defined as







≠

jiif

Page 10-2

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:

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:

5.01.02

8.29.13.0

0.22.45.2













−

Page 10-3

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 10-4

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 10-5

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 10-6

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 A(2,3) and

pressing `. In RPN mode, this exercise is performed by entering @@@A@@@ `

3 ` GET, or by using A(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: „ì³A(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 10-7

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 1.

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 .

Page 10-23

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 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 ROW 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:

ROW([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 ROW places the

vectors as rows in the resulting matrix. The following figure shows the RPN

stack before and after using function ROW.

Page 10-24

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+.

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-.

Page 10-25

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 CSWP 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

CSWP to matrix A in order to swap rows 2 and 3:

As you can see, the columns 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.

Page 10-26

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:

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 11-4

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.

Matrix multiplication

Matrix multiplication is defined by C

= A

⋅B

, where A = [a

]

, B =

]

, and C = [c

]

. Notice that matrix multiplication is only possible if the

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

.,,2,1;,,2,1,

njmiforbac

kjikij

KK ==⋅=

∑

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.

Page 11-5

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

(another row vector). For the calculator to identify a row vector, you must use

double brackets to enter it, e.g.,

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:

The identity matrix

In Chapter 9 we introduce the identity matrix as the matrix I = [δ

]

, where δ

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:

Page 11-6

The inverse matrix

The inverse of a square matrix A is the matrix A

-1

such that A⋅A

-1

= A

-1

⋅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:

To verify the properties of the inverse matrix, we present 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:

Page 11-7

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.

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

∑∑

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

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

i+1

, for i = 1, 2, …, n-1. The columns [u

] of U and [v

] of

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. Examples,

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.,

cond(A) = ||A||×||A

-1

||. We will choose as the matrix norm, ||A||, the

maximum of its row norm (RNRM) and column norm (CNRM), while the norm

of the inverse, ||A

-1

||, will be selected as the minimum of its row norm and

column norm. Thus, ||A|| = max(RNRM(A),CNRM(A)), and ||A

-1

|| =

min(RNRM(A

-1

), CNRM(A

-1

)).

Page 11-10

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).

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

], where c

(i = 1, 2, …, n) are vectors

representing the columns of the matrix A, then, if any of those columns, say c

can be written as ,

},...,2,1{,

∑

∈≠

⋅=

njkj

jjk

d cc

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.,

333231

232221

131211

2221

1211

aaa

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,

21122211

2221

1211

aaaa

⋅−⋅=

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

∑

atr

)(A .

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 , 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

Page 11-15

Function AXL

Function AXL converts an array (matrix) into a list, and vice versa. For

examples,

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. For example,

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

1,m-1

⋅x

m-1

+ a

1,m

⋅x

= b

⋅x

+ a

⋅x

+ a

⋅x

+ …+ a

2,m-1

⋅x

m-1

+ a

2,m

⋅x

= b

⋅x

+ a

⋅x

+ a

⋅x

+ …+ a

3,m-1

⋅x

m-1

+ a

3,m

⋅x

= b

. . . … . . .

n-1,1

⋅x

+ a

n-1,2

⋅x

+ a

n-1,3

⋅x

+ …+ a

n-1,m-1

⋅x

m-1

+ a

n-1,m

⋅x

= b

n-1

⋅x

+ a

⋅x

+ a

⋅x

+ …+ a

n,m-1

⋅x

m-1

+ a

n,m

⋅x

= b

This system of linear equations can be written as a matrix equation, A

⋅x

= b

, if we define the following matrix and vectors:

nmnn

aaa













MOMM

22221

11211

























Page 11-17

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 provide (right):

To solve the linear system A⋅x = b, enter the matrix A, in the format [[ a

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

+ 3x

–5x

= 13,

– 3x

+ 8x

= -13,

– 2x

+ 4x

= -6,

can be written as the matrix equation A⋅x = b, if

422

831

532













−

−=

























−

= bxA and

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

) represented by the three equations.

Page 12-1

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 12-2

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,

)

exp(

)(

xf −=

• 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

Page 12-40

• 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.

• Keep 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.

• Press @EDITL @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.

Page 12-41

• 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 =

-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@@@.

• 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

• 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.

Page 12-42

• 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:

• 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.

Page 12-43

• 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)) i.e., F(z) = sin(z) (2)(X,Y)^2 i.e., F(z) = z

(3) EXP((X,Y)) i.e., F(z) = e

(4) SINH((X,Y)) i.e., F(z) = sinh(z)

(5) TAN((X,Y)) i.e., F(z) = tan(z) (6) ATAN((X,Y)) i.e., F(z) = tan

-1

(z)

(7) (X,Y)^3 i.e., F(z) = z

(8) 1/(X,Y) i.e., F(z) = 1/z

(9) √ (X,Y) i.e., F(z) = z

1/2

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.

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 @EDITL @LABEL @MENU to see the graph with labels and ranges.

Page 12-46

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

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.

Page 12-47

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.

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.

Page 12-48

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

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.

PICT

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.

X,Y

This command copies the coordinates of the current cursor position, in user

coordinates, in the stack.

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.

Page 12-49

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.

You can always return to the very last zoom window by using @ZLAST.

Page 12-50

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.

Page 13-7

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.

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

Page 13-8

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 TYPE to 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 13-9

• 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 13-10

This result indicates that the range of the function

)(

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:

• Level 3: the function f(VX)

Page 13-11

• 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 13-22

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.

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,

Page 13-23

Infinite series

An infinite series has the form

axnh )()(

1,0

−

∑

∞

. The infinite series typically

starts with indices n = 0 or n = 1. Each term in the series has a coefficient

h(n) that depends on the index n.

Taylor and Maclaurin’s series

A function f(x) can be expanded into an infinite series around a point x=x

using a Taylor’s series, namely,

∑

∞

−⋅=

)(

where f

(n)

(x) represents the n-th derivative of f(x) with respect to x, f

(0)

(x) = f(x).

If the value x

is zero, the series is referred to as a Maclaurin’s series, i.e.,

∑

∞

⋅=

)(

)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

Page 13-24

∑∑

∞

+==

−⋅+−⋅=

)(

i.e.,

).()()( xRxPxf

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

)1(

)(

⋅=

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

reference to h, i.e., we say that R

(x) has an error of order h

n+1

, or R ≈ O(h

k+1

If h is a small number, say, h<<1, then h

k+1

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,

Page 13-25

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

increment h. The list returned as the first output object includes the following

items:

1 - Bi-directional limit of the function at point of expansion, i.e.,

)(lim xf

ax→

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:

Page 13-26

In the right-hand side figure above, we are using the line editor to see the

series expansion in detail.

Page 14-1

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)

= (x

)

1/2

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 14-2

yxfyhxf

),(),(

lim

−+

∂

→

Similarly,

yxfkyxf

),(),(

lim

−+

∂

→

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 h0. 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,

() ()

)sin()cos(),cos()cos( yxyx

yyx

−=

∂

which are the same results as found with the limits calculated earlier.

Consider another example,

()

xyyxyyx

202

=+=+

∂

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 14-3

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













∂













∂













∂

∂∂

∂













∂

∂∂

∂

xyx

yxy

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

∂∂

∂

∂∂

∂

Page 14-4

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

∂

⋅

∂

⋅

∂

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 14-5

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:

∂

⋅

∂

⋅

∂

⋅

∂

⋅

∂

Determining extrema in functions of two variables

In order for the function z = f(x,y) to have an extreme point (extrema) at (x

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

) are ∂f/∂x = 0, ∂f/∂y = 0, and ∆ = (∂

f/∂x

)⋅ (∂

f/∂y

)-[∂

f/∂x∂y]

> 0. The point (x

) is a relative maximum if ∂

f/∂x

< 0, or a relative

minimum if ∂

f/∂x

> 0. The value ∆ is referred to as the discriminant.

If ∆ = (∂

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 16-4

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:

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

ILAP: Inverse LAPlace transform, L

-1

[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.

Page 16-5

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

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. 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

y = K

⋅e

–3x

+ 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

Page 16-6

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:

y/dx

-4⋅(d

y/dx

)-11⋅(dy/dx)+30⋅y = x

Enter:

'X^2' ` 'X^3-4*X^2-11*X+30' ` LDEC

The solution, shown partially here in the Equation Writer, is:

Replacing the combination of constants accompanying the exponential terms

with simpler values, the expression can be simplified to y = K

⋅e

–3x

+ 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

homogeneous equation, i.e., y

= K

⋅e

–3x

+ 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 EVAL

Page 16-7

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:

’(t) + 2x

’(t) = 0,

’(t) + x

’(t) = 0.

In algebraic form, this is written as: A⋅x’(t) = 0, where













. The

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:

Page 16-12

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

}sin{)(

+⋅−

=⋅=

tesF

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)

= L

-1

{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

interpreted as L

-1

{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

). The transform is

interpreted as follows: L {cos(a⋅t+b)} = (s⋅cos b – a⋅sin b)/(s

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

Page 16-13

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|

t=0

, then L{d

f/dt

} = s

⋅F(s) - s⋅f

– (df/dt)

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|

t=0

, then the Laplace

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

Let f

(k)

= d

f/dx

t = 0

, and f

= f(0), then

L{d

f/dt

} = s

⋅F(s) – s

n-1

⋅f

−…– s⋅f

(n-2)

– f

(n-1)

• 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

F/ds

= L{(-t)

⋅f(t)}.

Example 3 – Let f(t) = e

–at

, 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

F/ds

= -6/(s

+4⋅a⋅s

+6⋅a

⋅s

+4⋅a

⋅s+a

Page 16-14

Now, use ‘(-X)^3*EXP(-a*X)’ ` LAP µ. The result is exactly the same.

• Integration theorem

. Let F(s) = L{f(t)}, then

• Convolution theorem

. Let F(s) = L{f(t)} and G(s) = L{g(t)}, then

{

}

==−

∫

)})(*{()()(

tgfduutguf

)()()}({)}({ sGsFtgtf ⋅=⋅ LL

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)}

= 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

L{f(t-a)}=e

–as

⋅L{f(t)} = e

–as

⋅F(s).

• Shift theorem for a shift to the left

. Let F(s) = L{f(t)}, and a >0, then

• Similarity theorem

. Let F(s) = L{f(t)}, and a>0, then L{f(a⋅t)} =

(1/a)⋅F(s/a).

• Damping theorem

. Let F(s) = L{f(t)}, then L{e

–bt

⋅f(t)} = F(s+b).

• Division theorem

. Let F(s) = L{f(t)}, then

{

}

).(

)(

duuf

⋅=

∫

.)()()}({













⋅⋅−⋅=+

∫

−

stas

dtetfsFeatfL

Page 16-15

• Laplace transform of a periodic function of period T

• Limit theorem for the initial value

: Let F(s) = L{f(t)}, then

• Limit theorem for the final value

: Let F(s) = L{f(t)}, then

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

Also, if f(x) is a continuous function, then

∫

∞

∞−

=− ).()()(

xfdxxxxf δ

∫

∞













duuF

.)(

)(

∫

⋅⋅⋅

−

dtetf

.)(

)}({L

)].([lim)(lim

sFstff

⋅==

∞→→

)].([lim)(lim

sFstff

⋅==

→∞→

∞

∫

∞

∞−

= .0.1)( dxxδ

Page 16-16

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







0,0

0,1

)(

Also, for a continuous function f(x),

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.

You can prove that L{H(t)} = 1/s,

from which it follows that L{U

⋅H(t)} = U

/s,

where U

is a constant. Also, L

-1

{1/s}=H(t),

and L

-1

{ U

/s}= U

⋅H(t).

Also, using the shift theorem for a shift to the right, L{f(t-a)}=e

–as

⋅L{f(t)} =

–as

⋅F(s), we can write L{H(t-k)}=e

–ks

⋅L{H(t)} = e

–ks

⋅(1/s) = (1/s)⋅e

–ks

∫∫

∞

∞−

∞

=−

.)()()(

dxxfdxxxHxf

Page 16-17

Another important result, known as the second shift theorem for a shift to the

right, is that L

-1

–as

⋅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,

because from L

-1

{1.0}= δ(t), it follows that L{δ(t)} = 1.0

Also, using the shift theorem for a shift to the right, L{f(t-a)}=e

–as

⋅L{f(t)} =

–as

⋅F(s), we can write L{δ(t-k)}=e

–ks

⋅L{δ(t)} = e

–ks

⋅1.0 = e

–ks

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,

L{d

f/dt

} = s

⋅F(s) – s

n-1

⋅f

−…– s⋅f

(n-2)

– f

(n-1)

are particularly useful in transforming an ODE into an algebraic equation.

Page 16-18

Example 1 – To solve the first order equation,

dh/dt + k⋅h(t) = a⋅e

–t

by using Laplace transforms, we can write:

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

–t

}=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 s⋅H(s)-h

+k⋅H(s) = a/(s+1).

Use the calculator to solve for H(s), by writing:

‘X*H-h0+k*H=a/(X+1)’ ` ‘H’ ISOL

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:

OBJ ƒ ƒµ Isolates right-hand side of last expression

ILAP Obtains the inverse Laplace transform

The result is

. Replacing X with t in this expression

and simplifying, results in h(t) = a/(k-1)⋅e

-t

+((k-1)⋅h

-a)/(k-1)⋅e

-kt

Check what the solution to the ODE would be if you use the function LDEC:

‘a*EXP(-X)’ ` ‘X+k’ ` LDEC µ

Page 16-19

The result is:

, i.e.,

h(t) = a/(k-1)⋅e

-t

+((k-1)⋅cC

-a)/(k-1)⋅e

-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),

representing the initial conditions f(0), f’(0), f”(0), …, f

(n-1)

(0).

Example 2

– Use Laplace transforms to solve the second-order linear equation,

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}.

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

⋅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)’.

Page 16-20

To find the solution to the ODE, y(t), we need to use the inverse Laplace

transform, as follows:

OBJ ƒ ƒ Isolates right-hand side of last expression

ILAPµ 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.

