An Introduction to Programming and Numerical Methods in MATLAB Springer 2005:An Introduction to Programming and Numerical Methods in MATLAB Springer 2005
S.R. Otto, BSc, PhD
The R & A
St Andrews
Fife
KY16 9JD
Scotland
J.P. Denier, BSc (Hons), PhD
School of Mathematical Sciences
The University of Adelaide
South Australia 5005
Australia
British Library Cataloguing in Publication Data
Otto, S. R. (Stephen Robert)
An introduction to programming and numerical methods in
MATLAB
1. MATLAB (Computer file) 2. Numerical analysis — Data
processing
I. Title II. Denier, J. P.
518′.02855
ISBN 1852339195
Library of Congress Control Number: 2005923332
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of
the publishers, or in the case of reprographic reproduction in accordance with the terms of licences
issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms
should be sent to the publishers.
ISBN-10: 1-85233-919-5
ISBN-13: 978-185233-919-7
Springer Science+Business Media
springeronline.com
© Springer-Verlag London Limited 2005
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement, that such names are exempt from the relevant laws and regulations and therefore
free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the information
contained in this book and cannot accept any legal responsibility or liability for any errors or
omissions that may be made.
Typesetting: Camera-ready by authors
Printed in the United States of America
12/3830-543210 Printed on acid-free paper SPIN 11317333
Preface
This text provides an introduction to the numerical methods that are typically
encountered (and used) in science and engineering undergraduate courses. The
material is developed in tandem with MATLAB which allows rapid prototyping
and testing of the methods. The package MATLAB (matrix laboratory) provides
an environment in which students can learn to programme and explore
the structure of the numerical methods. The methods included here are of a
basic nature and only rely on material which should have been explored prior
to the first year undergraduate stage.
The methods presented are supplemented with a set of tasks at the end
of each chapter (full solutions of these are given in Appendix C). The tasks
are introduced in such a way as to allow students to explore the topics as
they evolve. Some are of a mathematical nature, but in the main they involve
manipulating codes which are given in the text of the chapter (or section).
Those tasks which we regard as being harder are marked with an asterisk.
Throughout the text MATLAB commands appear using this font in the text.
In the main the names of MATLAB commands reflect the commands rˆole. We
have taken particular care to highlight many of the problems that occur with
interpreting the syntax of MATLAB commands. In Appendix B we provide
a glossary of all MATLAB commands within the text and short examples of
how these commands can be used. Reference is made to the comprehensive
help facility within MATLAB: however examples are given which are similar
to those given in other places within the text.
Throughout the text we derive the numerical techniques we use, but also
emphasise that MATLAB’s rich vocabulary provides commands for performing
most of the fundamental tasks encountered in numerical methods. This
approach serves to introduce students to the methods and also provides an
viii Preface
understanding of their inner workings.
Why Do We Need Numerical Methods?
Mathematics is an elegant and precise subject: however when numerical answers
are required one sometimes needs to rely on approximate methods to
obtain useable answers. There are many problems which simply do not have
analytical solutions, or those whose exact solution is beyond our current state
of knowledge. There are also many problems which are too long (or tedious)
to solve by hand. When such problems arise we can exploit numerical analysis
to reduce the problem to one involving a finite number of unknowns and use a
computer to solve the resulting equations.
The text starts with a description of how we could perform some very
basic calculations (that is, simply using the computer as a calculator). It then
moves on to solving problems which cannot, in practice, be solved by hand.
Sometimes the solution of these problems can become as intricate and involved
as the original problems and requires almost as much finesse and care to obtain
a solution. There are several options available to us, both in terms of language
and also overall approach. In this book we elect to express our ideas in terms
of the syntax of the computer package MATLAB. Once you have mastered the
syntax of MATLAB it will be easier for you to learn other languages, if you
should decide you need to.
The Structure of This Text
This text is designed to be used as a source of reference forMATLAB commands
(mainly through the glossary in Appendix B) and the examples given in the
chapters. This is an approach we have found works well with our students. The
text gives an introduction to numerical methods and the manipulation of the
quantities used therein (for instance matrices). At each stage, short codes are
given to allow the reader to try their own examples.
