Numbered Examples: Chapters One to Five Number and Topic Number and Topic Chapter One Chapter 1.1-1 1.3-1 1.4-1 1.6-1 4.2-1 4.4-1 4.4-2 4.4-3 4.7-1 4.7-2 Volume of a circular cylinder Analysis of temperature data Growth of a bank account Piston motion Chapter Two 2.3-1 2.3-2 2.3-3 2.3-4 2.3-5 2.3-6 2.3-7 2.4-1 2.4-2 2.5-1 3.6-1 3.7-1 Height and speed of a projectile Data sorting Flight of an instrumented rocket Time to reach a specified height A college enrollment model: Part I A college enrollment model: Part II Vectors and relative velocity Vectors and displacement Chapter Five Aortic pressure model 5.2-1 Load-line analysis of electrical circuits Current and power dissipation in resistors 5.3-1 Frequency-response low-pass filter A batch distillation process 5.3-2 5.5-1 5.5-2 5.5-3 5.6-1 5.6-2 5.6-3 Transportation route analysis Height versus velocity Manufacturing cost analysis Product cost analysis Earthquake-resistant design building Chapter Three 3.3-1 3.5-1 3.5-2 Four Design of a parallel-plate capacitor Using global variables Optimization of an irrigation channel An environmental database A student database 5.6-4 plot of a Plotting orbits A cantilever beam-deflection Temperature dynamics Hydraulic resistance Estimation of traffic flow Modeling bacteria growth Breaking strength and alloy composition Response of a biomedical instrument model Numbered Examples: Chapters Six to Nine Number and Topic Number and Topic Chapter Six Chapter Eight 6.1-1 6.2-1 Gauss elimination 6.2-2 6.2-3 6.2-4 6.2-5 6.4-1 6.4-2 6.4-3 6.4-4 6.4-5 6.4-6 6.5-1 6.5-2 An electrical-resistance Left-division method with three unknowns network Ethanol production 8.5-1 8.5-2 Calculation of cable tension The matrix inverse method A set having a unique solution An underdetermined set A statically-indeterminate problem 8.5-3 8.6-1 8.6-2 8.9-1 A singular set Production planning The least squares method An overdetermined set Breaking strength of thread Mean and standard deviation of heights Estimation of height distribution 7.2-2 7.3-1 7.3-2 Optimal production quantity 7.4-1 Robot path control using three Statistical analysis and manufacturing tolerances knot points Velocity from an accelerometer Evaluation of Fresnel's cosine integral Response of an RC circuit Decaying sine voltage applied to an RC circuit Liquid height in a spherical tank A nonlinear pendulum model Trapezoidal profile for a DC motor A Simulink model of dead-zone response 8.9-2 Simulink model of a nonlinear system 8.9-3 Simulink model of a DC motor Traffic engineering Chapter Seven 7.1-1 7.2-1 8.2-1 8.2-2 Chapter Nine 9.2-1 9.2-2 9.3-1 Intersection of two circles Positioning a robot arm Topping the Green Monster Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto INTRODUCTION TO MATLAB FOR ENGINEERS Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2001 by The McGraw-Hill Companies, Inc All rights reserved No part ofthis publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper 34567890DOCmOC0987654321 ISBN 0-07-234983-2 Publisher: Thomas E Casson Executive editor: Eric M Munson Editorial coordinator: Zuzanna Borciuch Senior marketing manager: John Wannemacher Project manager: Joyce Watters Media technology senior producer: Phillip Meek Production supervisor: Sherry L Kane Coordinator of freelance design: Michelle D Whitaker Cover images: ©PhotoDisc, Inc Senior photo research coordinator: Carrie K Burger Digital content specialist: Candy M Kuster Compositor: Techsetters, Inc Typeface: 10 / 12 New Century Schoolbook Printer: R R Donnelley & Sons Company / Crawfordsville, IN Library of Congress Cataloging-in-Publication Data Palm, William J (William John), 1944 Introduction to MATLAB for engineers / William J Palm IlL-1st ed p cm.