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

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)},

Page 16-37

Press `` to copy this result to the screen. Then, reactivate the Equation

Writer to calculate the second integral defining the coefficient c

, namely,

Once again, replacing e

= (-1)

, and using e

2in

= 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:

Page 16-38

Once again, replacing e

= (-1)

, results in

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:

Page 16-39

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):

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











43,0

31,1

10,0

)(

xif

Page 16-40

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

5.01













⋅⋅=

∫

c ,

and

We can simplify this expression by using e

= i

and e

3in

= (-i)

to get:

Page 16-41

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:

Page 16-42

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):

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:

Page 16-43

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:

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

Page 16-44

integration of the form

∫

⋅⋅=

dttftssF .)(),()( κ

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

()

,sincos)(

∑

∞

⋅+⋅+=

nnnn

xbxaatf ωω

representing a periodic function with a period T. This Fourier series can be

re-written as

∑

∞

+⋅+=

),cos()(

nnn

xAaxf φϖ where

,tan,

122













=+=

−

nnnn

baA φ

for n =1,2, …

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

∑

∞

+⋅+=

).cos()(

nnn

xAaxf φω

()

∑

∞

⋅+⋅+=

sincos

nnnn

xbxaa ωω

Page 16-57

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

(1)

(x) = J

(x)+i⋅Y

(x), and H

(2)

(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 functions of the first kind of order ν defined as I

(x)= i

⋅J

(i⋅x), where i is

the unit imaginary number. These functions are solutions to the differential

equation x

⋅(d

y/dx

) + x⋅ (dy/dx)- (x

+ν

) ⋅y = 0.

The modified Bessel functions of the second kind,

(x) = (π/2)⋅[I

(x)−I

(x)]/sin νπ,

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

The functions T

(x) = cos(n⋅cos

-1

x), and U

(x) = sin[(n+1) cos

-1

x]/(1-x

)

1/2

, 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

Page 16-58

(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, i.e., T

(x) = 1.0.

-0 TCHEBYCHEFF, result: 1, i.e., U

(x) = 1.0.

1 TCHEBYCHEFF, result: ‘X’, i.e., T

(x) = x.

-1 TCHEBYCHEFF, result: 1, i.e., U

(x) =1.0.

2 TCHEBYCHEFF, result: ‘2*X^2-1, i.e., T

(x) =2x

-1.

-2 TCHEBYCHEFF, result: ‘2*X’, i.e., U

(x) =2x.

3 TCHEBYCHEFF, result: ‘4*X^3-3*X’, i.e., T

(x) = 4x

-3x.

-3 TCHEBYCHEFF, result: ‘4*X^2-1’, i.e., U

(x) = 4x

-1.

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

,...2,1,

)(

)(,1)(

⋅

⋅==

−

exd

xLxL

xnnx

are solutions to Laguerre’s equation. Laguerre’s polynomials can also be

calculated with:

)1(

)(

xL ⋅













⋅

−

∑

The term

),(

)!(!

mnC

mnm

−













xn ⋅

−

++−⋅

−

+⋅−=

)1(

.......

)1(

Page 16-59

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:

(x) = .

(x) = 1-x.

(x) = 1-2x+ 0.5x

(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:

,..2,1),()1()(*,1*

=−==

−

exHH

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:

Page 16-60

0 HERMITE, result: 1, i.e., H

= 1.

1 HERMITE, result: ’2*X’, i.e., H

= 2x.

2 HERMITE, result: ’4*X^2-2’, i.e., H

= 4x

-2.

3 HERMITE, result: ’8*X^3-12*X’, 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.

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:

Page 16-61

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:

t v

0.00 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 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…)

@@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):

Page 16-62

0.00 4.000

0.25 3.285

0.50 2.640

0.75 2.066

1.00 1.562

1.25 1.129

1.50 0.766

1.75 0.473

2.00 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 Eq in the TYPE field 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:

• „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

t v

Page 16-63

• 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:

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.

Page 16-64

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 equation is:

⋅−⋅−= 962.175.18

or, 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.













⋅













−−













'962.175.18

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 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]):

Page 16-65

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:

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

(Changes initial value of t to 0.75, and final value of t to 1, solve again for

w(1) = [-0.469 -0.607])

t x x'

0.00 0.00 6.00

0.25

… … …

2.00

Page 16-66

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:

t x x' t x x'

0.00 0.000 6.000 1.25 -0.354 1.281

0.25 0.968 1.368 1.50 0.141 1.362

0.50 0.748 -2.616 1.75 0.227 0.268

0.75 -0.015 -2.859 2.00 0.167 -0.627

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 16-67

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:

Press LL @PICT @CANCL $ to return to normal calculator display.

Page 16-68

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)’ `/.

The result is: ‘

((t+1)*EXP(100*t)+C)/EXP(100*t)’, i.e., y(t) = 1+ t +C⋅e

100t

. 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 16-69

Here we are trying to obtain the value of y(2) given y(0) = 1. With the

Soln:

Final

field 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, the general solution, y(t) = 1+ t +C⋅e

100t

, contains the components ‘t’

and ‘C⋅e

100t

’, which vary at very different rates, except for the cases C=0 or

C≈0 (e.g., for C = 1, t =0.1, C⋅e

100t

=22026).

The calculator’s ODE numerical solver allows for the solution of stiff ODEs by

selecting the option

_Stiff in 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.

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,

Page 16-70

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.

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

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, ∆x, to be used by the function. To run this function you

will prepare your stack as follows:

3: {‘x’, ‘y’, ‘f(x,y)’}

2: { ε ∆x }

1: x

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

final

= f

final

), where f

(x)

represents the solution to the differential equation. The second stack level may

contain only the value of ε, and the step ∆x will be taken as a small default

value. After running function @@RKF@@, the stack will show the lines:

2: {‘x’, ‘y’, ‘f(x,y)’}

1: ε

The value of the solution, y

final

, will be available in variable @@@y@@@. This

function 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

init

at y = y

init

, you can always add these values

to the solution provided by RKF, keeping in mind the following relationship:

Page 16-71

The following screens show the RPN stack before and after applying function

RKF for the differential equation dy/dx = x+y, ε = 0.001, ∆x = 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:

3: {‘x’, ‘y’, ‘f(x,y)’ ‘∂f/∂x’ ‘∂f/∂y’ }

2: { ε ∆x }

1: x

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

final

= f

final

), where f

(x)

represents the solution to the differential equation. The second stack level may

contain only the value of ε, and the step ∆x 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’ }

1: { ε ∆x }

The value of the solution, y

final

, will be available in variable @@@y@@@.

This function can be used to solve so-called “stiff” differential equations.

RKF solution Actual solution

x y x y

0 0 x

init

final

init

+ x

final

init

+ y

final

Page 17-13

)

(

)1()

()

(

)()

(

)(

FNDN

NDN

νν

ννν

−

⋅

−⋅Γ⋅Γ

⋅⋅

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,

∫∫

∞−

∞

≤ℑ−=−==

FPdFFfdFFfFDNUTPF )(1)(1)(),,( νν

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 find the value of x, such that x = F

-1

(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:

Page 17-14

• 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.,

Exponential: Weibull:

For the Gamma and Beta distributions

the expressions to solve will be more

complicated due to the presence of integrals, i.e.,

• Gamma,

∫

−⋅⋅

−

)exp(

)(

αβ

• Beta,

∫

−−

−⋅⋅

Γ⋅Γ

+Γ

dzzzp

)1(

)()(

)(

βα

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

Page 17-15

of the complicated nature of function Y(X), it will take some time before the

graph is produced. Be patient.)

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.,

Page 17-16

For the normal, Student’s t, Chi-square (χ

), 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)

Notice that the second parameter in the UTPN function is σ2, not σ

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,

= 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, σ

, 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.

Page 17-17

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:

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 σ

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 17-18

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 18-1

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:

« OBJ 1 2 LIST ARRY »

Page 18-2

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.

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

Type field 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.

Example 1

-- For the data stored in the previous example, the single-variable

statistics results are the following:

Mean: 2.133, Std Dev: 0.964, Variance: 0.929

Total: 25.6, Maximum: 4.5, Minimum: 1.1

Page 18-3

Definitions

The definitions

used for these quantities are the following:

Suppose that you have a number data points 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,

∑

⋅=

The value labeled

Total obtained 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:

∑

=⋅=

xxxx L

Examples of calculation of these measures, using lists, are available in

Chapter 8.

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

Page 18-4

(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

OBJ 1 + ARRY END OBJ OBJ 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

The variance

(Var) of the sample is defined as

∑

−⋅

−

)(

The standard deviation

(St Dev) of the sample is just the square root of the

variance, i.e., s

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.

Page 18-5

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: Population in the

SINGLE-VARIABLE STATISTICS screen. 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

Population rather than Sample as 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: the matrix containing the data of interest.

Col: the column of ΣDAT that is under scrutiny.

X-Min: the minimum class boundary (default = -6.5).

Bin Count: the number of classes(default = 13).

Bin Width: the uniform width of each class (default = 1).

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

counting the frequency

or number of values corresponding to each class.

(Note: the calculators refers to classes as bins).

Page 18-6

Suppose that the classes, or bins, will be selected by dividing the interval (x

bot

top

), into k = Bin Count classes by selecting a number of class boundaries,

i.e., {xB

, xB

, …, xB

k+1

}, so that class number 1 is limited by xB

-xB

, class

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

i+1

)/2, for i = 1, 2, …, k.

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 = ∆x = (x

max

- x

min

) / k,

and the class boundaries can be calculated as xB

= x

bot

+ (i - 1) * ∆x.

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).

• 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.

Page 18-7

• 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

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

Page 18-8

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

No.

Class Bound. Class

mark.

Frequency Cumulative

i XB

i+1

frequency

< XB

outlier below

range

1 10 20 15 18 18

2 20 30 25 14 32

3 30 40 35 17 49

4 40 50 45 17 66

5 50 60 55 22 88

6 60 70 65 22 110

7 70 80 75 24 134

k = 8 80 90 85 19 153

>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:

« 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.

Page 18-35

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

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.

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

: 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:

Page 18-36

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.

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:

Page 18-37

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 z-statistic:

/σ

µ−

• If n > 30, and σ is known, use z

as above. If σ is not known,

replace s for σ in z

, i.e., use

µ−

• If n < 30, and σ is unknown, use the t-statistic

µ−

, with ν =

n - 1 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

|).

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

Page 18-41

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

Pr[Z> z

] = 1-Φ(z

) = α/2, or Φ(z

) = 1- α/2,

where Φ(z) is the cumulative distribution function (CDF) of the standard normal

distribution (see Chapter 17).

Reject the null hypothesis, H

, if z

, or if z

< - z

In other words, the rejection region is R = { |z

| > z

}, while the

acceptance region is A = {|z

| < z

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

, and H

: p>p

, or if z

< - z

, and H

p<p

Page 18-42

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

, and p

’ = k

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

And the variance of the difference of proportions is estimated from: s

= s

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

)/s

Two-tailed test

If using a two-tailed test we will find the value of z

, from

Pr[Z> z

] = 1-Φ(z

) = α/2, or Φ(z

) = 1- α/2,

where Φ(z) is the cumulative distribution function (CDF) of the standard normal

distribution.

Reject the null hypothesis, H

, if z

, or if z

< - z

In other words, the rejection region is R = { |z

| > z

}, while the

acceptance region is A = {|z

| < z

One-tailed test

If using a one-tailed test we will find the value of z

, from

Page 18-43

Pr[Z> z

] = 1-Φ(z

) = α, or Φ(z

) = 1- α,

Reject the null hypothesis, H

, if z

, and H

: p

-p

> p

, or if z

< - z

, and

: 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:

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

, 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.

Page 18-44

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:

Then, we reject H

: µ = 150, against H

: µ ≠ 150. The test z value is z

5.656854. The P-value is 1.54×10

-8

. The critical values of ±z

±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:

Page 18-51

Additional equations for linear regression

The summary statistics such as Σx, Σx

, etc., can be used to define the

following quantities:













−=⋅−=−=

∑∑∑

===

ixx

xsnxxS

)1()(













−=⋅−=−=

∑∑∑

===

ysnyyS

























−=⋅−=−−=

∑∑∑∑

====

iixy

yxsnyyxxS

1111

)1())((

From which it follows that the standard deviations of x and y, and the

covariance of x,y are given, respectively, by

1−

, and

1−

Also, the sample correlation coefficient is

yyxx

⋅

In terms of x, y, S

, S

, and S

, the solution to the normal equations is:

xbya −= ,

b ==

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

Page 18-52

Let y

= actual data value,

= a + b⋅x

= least-square prediction of the data.

Then, the prediction error is: e

= y

- (a + b⋅x

An estimate of σ

is the, so-called, standard error of the estimate,

)1(

/)(

)]([

xyy

xxxyyy

SSS

bxay

s −⋅⋅

−

=+−

−

∑

Confidence intervals and hypothesis testing in linear regression

Here are some concepts and equations related to statistical inference for

linear regression:

• Confidence limits for regression coefficients:

For the slope (Β): b − (t

n-2,

)⋅s

/√S

< Β < b + (t

n-2,

)⋅s

/√S

For the intercept (Α):

a − (t

n-2,

)⋅s

⋅[(1/n)+x

]

1/2

< Α < a + (t

n-2,

)⋅s

⋅[(1/n)+x

]

1/2

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

or if t

< - t

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.

Page 18-53

• Hypothesis testing on the intercept , Α:

Null hypothesis, H

: Α = Α

, tested against the alternative hypothesis, H

Α ≠ Α

. The test statistic is t

= (a-Α

)/[(1/n)+x

]

1/2

, 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

or if t

< - t

• Confidence interval for the mean value of Y at x = x

, i.e., α+βx

a+b⋅x−(t

n-2,

)⋅s

⋅[(1/n)+(x

-x)

]

1/2

< α+βx

a+b⋅x+(t

n-2,

)⋅s

⋅[(1/n)+(x

-x)

]

1/2

• Limits of prediction: confidence interval for the predicted value Y

=Y(x

a+b⋅x−(t

n-2,

)⋅s

⋅[1+(1/n)+(x

-x)

]

1/2

< Y

a+b⋅x+(t

n-2,

)⋅s

⋅[1+(1/n)+(x

-x)

]

1/2

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

)1(

xxx

snS ⋅−= , )1(

222

xyye

s −⋅⋅

−

6) For either confidence intervals or two-tailed tests, obtain t

, with (1-

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.