Examples of code which can be typed at the MATLAB prompt will appear
within the general text in this font (this font is also used for the names of
MATLAB codes), while longer examples will be written
>> commands to be entered
Results
Preface ix
Those codes which are designed to be saved to a file will appear in boxes
a = 1;
(sometimes wider boxes will be used for codes with longer lines).
At the end of each section, or chapter, there are a variety of tasks which are
designed to help the reader understand the topics they have just read. Solutions
of these are given in Appendix C.
Chapter 2 concentrates on some aspects of programming. We also introduce
another of MATLAB’s powerful tools, namely its ability to produce high quality
plots of data. Students are shown how to create codes and functions, which
serve to augment MATLAB’s built-in vocabulary. This chapter concludes with
a discussion of the important topic of errors, both from the perspective of
classical numerical analysis and also from human interaction which arise in
the action of coding. We have found that students benefit enormously from
debugging programmes and it is hoped that this will help them to hone these
skills, which we consider to be essential.
The third chapter contains a discussion of looping and logical structures
within MATLAB. Again the mathematics is developed in tandem. Finally concluding
the first part of the book we give some examples of how MATLAB can
be used to solve problems (using just algebra and exploiting MATLAB as an
advanced calculator).
In Chapters 4 and 5 we meet some classical numerical methods, in the form
of root finding and interpolation (and extrapolation). Although MATLAB has
intrinsic functions which will perform most of these operations (fzero and
polyfit) we have included a detailed description of both topics. This is aimed
at helping the students to understand how these methods work, and where
they can potentially fail. In Chapter 5 we discuss the general form of Newton
forward differences, which are used in this chapter and subsequent ones (for
instance to derive the formula for integration). In the next two chapters we
explore the numerical analysis associated with integration and differentiation.
In Chapter 6 we return to the topic of matrices. We start by discussing
the mathematical concepts of rank and linear independence. We also discuss
eigenvalues and eigenvectors (and their interpretation for 2-by-2 matrices). The
topic of numerical integration is taken up in Chapter 7 and in Chapter 8 methods
for the numerical solution of ordinary differential equations are explored.
Finally in Chapter 9 we use MATLAB to calculate some basic statistical quantities
and also to explore some maps, some of which may be exhibit chaotic
behaviour.
The text finishes with appendices containing an introduction to the basics
x Preface
of matrix algebra, a glossary of useful terms and solutions to all the tasks
contained within the text. These appendices have deliberately been made quite
wordy, since we believe that the material they contain is one of the important
aspects of the book.
At the outset we expect readers to be mathematically literate to the level of
being able to: and solve algebraic equations (for instance determine the roots
of a quadratic and solve simultaneous equations); integrate and differentiate
simple functions; solve separable differential equations (although examples are
given within the text). Although exposure to complex numbers and matrices
would be useful at the start of the text, it is not necessary as these concepts
are introduced in some detail in the introductory chapter.
We would like to thank students of the University of Birmingham whose
patience and comments have been invaluable in the construction of this text.
In particular we would like to identify Sukhjinder Chana and Rob Ackeroyd
for their careful proof reading and comments.