-(McGraw-Hill best series for introductory engineering.) Includes index ISBN 0-07-234983-2 Engineering mathematics-Data processing MATLAB Numerical analysis-Data processing I Title II Series TA345 P35 2001 620'.001'51-dc21 www.mhhe.com 00-064718 CIP To my sisters: Linda and Chris WILLIAM J PALM III is Professor of Mechanical Engineering at the University of Rhode Island He received the Ph.D in Mechanical Engineering and Astronautical Sciences from Northwestern University in 1971 During his 30 years as a faculty member, he has taught 19 courses and written three other textbooks, Modeling, Analysis, and Control of Dynamic Systems (1999) and Control Systems Engineering (1986), both published by John Wiley and Sons, and Introduction to MATLABfor Engineers (1998), published by McGraw-Hill His research and industrial experience are in control systems, robotics, vibrations, and system modeling He helped found the Robotics Research Center at the University of Rhode Island and was its director from 1985 to 1993 He is also one of the developers of the university's freshman engineering course, which is based on MATLAB McGraw-Hill's BEST-BASIC ENGINEERING SERIES AND TOOLS Bertoline: Introduction to Graphics Communications for Engineers Burghardt: Introduction to Engineering Design and Problem Solving Chapman: Introduction to Fortran 90/95 Donaldson: The Engineering Student Survival Guide Eide et al.: Introduction to Engineering Design Eide et al.: Introduction to Engineering Problem Solving Eisenberg: A Beginner's Guide to Technical Communication Gottfried: Spreadsheet Tools for Engineers: Excel '97 Version Greenlaw and Hepp: Introduction to the Internet for Engineers Mathsoft's Student Edition of Mathcad 7.0 Palm: Introduction to MATLAB for Engineers Palm: Introduction to MATLAB for Engineers Pritchard, Mathcad: A Tool for Engineering Problem Solving Smith: Project Management and Teamwork Tan and D'Orazio: C Programming for Engineering and Computer Science Contents Preface xv MATLAB and Problem Solving Remote Exploration 1.1 MATLABInteractive Sessions 1.2 Menus and the Toolbar 16 1.3 Computing with MATLAB 19 1.4 Script Files and the Editor/Debugger 1.5 The MATLABHelp System 42 1.6 Problem-SolvingMethodologies 48 1.7 Summary 56 Problems 58 Array and Matrix Operations Innovative Construction 64 2.1 Arrays 66 2.2 Multidimensional Arrays 2.3 Array Operations 78 2.4 Matrix Operations 91 2.5 Polynomials 100 2.6 Summary 104 Problems 105 32 65 76 Files, Functions, and Data Structures Nanotechnology 116 3.1 Using Files 118 3.2 The Import Wizard 122 3.3 Controlling Input and Output 126 3.4 Elementary Mathematical Functions 3.5 User-Defined Functions 137 3.6 Cell Arrays 149 117 130 3.7 Structure Arrays 3.8 Summary 159 Problems 160 154 Programming with MATLAB 167 Robot-Assisted Microsurgery 166 4.1 Relational Operators 168 4.2 Logical Operators and Functions 169 4.3 Conditional Statements 175 4.4 Loops 187 4.5 The swi tch Structure 199 4.6 Debugging MATLABPrograms 201 4.7 Applications to Simulation 208 4.8 Summary 213 Problems 215 Plotting and Model Building 229 Low-SpeedAeronautics 228 5.1 xy Plotting Functions 230 5.2 Subplots and Overlay Plots 239 5.3 Special Plot Types 250 5.4 The Plot Editor 259 5.5 Function Discovery 262 5.6 Regression 274 5.7 The Basic Fitting Interface 293 5.8 Three-Dimensional Plots 296 5.9 Summary 301 Problems 302 Linear Algebraic Equations Virtual Prototyping 316 6.1 