Page 18-54

Example 1 -- For the following (x,y) data, determine the 95% confidence

interval for the slope B and the intercept A

2.0 2.5 3.0 3.5 4.0

5.5 7.2 9.4 10.0 12.2

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

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

5.2427905694150.0)15()1(

=⋅−=⋅−=

xxx

snS

=−⋅⋅

−

= )1(

222

xyye

...1826.0)...9897.01(...5880.2

=−⋅⋅

−

Page 18-55

Confidence intervals for the slope (Β) and intercept (A):

• First, we obtain t

n-2,

= t

0.025

= 3.18244630528 (See chapter 17 for a

program to solve for t

• Next, we calculate the terms

n-2,

)⋅s

/√S

= 3.182…⋅(0.1826…/2.5)

1/2

= 0.8602…

n-2,

)⋅s

⋅[(1/n)+x

]

1/2

3.1824…⋅√0.1826…⋅[(1/5)+3

/2.5]

1/2

= 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).

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.

The test statistic is t

= (a-0)/[(1/n)+x

]

1/2

= (-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

n-2,

= t

3,0.025

. Also, for γ = 3 and α =

0.025, t

n-2,

= t

3,0.025

= 3.18244630528. Because t

> - t

n-2,

, we cannot

reject the null 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.

Page 18-56

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

n-2,

= t

3,0.025

= 3.18244630528. Because, t

, we must reject the null hypothesis H

: Β ≠ 0, at the level of significance α

= 0.05, for the linear fitting of Example 1.

Multiple linear fitting

Consider a data set of the form

… x

. . . . .

. . . . . .

1,m-1

2,m-1

3,m-1

… x

n,m-1

m-1

1,m

2,m

3,m

… x

n,m

Suppose that we search for a data fitting of the form y = b

+ b

⋅x

+ b

⋅x

+ … + b

⋅x

. You can obtain the least-square approximation to the

values of the coefficients b = [b

… b

], by putting together the

matrix X:

_ _

1 x

… x

1 x

… x

1 x

… x

. . . . .

. . . . . .

1 x

1,m

2,m

3,m

… x

n,m

_ _

Page 18-57

Then, the vector of coefficients is obtained from b = (X

⋅X)

-1

⋅X

⋅y, where y is

the vector y = [y

… y

]

For example

, use the following data to obtain the multiple linear fitting

y = b

+ b

⋅x

+ b

⋅x

+ b

⋅x

1.20 3.10 2.00 5.70

2.50 3.10 2.50 8.20

3.50 4.50 2.50 5.00

4.00 4.50 3.00 8.20

6.00 5.00 3.50 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.,

y = -2.1649–0.7144⋅x

-1.7850×10

-2

⋅x

+ 7.0941⋅x

Page 18-58

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.., 8.23.., 9.45..].

Compare these fitted values with the original data as shown in the table

below:

y y-fitted

1.20 3.10 2.00 5.70 5.63

2.50 3.10 2.50 8.20 8.25

3.50 4.50 2.50 5.00 5.03

4.00 4.50 3.00 8.20 8.23

6.00 5.00 3.50 9.50 9.45

Polynomial fitting

Consider the x-y data set {(x

), (x

), …, (x

)}. 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

], by putting together the matrix X

_ _

1 x

… x

p-1

1 x

… x

p-1

1 x

… x

p-1

. . . . . .

. . . . . . .

1 x

… x

p-1

_ _

Then, the vector of coefficients is obtained from b = (X

⋅X)

-1

⋅X

⋅y, where y is

the vector y = [y

… y

]

In Chapter 10, we defined the Vandermonde matrix corresponding to a

vector 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:

Page 18-59

If p = n-1, X = V

If p < n-1, then remove columns p+2, …, n-1, n from V

to form X.

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

]. If we were to use a list of data values for x

rather than a vector, i.e., x = { x

… x

}, we can easily calculate the

sequence { 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)

Page 18-60

Here is the translation of the algorithm

to a program in User RPL language.

(See Chapter 21 for additional information on programming):

« Open program

 x y p Enter lists x and y, and p (levels 3,2,1)

« Open subprogram 1

x SIZE  n Determine size of x list

« Open subprogram 2

x VANDERMONDE Place x in stack, obtain V

IF ‘p<n-1’ THEN This IF implements step 3 in algorithm

n Place n in stack

p 2 + Calculate p+1

FOR j Start loop j = n-1, n-2, …, p+1, step = -1

j COL− DROP Remove column and drop it from stack

-1 STEP Close FOR-STEP loop

ELSE

IF ‘p>n-1’ THEN

n 1 + Calculate n+1

p 1 + Calculate p+1

FOR j Start a loop with j = n, n+1, …, p+1.

x j ^ Calculate x

, as a list

OBJ ARRY Convert list to array

j COL+ Add column to matrix

NEXT Close FOR-NEXT loop

END Ends second IF clause.

END Ends first IF clause. Its result is X

y OBJ ARRY Convert list y to an array

MTREG X and y used by program MTREG

NUM Convert to decimal format

» Close sub-program 2

» Close sub-program 1

» Close main program

Save it into a variable called POLY (POLYnomial fitting).

Page 18-61

As an example

, use the following data to obtain a polynomial fitting with p =

2, 3, 4, 5, 6.

x y

2.30 179.72

3.20 562.30

4.50 1969.11

1.65 65.87

9.32 31220.89

1.18 32.81

6.24 6731.48

3.45 737.41

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 = 4527.73-39.58x+742.23x

@@xx@@ @@yy@@ 3 @POLY, Result: [ –998.05 1303.21 -505.27 79.23]

i.e., 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 = -16.73+67.17x-48.69x

+21.11x

+1.07x

+0.19x

+0.0058x

Page 18-62

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:

• 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

r = [1-(SSE/SST)]

1/2

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):

Page 18-63

« Open program

 x y p Enter lists x and y, and number p

« Open subprogram1

x SIZE  n Determine size of x list

« Open subprogram 2

x VANDERMONDE Place x in stack, obtain V

IF ‘p<n-1’ THEN This IF is step 3 in algorithm

n Place n in stack

p 2 + Calculate p+1

FOR j Start loop, j = n-1 to p+1, step = -1

j COL− DROP Remove column, drop from stack

-1 STEP Close FOR-STEP loop

ELSE

IF ‘p>n-1’ THEN

n 1 + Calculate n+1

p 1 + Calculate p+1

FOR j Start loop with j = n, n+1, …, p+1.

x j ^ Calculate x

, as a list

OBJ ARRY Convert list to array

j COL+ Add column to matrix

NEXT Close FOR-NEXT loop

END Ends second IF clause.

END Ends first IF clause. Produces X

y OBJ ARRY Convert list y to an array

 X yv Enter matrix and array as X and y

« Open subprogram 3

X yv MTREG X and y used by program MTREG

NUM If needed, converts to floating point

 b Resulting vector passed as b

« Open subprogram 4

b yv Place b and yv in stack

X b * Calculate X⋅b

- Calculate e = y - X⋅b

ABS SQ DUP Calculate SSE, make copy

y ΣLIST n / Calculate y

n 1 LIST SWAP CON Create vector of n values of y

Page 18-64

yv − ABS SQ Calculate SST

/ Calculate SSE/SST

NEG 1 + √ Calculate r = [1–SSE/SST ]

1/2

“r” TAG Tag result as “r”

SWAP Exchange stack levels 1 and 2

“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:

p r SSE

2 0.9971908 10731140.01

3 0.9999768 88619.36

4 0.9999999 7.48

5 0.9999999 8.92

6 0.9999998 432.61

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 19-1

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

…c

)

. The “point” separates n

“integer” digits from m “decimal” digits. The value of the number, converted

to our customary decimal system, is calculated by using n = a

⋅bn

-1

+ a

⋅b

n-2

+ … + a

+ c

⋅b

-1

+ c

⋅b

-2

+ … +c

⋅b

-m

. For example, (15.234)

= 1⋅10

5⋅10

+ 2⋅10

-1

+ 3⋅10

-2

+ 4⋅10

-3

, and (101.111)

= 1⋅2

+ 0⋅2

+ 1⋅2

1⋅2

-1

+ 1⋅2

-2

+ 1⋅2

-3

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 21-6

DO: DO-UNTIL-END construct for loops

WHILE: WHILE-REPEAT-END construct for loops

TEST: Comparison operators, logical operators, flag testing functions

TYPE: Functions for converting object types, splitting objects, etc.

LIST: 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: To set and un-set flags and check their status

KEYS: To define and activate user-defined keys (Chapter 20)

MENU: To define and activate custom menus (Chapter 20)

MISC: Miscellaneous mode changes (beep, clock, etc.)

IN: Functions for program input

OUT: Functions for program output

TIME: Time-related functions

ALRM: Alarm manipulation

ERROR: Functions for error handling

IFERR: IFERR-THEN-ELSE-END construct for error handling

RUN: 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@@ ).

Functions listed by sub-menu

The following is a listing of the functions within the PRG sub-menus listed by

sub-menu.

Page 21-7

STACK MEM/DIR BRCH/IF BRCH/WHILE TYPE

DUP PURGE IF WHILE OBJ

SWAP RCL THEN REPEAT ARRY

DROP STO ELSE END LIST

OVER PATH END

STR

ROT CRDIR

TEST

TAG

UNROT PGDIR

BRCH/CASE

== UNIT

ROLL VARS CASE

≠

CR

ROLLD TVARS THEN < RC

PICK ORDER END > NUM

UNPICK

≤

CHR

PICK3

MEM/ARITH BRCH/START

≥

DTAG

DEPTH STO+ START AND EQ

DUP2 STO- NEXT OR TYPE

DUPN STOx STEP XOR VTYPE

DROP2 STO/ NOT

DROPN INCR

BRCH/FOR

SAME

LIST

DUPDU DECR FOR TYPE OBJ

NIP SINV NEXT SF LIST

NDUPN SNEG STEP CF SUB

SCONJ FS? REPL

MEM

BRCH/DO

FC?

PURGE

BRCH

DO FS?C

MEM IFT UNTIL FC?C

BYTES IFTE END LININ

NEWOB

ARCHI

RESTO

Page 21-8

LIST/ELEM GROB CHARS MODES/FLAG MODES/MISC

GET GROB SUB SF BEEP

GETI BLANK REPL CF CLK

PUT GOR POS FS? SYM

PUTI GXOR SIZE FC? STK

SIZE SUB NUM FS?C ARG

POS REPL CHR FS?C CMD

HEAD LCD OBJ FC?C INFO

TAIL LCD STR STOF

SIZE HEAD RCLF

LIST/PROC

ANIMATE TAIL RESET INFORM

DOLIST

SREPL NOVAL

DOSUB

PICT

MODES/KEYS

CHOOSE

NSUB PICT

MODES/FMT

ASN INPUT

ENDSUB PDIM STD STOKEYS KEY

STREAM LINE FIX RECLKEYS WAIT

REVLIST TLINE SCI DELKEYS PROMPT

SORT BOX ENG

SEQ ARC FM,

MODES/MENU OUT

PIXON ML MENU PVIEW

PIXOF CST TEXT

PIX?

MODES/ANGLE

TMENU CLLCD

PVIEW DEG RCLMENU DISP

PXC RAD

FREEZE

CPX GRAD

MSGBOX

RECT

BEEP

CYLIN

SPHERE

Page 21-9

TIME ERROR RUN

DATE DOERR DBUG

DATE ERRN SST

TIME ERRM SST↓

TIME ERR0 NEXT

TICKS LASTARG HALT

KILL

TIME/ALRM ERROR/IFERR

OFF

ACK IFERR

ACKALARM THEN

STOALARM ELSE

RCLALARM END

DELALARM

FINDALARM

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 21-11

@)STACK

DUP „°@)STACK @@DUP@@

SWAP „°@)STACK @SWAP@

DROP „°@)STACK @DROP@

@)@MEM@@ @)@DIR@@

PURGE „°@)@MEM@@ @)@DIR@@ @PURGE

ORDER „°@)@MEM@@ @)@DIR@@ @ORDER

@)@BRCH@ @)@IF@@

IF „°@)@BRCH@ @)@IF@@ @@@IF@@@

THEN „°@)@BRCH@ @)@IF@@ @THEN@

ELSE „°@)@BRCH@ @)@IF@@ @ELSE@

END „°@)@BRCH@ @)@IF@@ @@@END@@

@)@BRCH@ @)CASE@

CASE „°@)@BRCH@ @)CASE@ @CASE@

THEN „°@)@BRCH@ @)CASE@ @THEN@

END „°@)@BRCH@ @)CASE@ @@END@

@)@BRCH@ @)START

START „°@)@BRCH@ @)START @START

NEXT „°@)@BRCH@ @)START @NEXT

STEP „°@)@BRCH@ @)START @STEP

@)@BRCH@ @)@FOR@

FOR „°@)@BRCH@ @)@FOR@ @@FOR@@

NEXT „°@)@BRCH@ @)@FOR@ @@NEXT@

STEP „°@)@BRCH@ @)@FOR@ @@STEP@

@)@BRCH@ @)@@DO@@

DO „°@)@BRCH@ @)@@DO@@ @@@DO@@

UNTIL „°@)@BRCH@ @)@@DO@@ @UNTIL

END „°@)@BRCH@ @)@@DO@@ @@END@@

Page 21-21

)12(32

SSS

⋅⋅⋅

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.,

)(2

byg

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 21-22

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

+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 ` Copies program name to stack level 1

„°LL @)@RUN@ @@DBG@ Starts debugger

@SST

↓@ Step-by-step debugging, result: “Enter a:”

@SST

↓@ Result: {“  a:” {2 0} V}

@SST

↓@ Result: user is prompted to enter value of a

2` Enter a value of 2 for a. Result: “:a:2”

Page 21-23

@SST↓@ Result: a:2

@SST

↓@ Result: empty stack, executing →a

@SST

↓@ Result: empty stack, entering subprogram «

@SST

↓@ Result: ‘2*a^2+3’

@SST

↓@ Result: ‘2*a^2+3’, leaving subprogram »

@SST

↓@ 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

+3 for 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:

J Recovers variables menu

³@FUNCa ` Copies program name to stack level 1

„°LL @)@RUN@ @@DBG@ Starts debugger

@SST

↓@ Step-by-step debugging, result: “Enter a:”

@SST

↓@ Result: {“ a:” {2 0} V}

@SST

↓@ Result: user is prompted to enter value of a

2` Enter a value of 2 for a. Result: “:a:2”

@SST

↓@ Result: a:2

@SST

↓@ Result: empty stack, executing →a

@SST

↓@ 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

<!> Interrupted

message 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.

