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An Introduction To Programming And Numerical Methods In Matlab book cover, ISBN 9781852339197

An Introduction To Programming And Numerical Methods In Matlab

by Otto, S. R.; Denier, J. P.

ISBN13: 9781852339197
ISBN10: 1852339195
eBook ISBN(s): 9781846281334
Format: Paperback
Copyright: 2005-06-30
Publisher: SPRINGER - VERLAG

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Summary

MATLAB is a powerful programme, which naturally lends itself to the rapid implementation of most numerical algorithms. This text, which uses MATLAB, gives a detailed overview of structured programming and numerical methods for the undergraduate student. The book covers numerical methods for solving a wide range of problems, from integration to the numerical solution of differential equations or the simulation of random processes. Each chapter includes extensive examples and tasks, at varying levels of complexity. For practice, the early chapters include programmes that require debugging by the reader, while full solutions are given for all the tasks. The book also includes: a glossary of MATLAB commands appendices of mathematical techniques used in numerical methods. Designed as a text for a first course in programming and algorithm design, as well as in numerical methods courses, the book will be of benefit to a wide range of students from mathematics and engineering, to commerce.

Author Biography

Steve Otto was formerly a lecturer at Birmingham University but now heads a research unit at St. Andrews, UK. This book is based on courses taught at Birmingham and at Adelaide to students in mathematics, engineering and economics.

Table of Contents

Simple Calculations with MATLAB

1

(26)

Introduction and a Word of Warning

1

(1)

Scalar Quantities and Variables

2

(10)

Rules for Naming of Variables

4

(1)

Precedence: The Order in Which Calculations Are Performed

5

(3)

Mathematical Functions

8

(4)

Format: The Way in Which Numbers Appear

12

(1)

Vectors in MATLAB

13

(4)

Initialising Vector Objects

13

(1)

Manipulating Vectors and Dot Arithmetic

14

(3)

Setting Up Mathematical Functions

17

(3)

Some MATLAB Specific Commands

20

(3)

Looking at Variables and Their Sizes

22

(1)

Accessing Elements of Arrays

23

(1)

Tasks

24

(3)

Writing Scripts and Functions

27

(36)

Creating Scripts and Functions

27

(9)

Functions

30

(5)

Brief Aside

35

(1)

Plotting Simple Functions

36

(13)

Evaluating Polynomials and Plotting Curves

41

(3)

More on Plotting

44

(5)

Functions of Functions

49

(2)

Errors

51

(6)

Numerical Errors

51

(3)

User Error

54

(3)

Tasks

57

(6)

Loops and Conditional Statements

63

(40)

Introduction

63

(1)

Loops Structures

63

(5)

Summing Series

68

(15)

Sums of Series of the Form NΣ j=4 jp, p ε N

73

(3)

Summing Infinite Series

76

(3)

Summing Series Using MATLAB Specific Commands

79

(3)

Loops Within Loops (Nested)

82

(1)

Conditional Statements

83

(7)

Constructing Logical Statements

85

(3)

The MATLAB Command switch

88

(2)

Conditional loops

90

(2)

The break Command

92

(1)

MATLAB Specific Commands

92

(2)

Error Checking

94

(3)

Tasks

97

(6)

Root Finding

103

(30)

Introduction

103

(1)

Initial Estimates

104

(5)

Fixed Point Iteration

109

(4)

Bisection

113

(4)

Newton--Raphson and Secant Methods

117

(6)

Derivation of the Newton--Raphson Method

117

(6)

Repeated Roots of Functions

123

(2)

Zeros of Higher-Dimensional Functions(*)

125

(3)

MATLAB Routines for Finding Zeros

128

(2)

Roots of a Polynomial

128

(1)

The Command fzero

128

(2)

Tasks

130

(3)

Interpolation and Extrapolation

133

(36)

Introduction

133

(1)

Saving and Reading Data

134

(5)

Which Points to Use?

139

(2)

Newton Forward Differences and Lagrange Polynomials

141

(7)

Linear Interpolation/Extrapolation

147

(1)

Calculating Interpolated and Extrapolated Values

148

(2)

Splines

150

(2)

Curves of Best Fit

152

(3)

Interpolation of Non-Smooth Data

155

(6)

Insufficient Data Points

158

(3)

Minimisation of Functions and Parameter Retrieval

161

(5)

Parameter Retrieval

163

(1)

Using fmins for Parameter Retrieval

164

(2)

Tasks

166

(3)

Matrices

169

(56)

Introduction

169

(17)

Initialising Matrices Within MATLAB

169

(5)

Matrix Operations

174

(6)

Operations on Elements of Matrices

180

(2)

More on Special Matrices

182

(3)

Matrices Containing Strings

185

(1)

Properties of Matrices and Systems of Equations

186

(5)

Determinants of Matrices

190

(1)

Elementary Row Operations

191

(8)

Solving Many Equations at Once

198

(1)

Matrix Decomposition

199

(5)

Eigenvalues and Eigenvectors

204

(4)

Specific MATLAB Commands

208

(4)

Characteristic Polynomials

212

(2)

Exponentials of Matrices

214

(3)

Tasks

217

(8)

Numerical Integration

225

(22)

Introduction

225

(1)

Integration Using Straight Lines

226

(4)

Errors in the Trapezium Method

229

(1)

Integration Using Quadratics

230

(5)

Integration Using Cubic Polynomials

235

(2)

Integrating Using MATLAB Commands

237

(1)

Specific Examples of Integrals

238

(4)

Infinite Integrals and Removable Singularities

238

(2)

Indefinite Integrals

240

(2)

Tasks

242

(5)

Solving Differential Equations

247

(44)

Introduction

247

(1)

Euler's Method and Crank--Nicolson

247

(12)

Analytical Comparisons

253

(6)

Banded Matrices

259

(4)

Runge--Kutta Methods

263

(3)

Higher-Order Systems

266

(12)

Second-Order Systems

266

(4)

Bessel's equation

270

(3)

Airy's Equation

273

(1)

Shooting Methods

274

(4)

Boundary-Value Problems

278

(4)

Population Dynamics

282

(3)

Eigenvalues of Differential Systems

285

(1)

Tasks

286

(5)

Simulations and Random Numbers

291

(32)

Introduction

291

(1)

Statistical quantities

291

(4)

Averages

291

(2)

Other Statistical Measures

293

(2)

Random Numbers and Distributions

295

(5)

Normal Distribution

298

(1)

Calculating Probabilities

299

(1)

Permutations

300

(1)

Maps and White Noise

300

(19)

Modelling Discrete Systems

307

(2)

Periodicity and Chaos

309

(7)

Random Motion

316

(3)

Tasks

319

(4)

A. Mathematical Introduction to Matrices

323

(12)

Special Matrices

330

(1)

Inverses of Matrices

331

(4)

B. Glossary of Useful Terms

335

(54)

Arithmetic and Logical Operators

335

(8)

Symbols

343

(4)

Plotting Commands

347

(18)

General MATLAB Commands

365

(24)

C. Solutions to Tasks

389

(70)

Solutions for Tasks from Chapter 1

389

(5)

Solutions for Tasks from Chapter 2

394

(6)

Solutions for Tasks from Chapter 3

400

(8)

Solutions for Tasks from Chapter 4

408

(6)

Solutions for Tasks from Chapter 5

414

(4)

Solutions for Tasks from Chapter 6

418

(11)

Solutions for Tasks from Chapter 7

429

(7)

Solutions for Tasks from Chapter 8

436

(18)

Solutions for Tasks from Chapter 9

454

(5)

Index

459

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