Elementary Solution Methods 318 6.2 Matrix Methods for Linear Equations 6.3 Cramer's Method 335 6.4 Underdetermined Systems 338 6.5 Overdetermined Systems 351 6.6 Summary 356 Problems 360 Probability, Statistics, and Interpolation International Engineering 370 7.1 Statistics, Histograms, and Probability 7.2 The Normal Distribution 381 7.3 Random Number Generation 390 317 323 371 372 7.4 Interpolation 398 7.5 Summary 411 Problems 412 Numerical Calculus, Differential Equations, and Simulink 419 Energy-Efficient Transportation 418 8.1 Review oflntegration and Differentiation 420 8.2 Numerical Integration 425 8.3 Numerical Differentiation 432 8.4 Analytical Solutions to Differential Equations 437 8.5 Numerical Methods for Differential Equations 444 8.6 Extension to Higher-Order Equations 463 8.7 ODE Solvers in the Control System Toolbox 472 8.8 Advanced Solver Syntax 481 8.9 Simulink 485 8.10 Summary 496 Problems 497 Symbolic Processing with MATLAB Rebuilding the Intrastructure 506 9.1 SymbolicExpressions and Algebra 509 9.2 Algebraic and Transcendental Equations 9.3 Calculus 525 9.4 Differential Equations 537 9.5 Laplace Transforms 543 9.6 SymbolicLinear Algebra 552 9.7 Summary 556 Problems 557 507 518 A Guide to Commands and Functions in This Text 568 B Some Project Suggestions 579 C Formatted Output in MATLAB 586 D References 589 Answers to Selected Problems 590 Index 593 The disp and format commands provide simple ways to control the screen output However, some users might require more control over the screen display In addition, some users might want to write formatted output to a data file The fprintf function provides this capability Its syntax is count = fprintf (fid, format, A, ), which formats the data in the real part of matrix A (and in any additional matrix arguments) under control ofthe specified format string format, and writes it to the file associated with file identifier f id A count ofthe number of bytes written is returned in the variable count The argument f id is an integer file identifier obtained from fopen (It may also be for standard output-the screen-or for standard error See fopen or more information.) Omitting fid from the argument list causes output to appear on the screen, and is the same as writing to standard output (fid = 1) The string format specifies notation, alignment, significant digits, field width, and other aspects of output format It can contain ordinary alphanumeric characters, along with escape characters, conversion specifiers, and other characters, organized as shown in the following examples Table C.1 summarizes the basic syntax of fprintf Consult MATLABhelp for more details Suppose the variable Speed has the value 63.2 To display its value using three digits with one digit to the right of the decimal point, along with a message, the session is »fprintf('The The speed is: speed 63.2 is: %3.1f\n' ,Speed) Here the "field width" is 3, because there are three digits in 63.2 You may want to specify a wide enough field to provide blank spaces or to accommodate an unexpectedly large numerical value The %sign tells MATLABto interpret the following text as codes The code \n tells MATLABto start a new line after displaying the number The output can have more than one column, and each column can have its own format For example, »r = [2.25:20:42.25] »circum = 2*pi*ri »y = [ricircum] i i »fprintf('%5.2f %11.5g\n' ,y) 2.25 14.137 22.25 139.8 42.25 265.46 Note that the fprintf function displays the transpose ofthe matrix y Format code can be placed within text For example, note how the period after the code %6.3f appears in the output at the end of the displayed text »fprintf('The first circumference The first circumference is 14.137 is %6.3f.