Page 21-24

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.

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` Enter directory name ‘PTRICKS’

„°@)@MEM@@ @)@DIR@@ @CRDIR Create directory

J 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

Page 21-25

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.

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

TVp ⋅=⋅⋅= )_662902.1(2.031451.8),(

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)’ »

Page 21-26

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

`. 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

,)_31451.8(),,(

nTVp

⋅

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 21-27

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

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 21-28

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

}

1: 1

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

Chezy’s formula: Q = C⋅(R⋅S)

1/2

, 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} }

3. Field format information: { } (an empty list, thus, default values used)

Page 21-29

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 21-30

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 OBJ 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:

OBJ 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 OBJ, 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 21-31

« “ 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 OBJ 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

}. 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 21-32

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:

2: v

1: 1

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. units or 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” TAG will

produced a tagged result in the screen. If the value in stack level 1 is zero,

Page 21-33

the 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 OBJ („°@)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 21-34

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:

« “Enter a: “ {“:a: “ {2 0} V } INPUT OBJ→ → a «

‘2*a^2+3‘ EVAL ”F” →TAG a SWAP» »

Page 21-35

(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 a actually 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 a before 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 ` Copies program name to stack level 1

„°LL @)@RUN@ @@DBG@ Starts debugger

@SST

↓@ Step-by-step debugging, result: “Enter a:”

@SST

↓@ Result: {“ a:” {2 0} V}

@SST

↓@ Result: user is prompted to enter value of a

2` Enter a value of 2 for a. Result: “:a:2”

@SST

↓@ Result: a:2

@SST

↓@ Result: empty stack, executing →a

@SST

↓@ Result: empty stack, entering subprogram «

@SST

↓@ Result: ‘2*a^2+3’

@SST

↓@ Result: empty stack, calculating

@SST

↓@ Result: 11.,

@SST

↓@ Result: “F”

@SST

↓@ Result: F: 11.

@SST

↓@ Result: a:2.

@SST

↓@ Result: swap levels 1 and 2

@SST

↓@ leaving subprogram »

@SST

↓@ leaving main program »

Page 21-36

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.

Page 21-37

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.

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 21-45

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:

p NOT p

1 0

0 1

p q p AND q

1 1 1

1 0 0

0 1 0

0 0 0

p q p OR q

1 1 1

1 0 1

0 1 1

0 0 0

p q p XOR q

1 1 0

1 0 1

0 1 1

0 0 0

Page 21-46

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.

Branching with IF

In this section we presents examples using the constructs IF…THEN…END and

IF…THEN…ELSE…END.

Page 21-47

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:

With the cursor  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 » »

Page 21-48

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:

0 @@@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:

Example

: Type in the following program:

Page 21-60

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

3 @@@S2@@ Result: S:14 4 @@@S2@@ Result: S:30

5 @@@S2@@ Result: S:55 8 @@@S2@@ Result: S:204

10 @@@S2@@ Result: S:385 20 @@@S2@@ Result: S:2870

30 @@@S2@@ Result: S:9455 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 increment of 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 STEP

returns 1)

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.

Page 21-61

• 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 ` Enter parameters 1 1.5 0.5

[‘] @GLIS2 ` Enter the program name in level 1

„°LL @)@RUN@ @@DBG@ Start the debugger.

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_statement must

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

The following program calculates the summation

∑

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 » »

Page 21-62

Store this program in a variable @@S3@@. Verify the following exercises: J

3 @@@S3@@ Result: S:14 4 @@@S3@@ Result: S:30

5 @@@S3@@ Result: S:55 8 @@@S3@@ Result: S:204

10 @@@S3@@ Result: S:385 20 @@@S3@@ Result: S:2870

30 @@@S3@@ Result: S:9455 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 ` Enter parameters 1 1.5 0.5

[‘] @GLIS3 ` Enter the program name in level 1

„°LL @)@RUN@ @@DBG@ Start the debugger.

Use @SST↓@ to step into the program and see the detailed operation of each

command.

The WHILE construct

The general structure of this command is:

WHILE logical_statement REPEAT program_statements END

The WHILE statement will repeat the program_statements while

logical_statement is true (non zero). If not, program control is passed to

the statement right after END. The program_statements must include a

Page 21-63

loop index that gets modified before the logical_statement is 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

∑

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

3 @@@S4@@ Result: S:14 4 @@@S4@@ Result: S:30

5 @@@S4@@ Result: S:55 8 @@@S4@@ Result: S:204

10 @@@S4@@ Result: S:385 20 @@@S4@@ Result: S:2870

30 @@@S4@@ Result: S:9455 100 @@@S4@@ Result: S:338350

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 » »

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:

Page 21-64

J1 # 1.5 # 0.5 ` Enter parameters 1 1.5 0.5

[‘] @GLIS4 ` Enter the program name in level 1

„°LL @)@RUN@ @@DBG@ Start the debugger.

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

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

Page 21-65

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:

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:

Page 21-66

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…].

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>()

Page 21-67

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:

Whereas, using RPL, there is no problem when loading this program in

algebraic mode:

Page 22-1

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@ @@RCLK@.

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 22-35

with respect to segment AB. The coordinates of point A’ will give the values

(σ’

,τ’

), while those of B’ will give the values (σ’

,τ’

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

Page 22-36

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

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,

Page 22-37

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

25˜ Enter σx = 25

75˜ Enter σy = 75

50` Enter τxy = 50, and finish data entry.

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).

Page 22-38

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 ∆φ = 1

, we miss the actual point where the shear

stresses become zero. If you press š once more, you find values of are

= 58

, and (σ’

, τ’

) = (1.06E2,5.51E-1) = (106, 0.551). What this

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)

INDAT Enter data as in program MOHRCIRC

CC&r Calculate σc, r, and fn, as in MOHRCIRC

“φn” TAG Tag angle for principal stresses

3 ROLLD Move tagged angle to level 3

RC DUP Convert σc and r to (σc, r), duplicate

CR + “σPx” TAG Calculate principal stress σPx, tag it

SWAP CR - “σPy” TAG Swap,calculate stress σPy, tag it.

» End program PRNST

To run the program use:

J@PRNST Start program PRNST

25˜ Enter σx = 25

75˜ Enter σy = 75

50` Enter τxy = 50, and finish data entry.

Page 22-39

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 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

12.5˜ Enter σx = 12.5

6.25\˜ Enter σy = -6.25

5\` Enter τxy = -5, and finish data entry.

The result is:

To draw Mohr’s circle, use the program @MOHRC, as follows:

Page 22-40

J@MOHRC Start program PRNST

12.5˜ Enter σx = 12.5

6.25\˜ Enter σy = -6.25

5\` Enter τxy = -5, and finish data entry.

The result is:

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:

Page 22-41

Press @@@OK@@@ to continue program execution. The result is the following figure:

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 23-1

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:

OBJ: 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

Page 23-2

Examples of application of these functions to strings are shown next:

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 23-3

The operation of NUM, CHR, OBJ, 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 “

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,

Page 23-4

say they line feed character  , you will see at the left side of the bottom of

the 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 αM, or ~„m.

Greek characters, such as σ, will show the code α

S, 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 24-1

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

_________________________________________________________________

0 Real Number -1.23E-5

1 Complex Number (-1.2,2.3)

2 String "Hello, world "

3 Real Array [[1 2][3 4]]

4 Complex Array [[(1 2) (3 4)]

[(5 6) (7 8)]

5 List {3 1 'PI'}

6 Global Name X

7 Local Name y

8 Program <<  a 'a^2' >>

9 Algebraic object 'a^2+b^2'

10 Binary Integer # A2F1E h

11 Graphic Object Graphic 131×64

12 Tagged Object R: 43.5

13 Unit Object 3_m^2/s

14 XLIB Name XLIB 342 8

15 Directory DIR … END

16 Library Library 1230"...

17 Backup Object Backup MYDIR

18 Built-in Function COS

19 Built-in Command CLEAR

Page 24-2

Number Type Example

____________________________________________________________________

21 Extended Real Number Long Real

22 Extended Complex Number Long Complex

23 Linked Array Linked Array

24 Character Object Character

25 Code Object Code

26 Library Data Library Data

27 External Object External

28 Integer 3423142

29 External Object External

30 External Object External

____________________________________________________________________

Function TYPE

This function, available in the PRG/TYPE () sub-menu, or through the command

catalog, is used to determine the type of an object. The function argument is

the object of interest. The function returns the object type as indicated by the

numbers specified above

Function VTYPE

This function operates similar to function TYPE, but it applies to a variable

name, returning the type of object stored in the variable.

Page 24-3

Calculator flags

A flag is a variable that can either be set or unset. The status of a flag affects

the behavior of the calculator, if the flag is a system flag, or of a program, if it

is a user flag. They are described in more detail next.

System flags

System flags can be accessed by using H @)FLAGS!. Press the down arrow

key to see a listing of all the system flags with their number and brief

description. The first two screens with system flags are shown below:

You will recognize many of these flags because they are set or unset in the

MODES menu (e.g., flag 95 for Algebraic mode, 103 for Complex mode,

etc.). Throughout this user’s manual we have emphasized the differences

between CHOOSE boxes and SOFT menus, which are selected by setting or

un-setting system flag 117. Another example of system flag setting is that of

system flags 60 and 61 that relate to the constant library (CONLIB, see

Chapter 3). These flags operate in the following manner:

• user flag 60: clear (default):SI units, set: ENGL units

• user flag 61: clear (default):use units, set: value only

Functions for setting and changing flags

These functions can be used to set, un-set, or check on the status of user flags

or system flags. When used with these functions system flags are referred to

with negative integer numbers. Thus, system flag 117 will be referred to as

flag -117. On the other hand, user flags will be referred to as positive integer

numbers when applying these functions. It is important to understand that user

flags have applications only in programming to help control the program flow.

Page 24-4

Functions for manipulating calculator flags are available in the

PRG/MODES/FLAG menu. The PRG menu is activated with „°. The

following screens (with system flag 117 set to CHOOSE boxes) show the

sequence of screens to get to the FLAG menu:

The functions contained within the FLAG menu are the following:

The operation of these functions is as follows:

SF Set a flag

CF Clear a flag

FS? Returns 1 if flag is set, 0 if not set

FC? Returns 1 if flag is clear (not set), 0 if flag is set

FS?C Tests flag as FS does, then clears it

FC?C Tests flag as FC does, then clears it

STOF Stores new system flag settings

RCLF Recalls existing flag settings

RESET Resets current field values (could be used to reset a flag)

User flags

For programming purposes, flags 1 through 256 are available to the user.

They have no meaning to the calculator operation.

Page 25-1

Chapter 25

Date and Time Functions

In this Chapter we demonstrate some of the functions and calculations using

times and dates.

The TIME menu

The TIME menu, available through the keystroke sequence ‚Ó (the 9 key)

provides the following functions, which are described next:

Setting an alarm

Option 2. Set alarm.. provides an input form to let the user set an alarm. The

input form looks like in the following figure:

The Message: input field allows you to enter a character string identifying the

alarm. The Time: field lets you enter the time for activating the alarm. The

Date: field is used to set the date for the alarm (or for the first time of

activation, if repetition is required). For example, you could set the following

alarm. The left-hand side figure shows the alarm with no repetition. The right-

hand figure shows the options for repetition after pressing @CHOOS. After

pressing @@@OK@@@ the alarm will be set.

Page 25-2

Browsing alarms

Option 1. Browse alarms... in the TIME menu lets you review your current

alarms. For example, after entering the alarm used in the example above, this

option will show the following screen:

This screen provides four soft menu key labels:

EDIT: For editing the selected alarm, providing an alarm set input form

NEW: For programming a new alarm

PURG: For deleting an alarm

OK : Returns to normal display

Setting time and date

Option 3. Set time, date… provides the following input form that let’s the user

set the current time and date. Details were provided in Chapter 1.

TIME Tools

Option 4. Tools… provides a number of functions useful for clock operation,

and calculations with times and dates. The following figure shows the

functions available under TIME Tools:

Page 25-3

The application of these functions is demonstrated below.

DATE: Places current date in the stack

DATE: Set system date to specified value

TIME: Places current time in 24-hr HH.MMSS format

TIME: Set system time to specified value in 24-hr HH.MM.SS format

TICKS: Provides system time as binary integer in units of clock ticks with 1

tick = 1/8192 sec

ALRM..:Sub-menu with alarm manipulation functions (described later)

DATE+: Adds or subtract a number of days to a date

DDAYS(x,y): Returns number of days between dates x and y

HMS: Converts time from decimal to HH.MMSS

HMS: Converts time from HH.MMSS to decimal

HMS+: Add two times in HH.MMSS format

HMS-: Subtract two times in HH.MMSS format

TSTR(time, date): converts time, date to string format

CLKADJ(x): Adds x ticks to system time (1 tick = 1/8192 sec )

Functions DATE, TIME, CLKADJ are used to adjust date and time. There

are no examples provided here for these functions.

Here are examples of functions DATE, TIME, and TSTR:

Calculations with dates

For calculations with dates, use functions DATE+, DDAYS. Here is an

example of application of these functions, together with an example of

function TICKS:

Page C-6

It is recommended that you select EXACT mode as default CAS mode, and

change to APPROX mode if requested by the calculator in the performance of

an operation.

For additional information on real and integer numbers, as well as other

calculator’s objects, refer to Chapter 2.

Complex vs. Real CAS mode

A complex number is a number of the form a+bi, where i, defined by

−=i is the unit imaginary number (electrical engineers prefer to use the

symbol j), and a and b are real numbers. For example, the number 2 + 3i is

a complex number. Additional information on operations with complex

numbers are presented in Chapter 4 of this guide.

When the _Complex CAS option is selected, if an operation results in a

complex number, then the result will be shown in the form a+bi or in the form

of an ordered pair (a,b). On the other hand, if the _Complex CAS option is

unset (i.e., the Real CAS option is active), and an operation results in a

complex number, you will be asked to switch to Complex mode. If you

decline, the calculator will report an error.

Please notice that, in COMPLEX mode the CAS is able to perform a wider

range of operations than in REAL mode, but it will also be considerably slower.