St Andrews, UK S. R. Otto
Adelaide, Australia J. P. Denier
Contents
1. Simple Calculations with MATLAB . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction and a Word of Warning . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scalar Quantities and Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Rules for Naming of Variables. . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Precedence: The Order in Which Calculations Are Performed
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Mathematical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Format: The Way in Which Numbers Appear . . . . . . . . . . . . . . . . 12
1.4 Vectors in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Initialising Vector Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Manipulating Vectors and Dot Arithmetic . . . . . . . . . . . . . 14
1.5 Setting Up Mathematical Functions . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Some MATLAB Specific Commands . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.1 Looking at Variables and Their Sizes. . . . . . . . . . . . . . . . . . 22
1.7 Accessing Elements of Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2. Writing Scripts and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Creating Scripts and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2 Brief Aside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Plotting Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Evaluating Polynomials and Plotting Curves . . . . . . . . . . . 41
2.2.2 More on Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Functions of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xii Contents
2.4.1 Numerical Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.2 User Error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3. Loops and Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Loops Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Summing Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 Sums of Series of the Form
N
j=1
jp, p ∈ N. . . . . . . . . . . . . . . 73
3.3.2 Summing Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.3 Summing Series Using MATLAB Specific Commands . . . 79
3.3.4 Loops Within Loops (Nested) . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4.1 Constructing Logical Statements . . . . . . . . . . . . . . . . . . . . . 85
3.4.2 The MATLAB Command switch . . . . . . . . . . . . . . . . . . . . 88
3.5 Conditional loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.1 The break Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6 MATLAB Specific Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7 Error Checking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.8 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4. Root Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Initial Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Fixed Point Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Bisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5 Newton–Raphson and Secant Methods . . . . . . . . . . . . . . . . . . . . . . 117
4.5.1 Derivation of the Newton–Raphson Method. . . . . . . . . . . . 117
4.6 Repeated Roots of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7 Zeros of Higher-Dimensional Functions(*) . . . . . . . . . . . . . . . . . . . 125
4.8 MATLAB Routines for Finding Zeros . . . . . . . . . . . . . . . . . . . . . . . 128
4.8.1 Roots of a Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.8.2 The Command fzero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.9 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5. Interpolation and Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 Saving and Reading Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Which Points to Use? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Newton Forward Differences and Lagrange Polynomials . . . . . . . 141
5.4.1 Linear Interpolation/Extrapolation . . . . . . . . . . . . . . . . . . . 147
Contents xiii
5.5 Calculating Interpolated and Extrapolated Values . . . . . . . . . . . . 148
5.6 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.7 Curves of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.8 Interpolation of Non-Smooth Data . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.8.1 Insufficient Data Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.9 Minimisation of Functions and Parameter Retrieval . . . . . . . . . . . 161
5.9.1 Parameter Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.9.2 Using fmins for Parameter Retrieval . . . . . . . . . . . . . . . . . . 164
5.10 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6. Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.1 Initialising Matrices Within MATLAB . . . . . . . . . . . . . . . . 169
6.1.2 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.1.3 Operations on Elements of Matrices . . . . . . . . . . . . . . . . . . 180
6.1.4 More on Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.1.5 Matrices Containing Strings . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.2 Properties of Matrices and Systems of Equations . . . . . . . . . . . . . 186
6.2.1 Determinants of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.3 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.3.1 Solving Many Equations at Once . . . . . . . . . . . . . . . . . . . . . 198
6.4 Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.6 Specific MATLAB Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.7 Characteristic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.8 Exponentials of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.9 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7. Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.2 Integration Using Straight Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.2.1 Errors in the Trapezium Method . . . . . . . . . . . . . . . . . . . . . 229
7.3 Integration Using Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.4 Integration Using Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 235
7.5 Integrating Using MATLAB Commands . . . . . . . . . . . . . . . . . . . . . 237
7.6 Specific Examples of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.6.1 Infinite Integrals and Removable Singularities . . . . . . . . . . 238
7.6.2 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
7.7 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
xiv Contents
8. Solving Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 Euler’s Method and Crank–Nicolson . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2.1 Analytical Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.3 Banded Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
8.4 Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
8.5 Higher-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.5.1 Second-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.5.2 Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.5.3 Airy’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.5.4 Shooting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
8.6 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.7 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.8 Eigenvalues of Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.9 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
9. Simulations and Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.2 Statistical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.2.1 Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.2.2 Other Statistical Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.3 Random Numbers and Distributions . . . . . . . . . . . . . . . . . . . . . . . . 295
9.3.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
9.3.2 Calculating Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
9.3.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.4 Maps and White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.4.1 Modelling Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 307
9.4.2 Periodicity and Chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
9.4.3 Random Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.5 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
A. A Mathematical Introduction to Matrices . . . . . . . . . . . . . . . . . . 323
A.1 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
A.2 Inverses of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
B. Glossary of Useful Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
B.1 Arithmetic and Logical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 335
B.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
B.3 Plotting Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
B.4 General MATLAB Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Contents xv
C. Solutions to Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
C.1 Solutions for Tasks from Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . 389
C.2 Solutions for Tasks from Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 394
C.3 Solutions for Tasks from Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 400
C.4 Solutions for Tasks from Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 408
C.5 Solutions for Tasks from Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . 414
C.6 Solutions for Tasks from Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . 418
C.7 Solutions for Tasks from Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . 429
C.8 Solutions for Tasks from Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . 436
C.9 Solutions for Tasks from Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . 454
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
:11bb
[hide]
[/hide]
看看！！！感谢楼主，辛苦了！！！
好书支持
Thank you!!!