\n' ,circum(l)) An apostrophe in displayed text requires two single quotes For example: • »fprintf('The second circle"s The second circle's radius radius %15.3e is large.\n' 2.225e+001 is large ,r(2)) A minus sign in the format code causes the output to be left justified within its field Compare the following output with the preceding example: »fprintf('The second circle"s radius The second circle's radius 2.225e+001 %-15.3e is is large.\n' large Control codes can be placed within the format string The following example uses the tab code (\ t) »fprintf('The The radii are: radii are:%4.2f \t 2.25 22.25 %4.2f \t 42.25 %4.2f\n' ,r) ,r(2)) Note that in Example 3.3-1, the disp function displays more digits than necessary for the number ofplates We can improve the display by using the fprintf function instead of disp Replacing the last three lines in the program with , fprintf('No.Plates fprintf('%2.0f E-" - \t Capacitance (F) X e12 %s\n' ,E) \t \t %4.2f\n' ,table') produces the following display: No.Plates 10 Capacitance 4.42 8.85 13.27 17.70 22.12 26.55 30.97 35.40 39.82 (F) X e12 The empty matrix E is used because the syntax of the fprintf statement requires that a variable be specified Because the first fprintf is needed to display the table title only, we need to fool MATLAB by supplying it with a variable whose value will not display Note that the fprintf command truncates the results, rather than rounding them Note also that we must use the transpose operation to interchange the rows and columns of the table matrix in order to display it properly Only the real part of complex numbers will be displayed with the fprintf command For example: »z = -4+9i; • »fprintf('Complex Complex number: number: -4.00 %2.2f \n' ,z) Instead you can display a complex number as a row vector For example, ifw = -4 + 9i: = [-4,9]; »fprintf('Real Real part is »w -4 part is %2.0f Imaginary Imaginary part is part is %2.0f \n' ,w) [Beer and Johnston, 1997] Beer, F P and E R Johnston Jr Vector Mechanics for Engineers: Dynamics 6th ed New York: McGraw-Hili, 1997 [Brown, 1994] Brown, T L.; H E LeMay, Jr.; and B E Bursten Chemistry: The Central Science 6th ed Upper Saddle River, NJ: Prentice Hall 1994 [Eide, 1998] Eide, A R.; R D Jenison; L H Mashaw; and L L Northup Introduction to Engineering Problem Solving New York: McGraw-Hili, 1998 [Felder, 1986] Felder, R M and R W Rousseau Elementary Principles of Chemical Processes New York: John Wiley & Sons, 1986 [Garber, 1999] Garber, N.J and L A Hoel Traffic and Highway Engineering 2nd ed Pacific Grove CA: PWS Publishing, 1999 [Jayaraman, 1991] Jayaraman, S Computer-Aided Problem Solving for Scientists and Engineers New York: McGraw-Hili, 1991 [Kreyzig, 1993] Kreyzig, E Advanced Engineering New York: John Wiley and Sons, 1993 Mathematics 7th ed [Kutz, 1986] Kutz, M., editor Mechanical Engineers' Handbook New York: John Wiley and Sons, 1986 [Palm, 1999] Palm, W Modeling, Analysis, and Control of Dynamic Systems 2nd ed New York: John Wiley and Sons, 1999 [Rizzoni, 1996] Rizzoni, G Principles and Applications {)f Electrical Engineering 2nd ed Homewood, IL: Irwin, 1996 [Starfield, 1990] Starfield, A M.; K A Smith; and A L Bleloch How to Model It: Problem Solving for the Computer Age New York: McGraw-Hili, 1990 [The MathWorks, Inc., 1997] Inc., 1997 Using MATLAB Natick, MA: The MathWorks, 589 Symbols + addition - subtraction * multiplication · * array multiplication 