Thus, it is recommended that you use the REAL mode as default mode and

switch to COMPLEX if requested by the calculator in the performance of an

operation.

The following example shows the calculation of the quantity

85 − using

the Algebraic operating mode, first with the Real CAS option selected. In this

case, you are asked if you want to change the mode to Complex:

Page C-7

If you press the OK soft menu key (), then the _Complex option is forced, and

the result is the following:

The keystrokes used above are the following:

R„Ü5„Q2+ 8„Q2`

When asked to change to COMPLEX mode, use:F. If you decide not to

accept the change to COMPLEX mode, you get the following error message:

Verbose vs. non-verbose CAS mode

When the _Verbose CAS option is selected, certain calculus applications are

provided with comment lines in the main display. If the _Verbose CAS option

is not selected, then those calculus applications will show no comment lines.

The comment lines will appear momentarily in the top lines of the display

while the operation is being calculated.

Step-by-step CAS mode

When the _Step/step CAS option is selected, certain operations will be

shown step at a time in the display. If the _Step/step CAS option is not

selected, then intermediate steps will not be shown.

Page C-8

For example, having selected the Step/step option, the following screens

show the step-by-step division of two polynomials, namely, (X

-5X

+3X-2)/(X-

2). This is accomplished by using function DIV2 as shown below. Press `

to show the first step:

The screen inform us that the calculator is operating a division of polynomials

A/B, so that A = BQ + R, where Q = quotient, and R = remainder. For the

case under consideration, A = X

-5X

+3X-2, and B = X-2. These polynomials

are represented in the screen by lists of their coefficients. For example, the

expression A: {1,-5,3,-2} represents the polynomial A = X

-5X

+3X-2, B:{1,-2}

represents the polynomial B = X-2, Q: {1} represents the polynomial Q = X,

and R:{-3,3,-2} represents the polynomial R = -3X

+3X-2.

At this point, press, for example, the ` key. Continue pressing ` the key

to produce additional steps:

Thus, the intermediate steps shown represent the coefficients of the quotient

and residual of the step-by-step synthetic division as would have been

performed by hand, i.e.,

−

−+−

−

−+−

233

235

XXX

Page C-9

−

−−−=

−

−−

+−

XXX

Increasing-power CAS mode

When the _Incr pow CAS option is selected, polynomials will be listed so that

the terms will have increasing powers of the independent variable. If the _Incr

pow CAS option is not selected (default value) then polynomials will be listed

so that the terms will have decreasing powers of the independent variable.

An example is shown next in Algebraic mode:

In the first case, the polynomial (X+3)

is expanded in increasing order of the

powers of X, while in the second case, the polynomial shows decreasing

order of the powers of X. The keystrokes in both cases are the following:

„Üx+3™Q5`

In the first case the _Incr pow option was selected, while in the second it was

not selected. The same example, in RPN notation, is shown below:

The same keystroke sequence was used to produce each of these results:

³„Üx+3™Q5`µ

Rigorous CAS setting

Page C-10

When the _Rigorous CAS option is selected, the algebraic expression |X|,

i.e., the absolute value, is not simplified to X. If the _Rigorous CAS option is

not selected, the algebraic expression |X| is simplified to X.

The CAS can solve a larger variety of problems if the rigorous mode is not set.

However, the result, or the domain in which the result are applicable, might

be more limited.

Simplify non-rational CAS setting

When the _Simp Non-Rational CAS option is selected, non-rational

expressions will be automatically simplified. On the other hand, if the _Simp

Non-Rational CAS option is not selected, non-rational expressions will not be

automatically simplified.

Using the CAS HELP facility

Turn on the calculator, and press the I key to activate the TOOL menu.

Next, press the Bsoft menu key, followed by the ` key (the key in the

lowest right corner of the keyboard), to activate the HELP facility. The display

will look as follows:

At this point you will be provided with a list of all CAS commands in

alphabetical order. You can use the down arrow key, ˜, to navigate

through the list. To move upwards in the list use the up arrow key, — . The

arrow keys are located on the right-hand side of the keyboard between the

first and fourth rows of keys.

Suppose that you want to find information on the command ATAN2S

(ArcTANgent-to-Sine function). Press the down arrow key, ˜, until the

command ATAN2S is highlighted in the list:

Page C-11

Notice that, in this instance, soft menu keys E and F are the only one

with associated commands, namely:

!!CANCL E CANCeL the help facility

!!@@OK#@ F OK to activate help facility for the selected command

If you press the !!CANCL E key, the HELP facility is skipped, and the calculator

returns to normal display.

To see the effect of using !!@@OK#@ in the HELP facility, let’s repeat the steps used

above from to the selection of the command ATAN2S in the list of CAS

commands: @HELP B` ˜ ˜ …(10 times)

Then, press the !!@@OK#@ F key to obtain information about the command

ATAN2S.

The help facility indicates that the command, or function, ATAN2S replaces

the value of atan(x), the arc tangent of a value x, by its equivalent in terms of

the function asin (arcsine), i.e.,

The fourth and fifth lines in the display provide an example of application of

the function ATAN2S. Line four, namely, ATAN2S(ATAN(X)), is the statement

of the operation to be performed, while line five, namely, ASIN(X/√(X^2+1)),

is the result.

The bottom line in the display, starting with the particle See:, is a reference

line listing other CAS commands related to the command ATAN2S.

Notice that there are six commands associated with the soft menu keys in this

case (you can check that there are only six commands because pressing the

Page C-12

L produces no additional menu items). The soft menu key commands are

the following:

@EXIT A EXIT the help facility

@ECHO B Copy the example command to the stack and exit

@@ SEE1@@ C See the first link (if any) in the list of references

@@SEE2@ D See the second link (if any) of the list of references

!@@SEE3@ E See the third link (if any) of the list of references

@!MAIN F Return to the MAIN command list in the help facility

In this case we want to ECHO the example into the stack by pressing

@ECHO B. The resulting display is the following:

There are now four lines of the display occupied with output. The first two

lines from the top correspond to the first exercise with the HELP facility in

which we cancel the request for help. The third line from the top shows the

most recent call to the HELP facility, while the last line shows the ECHO of the

example command. To activate the command press the ` key. The result

is:

Notice that, as new lines of output are produced, the display (or stack) pushes

the existing lines upwards and fills the bottom of the screen with more output.

The HELP facility, described in this section, will be very useful to refer to the

definition of the many CAS commands available in the calculator. Each entry

in the CAS help facility, whenever appropriate, will have an example of

application of the command, as well as references as shown in this example.

Page C-13

To navigate quickly to a particular command in the help facility list without

having to use the arrow keys all the time, we can use a shortcut consisting of

typing the first letter in the command’s name. Suppose that we want to find

information on the command IBP (Integration By Parts), once the help facility

list is available, use the ~ key (first key in the fourth row from the bottom of

the keyboard) followed by the key for the letter i (the same as the key I) ,

i.e., ~i. This will take you automatically to the first command that starts

with an i, namely, IBASIS. Then, you can use the down arrow key ˜ ,

twice, to find the command IBP. Pressing the !!@@OK#@ F key, we activate the

help facility for this command. Press @!MAIN F to recover the main list of

commands, or @EXIT A to exit the facility.

References for non-CAS commands

The help facility contains entries for all the commands developed for the CAS

(Computer Algebraic System). There is a large number of other functions and

commands that were originally developed for the HP 48G series calculators

that are not included in the help facility. Good references for those

commands are the HP 48G Series user’s guide (HP Part No. 00048-90126)

and the HP 48G Series Advanced User’s Reference Manual (HP Part No.

00048-90136) both published by Hewlett-Packard Company, Corvallis,

Oregon, in 1993.

CAS End User Term and Conditions

Use of the CAS Software requires from the user an appropriate mathematical

knowledge. There is no warranty for the CAS Software, to the extent permitted

by applicable law. Except when otherwise stated in writing the copyright

holder provides the CAS Software "As Is" without warranty of any kind, either

expressed or implied, including, but not limited to, the implied warranties of

merchantability and fitness for a particular purpose. The entire risk as to the

quality and performance of the CAS Software is with you. Should the CAS

Software prove defective, you assume the cost of all necessary servicing,

repair or correction.

Page C-14

In no event unless required by applicable law will any copyright holder be

liable to you for damages, including any general, special, incidental or

consequential damages arising out of the use or inability to use the

CAS Software (including but not limited to loss of data or data being rendered

inaccurate or losses sustained by you or third parties or a failure of the CAS

Software to operate with any other programs), even if such holder or other

party has been advised of the possibility of such damages. If required by

applicable law the maximum amount payable for damages by the copyright

holder shall not exceed the royalty amount paid by Hewlett-Packard to the

Page D-1

Appendix D

Additional character set

While you can use any of the upper-case and lower-case English letter from

the keyboard, there are 255 characters usable in the calculator. Including

special characters like θ, λ, etc., that that can be used in algebraic

expressions. To access these characters we use the keystroke

combination …± (associated with the EVAL key). The result is the

following screen:

By using the arrow keys, š™˜—, we can navigate through the

collection of characters. For example, moving downwards in the screen

produces more characters in the display:

Moving farther down, we see these characters:

There will be one character highlighted at all times. The lower line in the

display will show the short cut for the highlighted character, as well as the

ASCII character code (e.g., see the screen above: the short cut is αDα9,

Page D-2

i.e., ~„d~…9, and the code is 240). The display also shows

three functions associated with the soft menu keys, f4, f5, and f6. These

functions are:

@MODIF: Opens a graphics screen where the user can modify highlighted

character. Use this option carefully, since it will alter the modified character

up to the next reset of the calculator. (Imagine the effect of changing the

graphic of the character 1 to look like a 2!).

@ECHO1: Copies the highlighted character to the command line or equation

writer (EQW) and exits the character set screen (i.e., echoes a single

character to the stack).

@ECHO: Copies the highlighted character to the command line or equation writer

(EQW), but the cursor remains in the character set screen to allow the user to

select additional characters (i.e., echoes a string of characters to the stack).

To exit the character set screen press `.

For example, suppose you have to type the expression: λ

+ 2µ + 5

Here is a suggested approach, using the stack in either Algebraic or RPN

mode:

Use the keystrokes: ³…±to get to the characters screen. Next, use the

arrow keys to highlight the character λ. Press @ECHO1 (i.e., the E key), and

continue with the keystrokes: + 2 *…±. Next, use the arrow

keys to highlight the character µ. Press @ECHO1 (i.e., the E key), and finish

the expression with the keystrokes: +5`. Here is the result of this

exercise in Algebraic and RPN modes, respectively:

Following, we list some of the most common ~‚keystroke combinations:

Page D-3

Greek letters

α (alpha) ~‚a

β (beta) ~‚b

δ (delta) ~‚d

ε (epsilon) ~‚e

θ (theta) ~‚t

λ (lambda) ~‚n

µ (mu) ~‚m

ρ (rho) ~‚f

σ (sigma) ~‚s

τ (tau) ~‚u

ω (omega) ~‚v

∆ (upper-case delta) ~‚c

Π (upper-case pi) ~‚p

Other characters

~ (tilde) ~‚1

! (factorial) ~‚2

? (question mark) ) ~‚3

\ (backward slash) ~‚5

 (angle symbol) ~‚6

@ (at) ~‚`

Some characters commonly used that do not have simple keystroke shortcuts

are: x (x bar), γ (gamma), η (eta), Ω (upper-case omega). These characters

can be “echoed” from the CHARS screen: …±.

Page E-1

Appendix E

The Selection Tree in the Equation Writer

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

an expression. The form of the expression tree is determined by a number of

rules known as the hierarchy of operation. The rules are as follows:

1. Operations in parentheses are executed first, from the innermost to

the outermost parentheses, and from left to right in the expression.

2. Arguments of functions are executed next, from left to right.

3. Functions are executed next, from left to right.

4. Powers of numbers are executed next, from left to right.

5. Multiplications and divisions are executed next, from left to right.

6. Additions and subtraction are executed last, from left to right.

Execution from left to right means that, if two operations of the same hierarchy,

say two multiplications, exist in an expression, the first multiplication to the left

will be executed before the second, and so on.

Consider, for example, the expression shown below in the equation writer:

The insertion cursor () at this point is located to the right of the 2 in the

argument of the SIN function in the denominator. Press the down arrow key

˜to trigger the clear, editing cursor () around the 2 in the denominator.

Next, press the left arrow key š, continuously, until the clear, editing cursor

is around the y in the first factor in the denominator. Then, press the upper-

arrow key to activate the selection cursor () around the y. By pressing the

upper arrow key —, continuously, we can follow the expression tree that will

take use from the y to the completion of the expression. Here is the sequence

of operations highlighted by the upper arrow key—:

Page G-2

• Set/clear system flag 117 (CHOOSE boxes vs. SOFT menus):

H @)FLAGS —„ —˜

• In ALG mode,

SF(-117) selects SOFT menus

CF(-117) selects CHOOSE BOXES.

• In RPN mode,

117 \` SF selects SOFT menus

117 \` CF selects SOFT menus

• Change angular measure:

o To degrees: ~~deg`

o To radian: ~~rad`

• Special characters:

o Angle symbol (∠): ~‚6

o Factorial symbol (!): ~‚2

o Degree symbol (

): ~‚(hold)6

• Lock/unlock alpha keyboard:

o Lock alpha keyboard (upper case): ~~

o Unlock alpha keyboard (upper case): ~

o Lock alpha keyboard (lower case): ~~„~

o Unlock alpha keyboard (lower case): „~~

• Greek letters:

Alpha (α): ~‚a Beta (β): ~‚b

DELTA (∆): ~‚c Delta (d): ~‚d

Epsilon (ε): ~‚e Rho (ρ): ~‚f

Mu (µ): ~‚m Lambda (λ): ~‚n

PI (Π): ~‚p Sigma (σ): ~‚s

Theta (θ): ~‚t Tau (t): ~‚u

Omega (ω): ~‚v

@CHK@

Page G-3

• System-level operation (Hold $, release it after entering second or

third key):

o $ (hold) AF: “Cold” restart - all memory erased

o $ (hold) B: Cancels keystroke

o $ (hold) C: “Warm” restart - memory preserved

o $ (hold) D: Starts interactive self-test

o $ (hold) E: Starts continuous self-test

o $ (hold) #: Deep-sleep shutdown - timer off

o $ (hold) A: Performs display screen dump

o $ (hold) D: Cancels next repeating alarm

• Menus not accessible through keyboard: In RPN, enter menu_number,

type MENU. In ALG mode, type MENU(menu_number).