:30bb :30bb :30bb
支持，非常感谢楼主的共享！！！！！
好书，感谢楼主分享:30bb :31bb
楼主辛苦了，新年开了！！！！！！！！！！！！
支持matlab
:27bb 学习一下---感谢楼主---
感谢
:30bb :30bb
楼主辛苦了，新年开了！！！！！！！！！！！！
楼主辛苦了,学习一下---感谢楼主---
楼主辛苦了，学习一下，十分感谢楼主的辛勤工作!
I'll have a look, and study. thinks
:27bb :27bb :27bb :27bb
支持支持！所有人都应该好好学学！
是本好书，非常感谢楼主的共享, 支持
gooooooooooooood
顶顶顶顶顶顶顶顶
:11bb :27bb :29bb
学习一下 谢谢楼主 辛苦 向楼主
好书好书，收藏了:11bb :11bb
一直在用matlab，感谢楼主惠赐。
好书，感谢楼主的辛勤工作！:11bb
楼主辛苦了，学习一下，十分感谢楼主的辛勤工作!
好书，谢谢楼主分享。。。。。。。我为人人，人人为我！！
想学习一下，谢谢分享:11bb
谢谢啦。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。
非常好的贴子，楼主辛苦了！
:11bb :11bb :11bb :11bb :11bb :11bb :11bb :11bb
楼主:30bb :30bb :30bb :30bb :30bb :30bb
谢谢。。。。。。。。。。。。。。。。
支持，非常感谢楼主的共享！！！！！
楼主辛苦了，学习一下，十分感谢楼主的辛勤工作!
感谢楼主，辛苦了！！！{:5_221:}
好东西绝对支持，下载学习！
谢谢楼主的分享。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。
:21bb:22bb:23bb
是本好书，非常感谢楼主的共享, 支持
看看,学学:22bb:31bb
谢谢楼主~~~~~学习一下。。。。
Unlocking Knowledge, Empowering Minds
',''╭⌒⌒╮.',''',,'.'',,','',.
'╱◥███◣'『微网』是个家
︱田︱ 田 田| ','''',.发贴把瓦加
╬╬╬╬╬╬╬╬╬╬╬╬╬
大作阿 一定要頂的
感谢楼主分享
谢谢谢谢谢谢
:31bb好书呀
入门的好书，感谢楼主
谢谢
看看呢
感谢楼主分享
:crackle.GIF
thank you very much
好书谢谢分享
好书支持!!!!!!!!!!!!!!!!!!!!!!!!!!
Thank you for your sharing
谢谢啊，更多好书期待ing。。。
MatLab is a good tool for study!
MatLab 在大学教育中要占主角！
支持支持！所有人都应该好好学学！
matlab数值分析的书，对设计算法原型很有帮助
看看{:7_1235:}
够噢噢噢噢噢噢
好书支持
謝謝分享
{:7_1234:}
{:7_1235:}
谢谢楼主无私分享！顶……
An Introduction to Programming and Numerical Methods in MATLAB Springer 2005
是本好书，非常感谢楼主的共享, 支持
学习学习！！
好书，顶一个！
very nice book，thanks
very nice book，thanks
very nice book ,thanks
thank you
好书。。。
xiexie 楼主
感謝分享！{:7_1234:}
Ok，OK
看看！！！感谢楼主，辛苦了！！！
感谢楼主分享
thanks
下来看看，可以当工具书
不错，下来看看！
看看！！！感谢楼主
看看。。。。谢谢分享。。。
学习学习
谢谢分享
感谢楼主分享！
好像还不错哈：）
好资料
好書一本
學習學習
感謝樓主大方分享
谢谢楼主分享
绝对是好书！
An Introduction to Programming and Numerical Methods in MATLAB Springer 2005: An Introduction to Programming and Numerical Methods in MATLAB Springer 2005.pdf