46 ~ exponentiation · ~ array exponentiation 46, 49 \ left division / right division · \ array left division 80 · / array right division 80 : colon array addressing 68, 72 vector generation 13 () parentheses function arguments 134 modifying precedence [] brackets 9, 69 · decimal point 13 · ellipsis 13 , comma column separation 13 statement separation 13 ; semicolon display suppression 13 row separation 69 % percent sign comment designation 33 format specification 586 ' apostrophe transpose 67 string designation 28 ' nonconjugated transpose 67 = assignment or replacement operator 11 == equal to 24 - = not equal to 24 < less than 24 greater than 24 >= greater than or equal to 24 & and 169 I or 169 not 169 » MATLAB prompt 593 A abs 131 acos 135 acosh137 acot135 acoth137 acsc 135 acsch 137 addpath 23 all 172 angle131 diary 121 diff 437, 528 digits 517 dir23 Dirac 557 disp126 dlmwri te 125 doc 46, 47 dot 100 double 517 dsol ve 543 ans 15 ~y1n asec 135 asech 137 asin135 asinh 137 atan 135 atan2135 atanh 137 axes 248 axis 234, 239 E eig 469, 496, 555 else41,180 elseif 41,182 end41,176 eps15 erf 388 exist 13 exp 131 expand 518 eye 99 ezplot 517 B bar 253, 378 break 190 C case 190 cat 75, 77 cd23 ceil 131 cell 149 celldisp 149 cellplot149 class 517 clc 13 clear 13 collect 518 conj 131 continue 190 contour 301 conv 101 cas 135 cosh 137 cot 135 coth137 cross 100 csc 135 csch137 cumsum 387 • F factor 518 fieldnames 157 find 25, 41, 75 findstr186 findsym 517 finite 172 fix 131 floor 131 fminbnd 147 fminsearch147 for 41 187 forma~ 126, 129 fplot 239 fprintf 126 function 137 fzero147 G gensig 480 get 121 getfield157 ginput 30 grid 30,239 global 139 gtext 30, 248 D H date 121 deal 149 deconv 101 delete 121 det 337, 555 Heaviside 557 help 46, 47 helpwin 47 hist 378 hold 248 I i 15 if 41,176 ilaplace 552 imag 131 impulse 476 lnf 15 ini tial 476 input 126 int 528 interpl 400, 406 interp2 402 inv 325, 334, 555 isce1l149 isempty 172 isfield 157 isinf 172 isnan 172 isreal 172 isstruct 157 ode15s 452 odeset 482 ones 99 otherwise 199 j 15 p path 23 pathtoo123 pi 15 pinv 337,357 plot 30, 239 plotyy 257 ploU 301 polar 253 poly21,555 poly2sym 518 polyder 437 polyfit26~278 polyva130, 101,278 pretty 518 print 30, 239 Pwd 23 L Q laplace 552 legend 248 length 75, 248 limi t 528 linspace 68, 75 load 121 quad 426 quad1426 qui t 13 J log 131 loglO 131 loglog 253 logspace 68, 75 lookfor46,47 lsim 476 M max 75 mean 372, 387 median 372, 387 menu 126,129 mesh 301 meshc 301 meshgrid301 meshz 301 75 N NaN 15 num2cel1149 numden 517 o ode23452 ode45452 odell3 452 ode23s 452 ode23t 452 ode23tb 452 R rand 392 randn 392 randperm 392 rank 346, 357 real 131 refresh 248 rmfield 157 rmpath23 roots 21,101 round 131 rref 347, 357 • S save 121 sawtooth 480 sec 135 sech 137 semilogx 253 semilogy 253 set 257 set field 157 sign 131 simple 518 simplify 518 simulink 485 sin 135 sinh 137 size 75 solve 518 sort 75 spline 406 sqrt 131 square 480 ss 472 ssdata 472 stairs 253 std 387 stem 253 step 476 stepfun 480 strcmp 186 struct 157 subplot 248 subs 518 sum 75 surf 301 surfc 301 switch 199 syrn 517 syrn2poly 518 syrns 517 syrnsum 528 U k 406 unm p V vpa 517 W waterfall 301 what 23 while41,195 who 13 whos 13 wklread 121 X xlabel 30, 239 xlsread 121 xor 171,172 T tan 135 137 taylor 528 text 248 tf 472 tfdata472 ti tle 30, 239 trapz 426 type 47,121 y ylabel 30, 239 Z zeros 99 zlabe1297 Topics A abscissa230 absolutefrequency375 array 19 additionand subtraction79 addressing72 cell 149 creatingan 67-69 division84 editor 71 empty 73 exponentiation86 multidimensional76 multiplication81 