Menu_number is one of the following:

o STAT soft menu: 96

o PLOT soft menu: 81

o SOLVE soft menu: 74, or use ‚(hold) 7

o UTILITY soft menu: 113

• Other menus:

o MATHS menu: ~~maths`

o MAIN menu: ~~main`

• Other keyboard short cuts:

o ‚(hold) 7 : SOLVE menu (menu 74)

o „ (hold) H : PRG/MODES menu (Chapter 21)

o „ (hold) ˜ : Starts text editor (Appendix L)

o „ (hold) § : HOME(), go to HOME directory

o „ (hold) « : Recover last active menu

o ‚ (hold) ˜ : List contents of variables or menu entries

o ‚(hold) ± : PRG/CHAR menu (Chapter 21)

Page H-1

Appendix H

The CAS help facility

The CAS help facility is available through the keystroke sequence I

L@HELP `. The following screen shots show the first menu page in the

listing of the CAS help facility.

The commands are listed in alphabetical order. Using the vertical arrow

keys —˜ one can navigate through the help facility list. Some useful

hints on navigating through this facility are shown next:

• You can hold down the down arrow key ˜ and watch the screen

until the command you’re looking for shows up in the screen. At this

point, you can release the down arrow key. Most likely the command

of interest will not be selected at this point (you may overshoot or

undershoot it). However, you can use the vertical keys —˜, one

stroke at a time, to locate the command you want, and then press

@@OK@@.

• If, while holding down the down arrow key ˜ you overshoot the

command of interest, you can hold down the up arrow key — to

move back towards that command. Refine the selection with the

vertical keys —˜, one stroke at a time.

• You can type the first letter of the command of interest, and then use

the down arrow key ˜ to locate that particular command. For

example, if you’re looking for the command DERIV. After activating

the help facility (I L@HELP `), type ~d. This will select

the first of the commands that start with D, i.e., DEGREE. To find

DERIV, press ˜, twice. To activate the command, press @@OK@@.

Page H-2

• You can type two or more letters of the command of interest, by

locking the alphabetic keyboard. This will take you to the command

of interest, or to its neighborhood. Afterwards, you need to unlock

the alpha keyboard, and use the vertical arrow keys —˜ to

locate the command, if needed. Press @@OK@@ to locate the to activate

the command. For example, to locate the command PROPFRAC,

you can use, one of the following keystroke sequences:

I L@HELP ` ~~pr ~ ˜˜@@OK@@

I L@HELP ` ~~pro ~ ˜@@OK@@

I L@HELP ` ~~prop ~ @@OK@@

See Appendix C for more information on the CAS (Computer Algebraic

System). Appendix C includes other examples of application of the CAS help

facility.

Page I-1

Appendix I

Command catalog list

This is a list of all commands in the command catalog (‚N). Those

commands that belong to the CAS (Computer Algebraic System) are listed

also in Appendix H. CAS help facility entries are available for a given

command if the soft menu key @HELP shows up when you highlight that

particular command. Press this soft menu key to get the CAS help facility

entry for the command. The first few screens of the catalog are shown below:

User-installed library commands would also appear on the command

catalog list, using italic font. If the library includes a help item, then the soft

menu key @HELP shows up when you highlight those user-created commands.

Page J-1

Appendix J

The MATHS menu

The MATHS menu, accessible through the command MATHS (available in the

catalog N), contains the following sub-menus:

The CMPLX sub-menu

The CMPLX sub-menu contains functions pertinent to operations with complex

numbers:

These functions are described in Chapter 4.

The CONSTANTS sub-menu

The CONSTANTS sub-menu provides access to the calculator mathematical

constants. These are described in Chapter 3:

Page J-2

The HYPERBOLIC sub-menu

The HYPERBOLIC sub-menu contains the hyperbolic functions and their

inverses. These functions are described in Chapter 3.

The INTEGER sub-menu

The INTEGER sub-menu provides functions for manipulating integer numbers

and some polynomials. These functions are presented in Chapter 5:

The MODULAR sub-menu

The MODULAR sub-menu provides functions for modular arithmetic with

numbers and polynomials. These functions are presented in Chapter 5:

Page J-3

The POLYNOMIAL sub-menu

The POLYNOMIAL sub-menu includes functions for generating and

manipulating polynomials. These functions are presented in Chapter 5:

The TESTS sub-menu

The TESTS sub-menu includes relational operators (e.g., ==, <, etc.), logical

operators (e.g., AND, OR, etc.), the IFTE function, and the ASSUME and

UNASSUME commands.

Relational and logical operators are presented in Chapter 21 in the context of

programming the calculator in User RPL language. The IFTE function is

introduced in Chapter 3. Functions ASSUME and UNASSUME are presented

next, using their CAS help facility entries (see Appendix C).

ASSUME UNASSUME

Page K-1

Appendix K

The MAIN menu

The MAIN menu is available in the command catalog. This menu include the

following sub-menus:

The CASCFG command

This is the first entry in the MAIN menu. This command configures the CAS.

For CAS configuration information see Appendix C.

The ALGB sub-menu

The ALGB sub-menu includes the following commands:

These functions, except for 0.MAIN MENU and 11.UNASSIGN are available

in the ALG keyboard menu (‚×). Detailed explanation of these

functions can be found in Chapter 5. Function UNASSIGN is described in

the following entry from the CAS menu:

Page K-2

The DIFF sub-menu

The DIFF sub-menu contains the following functions:

These functions are also available through the CALC/DIFF sub-menu (start with

„Ö). These functions are described in Chapters 13, 14, and 15, except

for function TRUNC, which is described next using its CAS help facility entry:

The MATHS sub-menu

The MATHS menu is described in detail in Appendix J.

The TRIGO sub-menu

The TRIGO menu contains the following functions:

These functions are also available in the TRIG menu (‚Ñ). Description of

these functions is included in Chapter 5.

Page K-3

The SOLVER sub-menu

The SOLVER menu includes the following functions:

These functions are available in the CALC/SOLVE menu (start with „Ö).

The functions are described in Chapters 6, 11, and 16.

The CMPLX sub-menu

The CMPLX menu includes the following functions:

The CMPLX menu is also available in the keyboard (‚ß). Some of the

functions in CMPLX are also available in the MTH/COMPLEX menu (start with

„´). Complex number functions are presented in Chapter 4.

The ARIT sub-menu

The ARIT menu includes the following sub-menus:

The sub-menus INTEGER, MODULAR, and POLYNOMIAL are presented in

detail in Appendix J.

Page K-4

The EXP&LN sub-menu

The EXP&LN menu contains the following functions:

This menu is also accessible through the keyboard by using „Ð. The

functions in this menu are presented in Chapter 5.

The MATR sub-menu

The MATR menu contains the following functions:

These functions are also available through the MATRICES menu in the

keyboard („Ø). The functions are described in Chapters 10 and 11.

The REWRITE sub-menu

The REWRITE menu contains the following functions:

Page K-5

These functions are available through the CONVERT/REWRITE menu (start

with „Ú). The functions are presented in Chapter 5, except for

functions XNUM and XQ, which are described next using the corresponding

entries in the CAS help facility (IL@HELP ):

XNUM XQ

Page L-1

Appendix L

Line editor commands

When you trigger the line editor by using „˜ in the RPN stack or in ALG

mode, the following soft menu functions are provided (press L to see the

remaining functions):

The functions are briefly described as follows:

SKIP: Skips characters to beginning of word.

SKIP: Skips characters to end of word.

DEL: Delete characters to beginning of word.

DEL: Delete characters to end of word.

DEL L: Delete characters in line.

INS: When selected inserts characters at cursor location. If not selected,

the cursor replaces characters (overwrites) instead of inserting

characters.

EDIT: Edits selection.

BEG: Move to beginning of word.

END: Mark end of selection.

INFO: Provides information on Command Line editor, e.g.,

Page L-2

The items show in this screen are self-explanatory. For example, X and Y

positions mean the position on a line (X) and the line number (Y). Stk Size

means the number of objects in the ALG mode history or in the RPN stack.

Mem(KB) means the amount of free memory. Clip Size is the number of

characters in the clipboard. Sel Size is the number of characters in the current

selection.

EXEC: Execute command selected.

HALT: Stop command execution.

The line editor also provide the following sub-menus:

SEARCH: Search characters or words in the command line. It includes the

following functions:

GOTO: Move to a desired location in the command line. It includes the

following functions:

Style: Text styles that can be used in the command line:

Page M-5

DIV2MOD, 5-12

DIV2MOD, 5-15

Divergence, 15-4

DIVIS, 5-10

DIVMOD, 5-12

DIVMOD, 5-15

DO construct, 21-61

DOERR, 21-64

DOLIST, 8-12

DOMAIN, 13-9

DOSUBS, 8-11

DOT, 9-11

Dot product, 9-11

DOT+ DOT-, 12-45

Double integrals, 14-6

DRAW, 12-21, 22-4

DRAW3DMATRIX, 12-54

Drawing functions programs, 22-22

DRAX, 22-4

DROITE, 4-9

DROP, 9-21

DTAG, 23-1

e, 3-16,

EDIT, 2-33

EDIT, L-1

Editor commands, L-1

EGCD, 5-19

EGDC, 5-11

EGV, 11-46

EGVL, 11-45

Eigenvalues, 11-9, 11-44

Eigenvectors, 11-9, 11-44

Electric units, 3-20

END, 2-26

ENDSUB, 8-11

Energy units, 3-19

Engineering format, 1-20

ENGL, 3-29

Entering vectors, 9-2

EPS, 2-35

EPSX0, 5-23

EQ, 6-28

Equation Writer (EQW), 2-10

Equation writer properties, 1-28

Equation Writer, Selection Tree,

E-1

Equations, linear systems, 11-16

EQW: BIG, 2-11

EQW: CMDS, 2-11

EQW: CURS, 2-11,

EQW: Derivatives, 2-30

EQW: EDIT, 2-11

EQW: EVAL, 2-11

EQW: FACTOR, 2-11

EQW: HELP, 2-11

EQW: Integrals, 2-31

EQW: SIMPLIFY, 2-11

EQW: Summations, 2-28

ERASE, 12-21, 12-58, 22-4

ERR0, 21-65

ERRM, 21-65

ERRN, 21-64

Error trapping in programming,

21-64

Errors in hypothesis testing, 18-35

Errors in programming, 21-64

EULER, 5-10

Euler constant, 16-56

Euler equation, 16-53

Euler formula, 4-1

Page M-6

EVAL, 2-5

Exact CAS mode, C-4

EXEC, L-2

EXP, 3-6

EXP2POW, 5-29

EXPAND, 5-5

EXPANDMOD, 5-12

EXPLN, 5-8

EXPLN, 5-29

EXPM, 3-9

Exponential distribution, 17-7

Extrema, 13-12

Extreme points, 13-12

EYEPT, 22-10

F distribution, 17-12

FACTOR, 2-10

Factorial, 3-14

Factorial symbol(!), G-2

Factoring an expression, 2-24

FACTORMOD, 5-12

FACTORS, 5-10

FANNING, 3-31

Fast 3D plots, 12-35

Fast Fourier transform, 16-49

Fast Replace All, L-3

FCOEF, 5-11

FDISTRIB, 5-29

FFT, 16-49

Fields, 15-1

File manager.. menu.., F-4

FILES, 2-32

Financial calculations, 6-9

Find next.., L-3

Finite arithmetic ring, 5-14

Finite population, 18-3

Fitting data, 18-10

Fixed format, 1-18

Flags, 24-1

FLOOR, 3-14

FOR construct, 21-59

Force units, 3-19

FOURIER, 16-27

Fourier series, 16-27

Fourier series and ODEs, 16-42

Fourier series for square wave, 16-40

Fourier series for triangular wave,

16-35

Fourier series, complex, 16-27

Fourier transforms, 16-43

Fourier transforms, convolution,

16-49

Fourier transforms, definitions,

16-46

FP, 3-14

Fractions, 5-24

Frequency distribution, 18-5

Frobenius norm, 11-17

FROOTS, 5-11

FROOTS, 5-26

Full pivoting, 11-34, 11-39

Function definition, 3-36

Function plot, 12-2

FUNCTION plot operation, 12-13

FUNCTION plots, 12-6

Function, table of values, 12-17,

12-26

Functions, multi-variate, 14-1

Fundamental theorem of algebra,

6-7

Page M-7

GAMMA, 3-15

Gamma distribution, 17-6

GAUSS, 11-53

Gaussian elimination, 11-28

Gauss-Jordan elimination, 11-28

11-37 11-39

GCD, 5-11, 5-19

GCDMOD, 5-12

Geometric mean, 18-3

Geometric mean, 8-16

GET, 10-6

GETI, 8-11

Global variable scope, 21-4

Global variable, 21-2

GOR, 22-32

Goto Line, L-4

GOTO menu, L-2 L-3

Goto Position, L-4

Grades, 1-22

Gradient, 15-1

Graphic objects, 22-30

Graphical solution of ODEs, 16-60

Graphics animation, 22-26

Graphics options, 12-1

Graphics programming, 22-1

Graphs, 12-1

Graphs polar, 12-19

Graphs conic curves, 12-21

Graphs parametric, 12-23

Graphs differential equations, 12-26

Graphs truth plots, 12-29

Graphs bar plots, 12-30

Graphs histograms, 12-30

Graphs scatterplots, 12-30

Graphs slope fields, 12-34

Graphs Fast 3D plots, 12-35

Graphs wireframe plots, 12-34

Graphs Ps-Contour plots, 12-39

Graphs Y-Slice plots, 12-41

Graphs Gridmap plots, 12-42

Graphs Pr-Surface plots, 12-43

Graphs Zooming, 12-49

Graphs SYMBOLIC menu, 12-51

Graphs saving, 12-7

GRD, 3-1

Greek letters, D-3, G-2

Gridmap plots, 12-42

GROB, 22-30

GROB programming, 22-33

GROB menu, 22-31

GROBADD, 12-52

Grouped data, 8-18

Grouped data statistics, 8-18

Grouped data variance, 8-18

GXOR, 22-32

HADAMARD, 11-5

HALT, L-2

Harmonic mean, 8-15

HEAD, 8-11

Header size, 1-29

Heaviside's step function, 16-15

HELP, 2-26

HERMITE, 5-11, 5-20

Hermite polynomials, 16-60

HESS, 15-2

Hessian matrix, 15-2

HEX, 19-2

HEX, 3-1

Hexadecimal numbers, 19-7

Page M-8

Higher-order derivatives, 13-13

Higher-order partial derivatives,

14-3

HILBERT, 10-14

Histograms, 12-30

HMS-, 25-3

HMS+, 25-3

HMS-->, 25-3

HORNER, 5-11, 5-20

H-VIEW, 12-20

Hyperbolic functions graphs, 12-

Hypothesis testing, 18-34

Hypothesis testing errors, 18-35

Hypothesis testing in linear

regression, 18-52

Hypothesis testing in the calculator,

18-42

Hypothesis testing on variances,

18-46

HZIN, 12-50

HZOUT, 12-50

i, 3-16

I/O functions menu, F-2

I→R, 5-28

IABCUV, 5-11

IBERNOULLI, 5-11

ICHINREM, 5-11

Identity matrix, 10-1

Identity matrix, 11-6

IDIV2, 5-11

IDN, 10-9

IEGCD, 5-11

IF...THEN..ELSE...END, 21-48

IF...THEN..END, 21-47

IFERR sub-menu, 21-65

IFTE, 3-35

ILAP, 16-11

Illumination units, 3-20

IM, 4-6

IMAGE, 11-54

Imaginary part, 4-1

Implicit derivatives, 13-7

Improper integrals, 13-21

Increasing-power CAS mode, C-9

INDEP, 22-6,

Independent variable in CAS, C-2,

Infinite series, 13-21, 13-23

INFO, 22-4

INPUT, 21-22 ,

Input forms programming, 21-21

Input forms use of A-1

Input string prompt programming,

21-21

Input-output functions menu, F-2

INS , L-1

INT, 13-14

Integer numbers, C-5

Integers, 2-1

Integrals, 13-14

Integrals definite, 13-15

Integrals step-by-step, 13-17

Integrals improper, 13-21

Integrals double, 14-6

Integrals multiple, 14-6

Integration change of variable, 13-19

Integration substitution, 13-18

Integration techniques, 13-18

Integration by partial fractions, 13-20

Integration by parts, 13-19

Page M-9

Interactive drawing, 12-43

Interactive input programming, 21-19

Interactive plots with PLOT menu,

22-15

Interactive self-test, G-3

INTVX, 13-14

INV, 4-4

INV, L-4

Inverse cdf’s, 17-14

Inverse cumulative distribution

functions, 17-13

Inverse function graph, 12-12

Inverse Laplace transforms, 16-10

Inverse matrix, 11-6

INVMOD, 5-12

IP, 3-14

IQUOT, 5-11

IREMAINDER, 5-11

Irrotational fields, 15-5

ISECT in plots, 12-7

ISOL, 6-1

ISOM, 11-54

ISPRIME? , 5-11

ITALI, L-4

Jacobian, 14-7

JORDAN, 11-47

KER, 11-55

Key Click, 1-23

Keyboard, 1-10, B-1

Keyboard main key functions B-2

Keyboard alternate key functions

B-4

Keyboard left-shift functions B-5

Keyboard right-shift functions B-8

Keyboard ALPHA characters B-9

Keyboard ALPHA-left-shift

characters B-10

Keyboard ALPHA-right-shift

characters, B-12

Keyboard ALPHA function, 1-12

Keyboard left-shift function, 1-12

Keyboard main function, 1-12

Keyboard right-shift function, 1-12

Kronecker's delta, 10-1

LABEL, 12-47

Labels, L-4

LAGRANGE, 5-11, 5-21

Laguerre's equation, 16-60

LAP, 16-11

LAPL, 15-4

Laplace transforms, 16-10

Laplace transforms inverse, 16-10

Laplace transforms and ODEs, 16-17

Laplace transforms theorems, 16-12

Laplace's equation, 15-4

Laplacian, 15-4

Last Stack, 1-24

LCM, 5-11

LCM, 5-21,

LCXM, 11-1

LDEC, 16-4

Least-square function, 11-23

Least-square method, 18-49

Left-shift functions, B-5

Page M-10

LEGENDRE, 5-11, 5-22

Legendre's equation, 16-54

Length units, 3-18

LGCD, 5-10

lim, 13-2

Limits, 13-1

LIN, 5-5

LINE, 12-46

Line editor commands, L-1

Line editor properties, 1-26

Linear algebra, 11-1

Linear applications, 11-54

Linear differential equations, 16-4

Linear regression confidence

intervals, 18-52

Linear regression hypothesis

testing, 18-52

Linear regression additional notes,

18-49

Linear regression prediction error,

18-51

Linear system of equations, 11-16

Linearized relationships, 18-11

LINSOLVE, 11-40

LIST, 2-33

LIST menu, 8-8

List of CAS help facility, H-1

List of command catalog, I-1

Lists, 8-1

LN, 3-5

Ln(X) graph, 12-9

LNCOLLECT, 5-5

LNP1, 3-9,

Local variables, 21-2

LOG, 3-5

LOGIC menu, 19-5

Logical operators, 21-43

Lower-triangular matrix, 11-49

LQ, 11-51

LQ decomposition, 11-51

LSQ, 11-23

LU, 11-49

LU decomposition, 11-49

LVARI, 7-12

Maclaurin series, 13-21

MAD, 11-47

Main diagonal, 10-1

MAIN menu, G-3 K-1

MAIN/ALGB menu, K-1

MAIN/ARIT menu, K-3

MAIN/CASCFG command, K-1

MAIN/CMPLX menu, K-3

MAIN/DIFF menu, K-2

MAIN/EXP&LN menu, K-4

MAIN/MATHS menu (MATHS

menu), J-1

MAIN/MATR menu, K-4

MAIN/REWRITE menu, K-4

MAIN/SOLVER menu, K-3

MAIN/TRIGO menu, K-2

Manning's equation, 21-19

MANT, 3-14

MAP, 8-12

MARK, 12-46

Mass units, 3-19

Math menu .., F-5

MATHS menu, G-3 J-1

MATHS/CMPLX menu, J-1

MATHS/CONSTANTS menu, J-1

MATHS/HYPERBOLIC menu, J-2

Page M-11

MATHS/INTEGER menu, J-2

MATHS/MODULAR menu, J-2

MATHS/POLYNOMIAL menu, J-3

MATHS/TESTS menu, J-3

Matrices, 10-1

Matrix, 10-1

Matrix augmented, 11-30

Matrix Jordan-circle decomposition

11-54

Matrix "division", 11-26

Matrix Writer, 10-2

Matrix factorization, 11-49

MATRIX menu, 10-3

Matrix multiplication, 11-2

Matrix operations, 11-1

Matrix quadratic forms, 11-51

Matrix term-by-term multiplication,

11-5,

Matrix transpose, 10-1,

Matrix writer, 9-3 ,

MATRIX/MAKE menu, 10-3,

Matrix-vector multiplication, 11-3,

MAX, 3-13,

Maximum, 13-13,

Maximum, 14-5,

MAXR, 3-16,

Mean, 18-3,

Measures of central tendency,

18-3

Measures of spreading, 18-3

Median, 18-3

MENU, 12-47

Menu numbers, 20-2

Menus, 1-3

Menus not accessible through

keyboard, G-3

MES, 7-10

Message box programming,

21-37

Method of least squares, 18-50

MIN, 3-13

Minimum, 13-12

Minimum, 14-5

MINIT, 7-13

MINR, 3-16

MITM, 7-12

MKSISOM, 11-55

MOD, 3-13

Mode, 18-4

MODL, 22-13

MODSTO, 5-12

Modular arithmetic, 5-12

Modular inverse, 5-17

Modular programming, 22-36

MODULO, 2-35

Modulus in CAS, C-3

Moment of a force, 9-17

MSGBOX, 21-31

MSLV, 7-4

MSOLV, 7-13

MTH menu, 3-7

MTH/LIST menu, 8-8

MTH/PROBABILITY menu, 17-1

MTH/VECTOR menu, 9-10

MTRW, 9-3

Multiple integrals, 14-6

Multiple linear fitting, 18-56

Multi-variate calculus, 14-1

MULTMOD, 5-12

NDIST, 17-10

Page M-12

NEG, 4-6

Nested IF...THEN..ELSE..END,

21-49

NEW, 2-33

NEXt eQuation, 12-6

NEXTPRIME, 5-11

Non-CAS commands, C-13

Non-linear differential equations,

16-4

Non-verbose CAS mode, C-7

NORM menu, 11-6

Normal distribution, 17-10

Normal distribution standard,

17-17

Normal distribution cdf, 17-10

NOT, 19-6

NSUB, 8-12

NUM, 23-1

NUM.SLV, 6-14

NUM.SLV input forms, A-1

Number in bases, 19-1

Number format, 1-17

Numeric CAS mode, C-3

Numeric solver menu, F-3

Numeric vs. symbolic CAS mode,

C-3

Numerical solution of ODEs,

16-60

Numerical solution to stiff ODEs,

16-68

Numerical solver, 6-5

NUMX, 22-10

NUMY, 22-10

OBJ-->, 9-19

Objects, 2-1

objects, 24-1

OCT, 19-2

Octal numbers, 3-2

ODEs Laplace transform

applications, 16-17

ODEs Fourier series, 16-46

ODEs Graphical solution, 16-60

ODEs Numerical solution, 16-60

ODEs (ordinary differential

equations), 16-1

ODETYPE, 16-8

OFF, 1-2

ON, 1-2

OPER menu, 11-14

Operating mode, 1-13

Operations with units, 3-25

Operators, 3-7

OR, 19-5

ORDER, 2-33

Organizing data, 2-32

Orthogonal matrices, 11-49

Other characters, D-3

Output tagging, 21-33

PA2B2, 5-11

Paired sample tests, 18-40

Parametric plots, 12-23

PARTFRAC, 5-5

Partial derivatives, 14-1

Partial derivatives higher-order 14-3

Partial derivatives chain rule 14-4

Partial fractions integration, 13-20

Partial pivoting, 11-33

PASTE, 2-26

Page M-13

PCAR, 11-44

PCOEF, 5-11, 5-22

PDIM, 22-20

Percentiles, 18-14,

PERIOD, 2-35 16-35

PERM, 17-2

Permutation matrix, 11-34

Permutations, 17-1

PEVAL, 5-23

PGDIR, 2-43

Physical constants, 3-28

PICT, 12-8

Pivoting, 11-33

PIX?, 22-22

Pixel coordinates, 22-25

Pixel references, 19-7

PIXOFF, 22-22

PIXON, 22-22

Plane in space, 9-18

PLOT, 12-52

PLOT environment, 12-3

Plot functions menu, F-1

PLOT menu, 22-1

PLOT menu interactive plots, 22-17

PLOT menu (menu 81), G-3

PLOT operations, 12-5

Plot setup, 12-52

PLOT SETUP environment, 12-3

PLOT WINDOW environment, 12-4

PLOT/FLAG menu, 22-14

PLOT/STAT menu, 22-11

PLOT/STAT/DATA menu, 22-12

PLOTADD, 12-52

Plots program-generated, 22-17

Poisson distribution, 17-5

Polar coordinate plot, 12-19

Polar coordinates double integrals,

14-9

Polar plot, 12-19

Polar representation, 4-1

POLY sub-menu, 6-30

Polynomial equations, 6-6

Polynomial fitting, 18-56

Polynomials, 5-18

Population, 18-3

POS, 8-11

POTENTIAL, 15-3

Potential function, 15-3 15-6

Potential of a gradient, 15-3

Power units, 3-20

POWEREXPAND, 5-29

POWMOD, 5-12

PPAR, 12-2, 12-11

Prediction error linear regression,

18-51

Pressure units, 3-20

PREVAL, 13-15

PREVPRIME, 5-11

PRG menu, 21-5

PRG menu shortcuts, 21-10

PRG/MODES/KEYS menu, 20-5

PRG/MODES/MENU menu, 20-1

PRIMIT, 2-35

Probability, 17-1

Probability density function, 17-6

Probability distributions discrete

17-4

Probability distributions continuous,

17-6

Probability distributions for

statistical inference, 17-9

Probability mass function, 17-4

Page M-14

Program branching, 21-46

Program loops, 21-53

Program-generated plots, 22-17

Programming, 21-1

Programming sequential 21-19

Programming interactive input, 21-19

Programming input string prompt,

21-21

Programming debugging, 21-22

Programming choose box, 21-31