operations78 pages 76 powers86 size 75 structure 154 ASCII files 118 assignmentoperator 11 augmentedmatrix340 autoscaling233 axis label 230 axis limits 234 B backwarddifferences,433 bar plots 257 Basic Fitting Interface293 bin 373 bin center374 block485 Booleanoperator 169 boundarycondition537 break point 203 C Cau?hyf~rm 463 cell Indexing150 cell array 149 centraldifferences434 characteristicequation439, 553 clearingvariables14 coefficientof determination281 colon use of 19 com~a, use of 19 command6 comment33 complexnumbers16 complexconjugate16 conditionalstatement38 175 contentindexing150 ' contourplots 298 ControlSystemtoolbox472 Cramer'sdeterminant335 Cramer'smethod335 cubic splines403 currentdirectory 16 curve fit, quality of 281 D data files 118 Data Statisticstool 378 data symbol230 debugging36, 201 defaultvariable511 derivative423 Desktop6 determinants324 diary 119 differentialequation Cauchyform 463 characteristicequationof 439 nonlinear440, 460, 464, 490 order of 438 ordinary437 partial 437 solvers452, 472, 475 stable441 state variableform 463, 492 stiff 484 symbolicsolutionof 537 unstable441 differentiation423 numerical432 polynomial436 symbolic423 directory22 E Edit menu 18 Editor/Debugger32,201 eigenvalue554 element-by-elementoperations78 ellipsis 13 emptyarray 73 error function388 Euclideannorm 342 Euler method445 extrapolation271 • F fl~ Id 154 File menu 17 files ASCII files 118 command32 data 118 diary,119 fun?tlon32 M-flies32 MAT-files119 script 32 spreads~eet 121 user-defined137 flowchart 178 forc~d respo.nse440 forcing.functlon438 formatting128 forward differences 433 free response 439 function definition line 137 function discovery 262 function file 32 functions arguments 134 elementary mathematical 130 hyperbolic 135 maximization of 144 minimization of 143 numeric 133 trigonometric 134 user-defined 137 zeros of 142 G Gauss elimination 319 global parameter 456 global variable 139 grid 234 H H-1 r 36 me H,elp system 42 histogram 372 , homogeneous equation 321 I !II-c~nditioned problem 321 Implied loop 191 importing data 120 Import Wizard 122 impulse function 477,549 initial condition 537 integral 420 definite 422 improper 422 indefinite 422 singularity 422 integration 420 adaptive 425 • integrand 420 limits 420 linearity property 421 panel 425 symbolic 530 trapezoidal 425 interpolation 271, 398 cubic spline 403 linear 399 2-D 402 inverse Laplace transform 545 K knot point 408 L Laplace transform 543 least squares 275, 352 left division method 30 legend 231 limits 535 line continuation 13 linear algebraic equations 30, 317 application of matrix rank 339 and augmented matrix 340 and Euclidean norm 342 homogeneous 321 ill-conditioned system of 321 and linearity 331 overdetermined system of 351 and rank 339 and reduced row echelon form 346 singular set of 320 solution by Cramer's method 335 solution by determinants 324 solution by Gauss elimination 319 solution by left divIsion method 30, 326 solution by matrix inverse 331 solution by pseudoinverse method 339 solution by successive elimination of variables 319 and subdeterminant 339 underdetermined system of 338 linear in parameters 292 local variable 35, 139 logarithmic plots 250 logical operator 169 loop 39,187 loop variable 39, 187 LTI object 473 M M-files 32 MAT-files 119 mathematical model 48 MathWorks website 47 matrix 68 augmented 340 creating a 69 division 99 exponentiation 99 identity 98 inverse 333 multiplication 92 null 98 operations 91 rank 339 special 99 symmetric 72 transpose 72 unity 98 matrix inverse 333 max-min problems 528 mean 372 median 372 methodology for developing a computer solution 52 for engineering problem solving 49 model 48 modified Euler method 448 multidimensional arrays 76 multiple linear regression 290 na~eds 112 188 nes e oop~ normal dlstn?utlon 381 normal function 383 null array 73 differentiation 436 division 102 evaluation 101 multiplication 102 plotting 29, 238 roots 20 precedence 9,170,516 predefined constants 15 predictor-corrector method 447 probability 378 program documentation 177 programming style 36 prompt o pseudocode 177 pseudoinverse method 339 N ODE See differential equation, ordinary operations research 208 ordinate 230 overdetermined system 351 overlay plots 28, 240 Q quadrature 429 R random number generator 390 p pages (in multidimensional arrays) 76 path 22 plot abscissa 230 anatomy 230 axis label 230 axis limits 234 bar 257 contour 298 data symbol 230 editor 259 grid 234 legend 231 line types 241 logarithmic 250 ordinate 230 overlay 28, 240 polar 257 requirements 231 scale 230 second y-axis 257 stairs 257 stem 275 surface mesh 298 text placement 244 three-dimensional line 297 tick mark labels 230 tick marks 230 tick mark spacing 254 title 230 plotting polynomials 238 with smart function plot command 236 symbolic expressions 515 polar plots 257 polynomial 20, 100 addition 101 algebra 101 coefficients 100 ~k~9 reduced row echelon form 346 regression analysis 274 relational operator 24, 168 relative frequency 375 replacement operator 11 residuals 275, 287 r-squared value 282 Runge-Kutta methods 451 S saving keystrokes 119 variables 119 scalar scale 230 scaled frequency histogram 381 script file 32 search path 22 semicolon use of 13, 69 session Simpson's rule 429 simulation 208 Simulink 485 singular problem 320 smart recall 14 solution structure 520 special constants 15 stack 204 stairs plots 257 standard deviation 383 state of random generator 390 state transition diagram 210 state variable form 463 statically indeterminate problem 344 stem plots 257 step function 477 step size 455 stiff equation 484 string 185 structure arrays 154 structure chart 178 subdeterminant 339 subplots 239 successive elimination of variables 319 summation 534 surface mesh plot 298 symbolic constant 509 differential equation solution 537 differentiation 525 equation solving 518 expression 510 integration 530 limits 535 linear algebra 552 processing 507 summation 534 Taylor series 533 T tab completion 14 Taylor series 533 three-dimensional plots 296 contour plots 298 line plots 297 surface mesh plots 298 tick mark 230 tick mark label 230 tick mark spacing 254 time constant 439 toolbox transfer function form 474 transpose 67 truth table 171 U underdetermined system 338 uniform distribution 390 user-defined functions 137 user input 129 V variance 384 variable vector 66 absolute value of 76 cross product 100 dot product 100 length of 76 magnitude of 76 multiplication 91 W working directory 23 workspace 12 workspace browser 70 ... Chapters Six to Nine Number and Topic Number and Topic Chapter Six Chapter Eight 6. 1-1 6. 2-1 Gauss elimination 6. 2-2 6. 2-3 6. 2-4 6. 2-5 6. 4-1 6. 4-2 6. 4-3 6. 4-4 6. 4-5 6. 4 -6 6.5-1 6. 5-2 An electrical-resistance... 3 .6 Cell Arrays 149 117 130 3.7 Structure Arrays 3.8 Summary 159 Problems 160 154 Programming with MATLAB 167 Robot-Assisted Microsurgery 166 4.1 Relational Operators 168 4.2 Logical Operators... Prototyping 3 16 6.1 Elementary Solution Methods 318 6. 2 Matrix Methods for Linear Equations 6. 3 Cramer's Method 335 6. 4 Underdetermined Systems 338 6. 5 Overdetermined Systems 351 6. 6 Summary 3 56 Problems