Programming output, 21-33

Programming tagged output, 21-34

Programming message box, 21-37

Programming using units, 21-37

Programming error trapping, 21-64

Programming graphics 22-1

Programming drawing functions,

22-24,

Programming modular, 22-36

Programming drawing commands,

22-19

Programming input forms, 21-27

Programming plots, 22-14

Programming with GROBs, 22-33

Programs with drawing functions,

22-24

PROOT, 5-22

PROPFRAC, 5-10, 5-25

Pr-Surface plots, 12-43

Ps-Contour plots, 12-39

PSI, 3-15

Psi, 3-15

PTAYL, 5-11, 5-22

PTYPE, 22-4

PUT, 8-10

PUTI, 10-6

PVIEW, 22-22

PX-->C, 19-7

QR, 11-51

QR decomposition, 11-51

QUADF, 11-52

Quadratic form diagonal

representation, 11-53

QUIT, 3-30

QUOT, 5-11

QUOTIENT, 5-23

QXA, 11-53

R→B, 19-3

R→C, 4-6

R→D, 3-14

R→I, 5-28

RAD, 3-1

Radians, 1-22

Radiation units, 3-20

RAND, 17-2

Random numbers, 17-2

RANK, 11-10

Rank of a matrix, 11-10

RANM, 10-11

RCI, 10-25

RCIJ, 10-26

RCLKEYS, 20-6

RCLMENU, 20-1

RCWS, 19-4

RDM, 10-10

RDZ, 17-3

RE, 4-6

Page M-15

Real CAS mode, C-6

Real numbers, C-6

Real numbers vs. Integer numbers,

C-6

Real objects, 2-1

Real part, 4-1

REALASSUME, 2-35

RECT, 4-3

RECV, 2-34,

Relational operators, 21-43

REMAINDER, 5-11, 5-23

RENAM, 2-34

REPL, 10-12

Replace, L-3

Replace All, L-3

Replace Selection, L-3

Replace/Find Next, L-3

RES, 22-6

RESET, 22-8

Restart calculator, G-3

RESULTANT, 5-11

Resultant of forces, 9-16

REVLIST, 8-9

REWRITE menu, 5-29

Right-shift functions, B-8 ,

Rigorous CAS mode, C-9

RISCH, 13-14

RKF, 16-71

RKFERR, 16-74

RKFSTEP, 16-72

RL, 19-6

RLB, 19-7

RND, 3-14

RNRM, 11-8

ROOT, 6-27

ROOT in plots, 12-6

ROOT sub-menu, 6-27

ROW-, 10-24

Row norm, 11-8

Row vectors, 9-19

ROW+, 10-24

ROW→, 10-23

RPN mode, 1-13

RR, 19-6

RRB, 19-7

REF. RREF, rref, 11-47

RRK, 16-71

RSBERR, 16-74

RSD, 11-43

RSWP, 10-25

R∠Z, 3-1

Saddle point, 14-5,

Sample correlation coefficient,

18-11

Sample covariance, 18-11

Sample vs. population, 18-5

Saving a graph, 12-7

Scalar field, 15-1

SCALE, 22-7

SCALEH, 22-7

SCALEW, 22-7

Scatterplots, 12-32

Scientific format, 1-20

Scope global variable 21-4

SEARCH menu, L-2 L-3

Selection tree in Equation Writer,

E-1

SEND, 2-34

SEQ, 8-11

Sequential programming, 21-15

Page M-16

Series, 13-23

Series Maclaurin, 13-23

Series Taylor, 13-23

SERIES, 13-23

Series Fourier, 16-27

Setting time and date, 25-2

SHADE in plots, 12-6

Shortcuts, G-1

SI, 3-29

SIGMA, 13-14

SIGMAVX, 13-14

SIGN, 3-14

SIGN, 4-6

SIGNTAB, 12-52 13-10

SIMP2, 5-10, 5-24

SIMPLIFY, 5-29

Simplify non-rational CAS setting,

C10

Simplifying an expression, 2-23

SIN, 3-7

Single-variable statistics, 18-2

Singular value decomposition, 11-8

SINH, 3-9

SIZE, 8-10

SIZE, 10-7

SKIP, L-1

SL, 19-6 ,

SLB, 19-7

Slope fields, 12-34

Slope fields for differential

equations, 16-3

SLOPE in plots, 12-7

SNRM, 11-7

SOFT menus, 1-3

SOLVE, 5-5

SOLVE, 6-2, 7-1,

SOLVE menu, 6-27

SOLVE menu (menu 74), G-3

SOLVE/DIFF menu, 16-69

SOLVEVX, 6-4

SOLVR menu, 6-28

SORT, 2-34

Special characters, G-2

Speed units, 3-19

SPHERE, 9-15

SQ, 3-5

Square root, 3-5

Square wave Fourier series 16-39

SR, 19-6

SRAD, 11-9

SRB, 19-7

SREPL, 23-3

SST, 21-35

Stack properties, 1-27

Standard deviation, 18-4

Standard format, 1-17

Standard normal distribution, 17-

17,

START ..STEP construct, 21-58

START...NEXT construct, 21-54

STAT menu, 18-15

STAT menu (menu 96), G-3

Statistical inference probability

distributions, 17-9

Statistics, 18-1

Step function (Heaviside's ), 16-15

Step-by-step CAS mode, C-7

Step-by-step derivatives, 13-16

Step-by-step integrals, 13-16

STEQ, 6-14

Stiff differential equations, 16-71

Stiff ODE, 16-68

Page M-17

Stiff ODEs numerical solution, 16-69

Strings, 23-1

STO, 2-46

STOALARM, 25-4

STOKEYS, 20-6

STREAM, 8-12

String concatenation, 23-2

Student t distribution, 17-11

STURM, 5-11

STURMAB, 5-11

STWS, 19-4

Style menu, L-4

SUB, 10-11

Sub-directories creating, 2-38

Sub-directories deleting, 2-42

Sub-expressions, 2-17

SUBST, 5-5

SUBTMOD, 5-12

SUBTMOD, 5-16

Sum of squared errors (SSE),

18-62

Sum of squared totals (SST), 18-62,

Summary statistics, 18-13

SVD, 11-50

SVL, 11-50

SYLVESTER, 11-53

SYMB/GRAPH menu, 12-52

Symbolic CAS mode, C-3

SYMBOLIC menu, 12-51

Synthetic division, 5-26,

SYST2MAT, 11-42,

System flag

(EXACT/APPROX), G-1,

System flag 117 (CHOOSE/SOFT),

1-4, G-2,

System flag 95 (ALG/RPN), G-1

system flags, 24-3

System level operation, G-3

System of equations, 11-16

System tests, G-3

Table, 12-17, 12-26

TABVAL, 12-52, 13-9

TABVAR, 12-52, 13-11

Tagged output programming,

21-34

TAIL, 8-11

TAN, 3-7

TANH, 3-9

Taylor polynomial, 13-23

Taylor series, 13-24

TAYLR, 13-25

TAYLR0, 13-24

TCHEBYCHEFF, 5-24

Tchebycheff polynomials, 16-58

TDELTA, 3-31

Techniques of integration, 13-18

Temperature units, 3-20

TEXPAND, 5-5

Text editor.. menu, F-5

Three-dimensional plot programs,

22-15

Three-dimensional vector, 9-13

TICKS, 25-3

Time setting 25-2

TIME, 25-3

Time & date... menu, F-3

Time functions, 25-1

TIME menu, 25-1

Time setting, 1-7

TIME tools, 25-2

Page M-18

Time units, 3-19

Times calculations 25-4

TINC, 3-32

TITLE, 7-15

TLINE, 12-46

TLINE, 22-20

TMENU, 20-1

TOOL menu, 1-6

TOOL menu: CASCMD, 1-7

TOOL menu: CLEAR, 1-7

TOOL menu: EDIT, 1-6

TOOL menu: HELP, 1-7

TOOL menu: PURGE, 1-7

TOOL menu: RCL, 1-7

TOOL menu: VIEW, 1-7

Total differential, 14-5

TPAR, 12-18

TRACE, 11-13

TRAN, 11-14

Transforms Laplace, 16-10

Transpose, 10-1,

Triangle solution, 7-10,

Triangular wave Fourier series 16-35,

TRIG menu, 5-8,

Trigonometric functions graphs,

12-16,

TRN, 10-8,

TRNC, 3-14

Truth plots, 12-29

TSTR, 25-3

TVM menu, 6-32

TVMROOT, 6-32

Two-dimensional plot programs,

22-14

Two-dimensional vector, 9-12

TYPE, 24-2

UBASE, 3-21

UFACT, 3-27

UNASSIGN, K-1

UNASUMME, J-3

UNDE, L-4

UNDO, 2-61

UNIT, 3-29

Unit prefixes, 3-24

Units, 3-17

Units in programming, 21-37

Upper-triangular matrix, 11-41,

User RPL language, 21-1

User-defined keys, 20-1

Using input forms, A-1,

UTILITY menu (menu 113), G-3

UTPC, 17-12

UTPF, 17-13

UTPN, 17-10

UTPT, 17-11

UVAL, 3-27

V, 9-12

VALUE, 3-30

VANDERMONDE, 10-14

VANDERMONDE, 18-59

Variable scope, 21-4

Variables, 26-1

Variance confidence intervals, 18-33

Variance, 18-4

Variance inferences, 18-47

Vector building, 9-12

Vector analysis, 15-1

Page M-19

Vector elements, 9-7

Vector fields, 15-1

Vector fields curl, 15-5

Vector fields divergence 15-4

VECTOR menu, 9-10

Vector potential, 15-6

Vectors, 9-1

Verbose CAS mode, C-6

Verbose vs. non-verbose CAS

mode, C-7

VIEW in plots, 12-6

Viscosity, 3-20

Volume units, 3-19

VPAR, 12-44

VPAR, 22-10

VPOTENTIAL, 15-6

VTYPE, 24-2

V-VIEW, 12-20

VX, 2-35

VX, 5-20

VZIN, 12-50

"Warm" calculator restart, G-3

Weber's equation, 16-59

Weibull distribution, 17-7

Weighted average, 8-17

WHILE construct, 21-62

Wireframe plots, 12-37

Wordsize, 19-4

XCOL, 22-13

XNUM, K-5

XOR, 19-5

XPON, 3-14

XQ, K-5

XRNG, 22-6

XROOT, 3-5

XSEND, 2-34

XVOL, 22-10

XXRNG, 22-10

XYZ, 3-1

YCOL, 22-13

YRNG, 22-6

Y-Slice plots, 12-41

YVOL, 22-10

YYRNG, 22-10

ZAUTO, 12-50

ZDECI, 12-50

ZDFLT, 12-50

ZEROS, 6-4

ZFACT, 12-49

ZFACTOR, 3-31

ZIN, 12-49

ZINTG, 12-51

ZLAST, 12-49

ZOOM, 12-19

Zooming, 12-49

ZOUT, 12-49

ZSQR, 12-51

ZTRIG, 12-51

ZVOL, 22-10

Other characters

!, 17-2

Page M-20

%, 3-12

%CH, 3-12

%T, 3-12

Σ, 2-28

ΣDAT, 18-5,

∆LIST, 8-9

ΣLIST, 8-9

ΠLIST, 8-9

ΣPAR, 22-13

ARRY, 9-21

ARRY, 9-6

BEG, L-1

COL, 10-18

DATE, 25-3

DEL, L-1

DIAG, 10-13

END, L-1

GROB, 22-31

HMS, 25-3

LCD, 22-32

LIST, 9-23

ROW, 10-22

SKIP, L-1

STK, 3-30

STR, 23-1

TAG, 21-33

TAG, 23-1

TIME, 25-3

UNIT, 3-27

V2, 9-12

V3, 9-13

Page W-1

Limited Warranty

hp 48gII graphing calculator; Warranty period: 12 months

HP warrants to you, the end-user customer, that HP hardware,

accessories and supplies will be free from defects in materials and

workmanship after the date of purchase, for the period specified

above. If HP receives notice of such defects during the warranty period,

HP will, at its option, either repair or replace products which prove to

be defective. Replacement products may be either new or like-new.

2. HP warrants to you that HP software will not fail to execute its

programming instructions after the date of purchase, for the period

specified above, due to defects in material and workmanship when

properly installed and used. If HP receives notice of such defects

during the warranty period, HP will replace software media which

does not execute its programming instructions due to such defects.

3. HP does not warrant that the operation of HP products will be

uninterrupted or error free. If HP is unable, within a reasonable time, to

repair or replace any product to a condition as warranted, you will be

entitled to a refund of the purchase price upon prompt return of the

product with proof of purchase.

4. HP products may contain remanufactured parts equivalent to new in

performance or may have been subject to incidental use.

5. Warranty does not apply to defects resulting from (a) improper or

inadequate maintenance or calibration, (b) software, interfacing, parts

or supplies not supplied by HP, (c) unauthorized modification or misuse,

(d) operation outside of the published environmental specifications for

the product, or (e) improper site preparation or maintenance.

6. HP MAKES NO OTHER EXPRESS WARRANTY OR CONDITION

WHETHER WRITTEN OR ORAL. TO THE EXTENT ALLOWED BY LOCAL

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PARTICULAR PURPOSE IS LIMITED TO THE DURATION OF THE

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This warranty gives you specific legal rights and you might also have

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Page W-2

7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS

WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE

REMEDIES. EXCEPT AS INDICATED ABOVE, IN NO EVENT WILL HP

OR ITS SUPPLIERS BE LIABLE FOR LOSS OF DATA OR FOR DIRECT,

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HP shall not be liable for technical or editorial errors or omissions

contained herein.

FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND NEW ZEALAND: THE

WARRANTY TERMS CONTAINED IN THIS STATEMENT, EXCEPT TO THE EXTENT

LAWFULLY PERMITTED, DO NOT EXCLUDE, RESTRICT OR MODIFY AND ARE IN

ADDITION TO THE MANDATORY STATUTORY RIGHTS APPLICABLE TO THE

SALE OF THIS PRODUCT TO YOU.

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Page W-3

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ROTC = Rest of the country

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Page W-4

Regulatory information

This section contains information that shows how the hp 48gII graphing calculator complies with

regulations in certain regions. Any modifications to the calculator not expressly approved by

Hewlett-Packard could void the authority to operate the 48gII in these regions.

USA

This calculator generates, uses, and can radiate radio frequency energy and may interfere with

radio and television reception. The calculator complies with the limits for a Class B digital device,

pursuant to Part 15 of the FCC Rules. These limits are designed to provide reasonable protection

against harmful interference in a residential installation.

However, there is no guarantee that interference will not occur in a particular installation. In the

unlikely event that there is interference to radio or television reception(which can be determined

by turning the calculator off and on), the user is encouraged to try to correct the interference by

one or more of the following measures:



Reorient or relocate the receiving antenna.

 Relocate the calculator, with respect to the receiver.

Connections to Peripheral Devices

To maintain compliance with FCC rules and regulations, use only the cable accessories provided.

Canada

This Class B digital apparatus complies with Canadian ICES-003. Cet appareil numerique de la

classe B est conforme a la norme NMB-003 du Canada.

Japan

この装置は、情報処理装置等電波障害自主規制協議会

(VCCI)

の基準

に基づく第二情報技術装置です。この装置は、家庭環境で使用することを目的としていますが、この装

置がラジオやテレビジョン受信機に近接して使用されると、受信障害を引き起こすことがあります。

取扱説明書に従って正しい取り扱いをしてください。

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

Union

This symbol on the product or on its packaging indicates that this product must

not be disposed of with your other household waste. Instead, it is your

responsibility to dispose of your waste equipment by handing it over to a

designated collection point for the recycling of waste electrical and electronic

equipment. The separate collection and recycling of your waste equipment at the

time of disposal will help to conserve natural resources and ensure that it is

recycled in a manner that protects human health and the environment. For more

information about where you can drop off your waste equipment for recycling, please contact

your local city office, your household waste disposal service or the shop where you purchased the

product.

HP 48gII Seite 861

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