This textbook is an introduction to Scientific Computing, in which several numerical methods for the computer solution of certain classes of mathematical problems are illustrated. The authors show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions by polynomials and construct accurate approximations for the solution of ordinary and partial differential equations. To make the presentation concrete and appealing, the programming environments Matlab and Octave, which is freely distributed, are adopted as faithful companions. The book contains the solutions to several problems posed in exercises and examples, often originating from specific applications. A specific section is devoted to subjects which were not addressed in the book and contains the bibliographical references for a more comprehensive treatment of the material. The second edition features many new problems and examples, as well as more numerical methods for linear and nonlinear systems and ordinary and partial differential equations. This book is presently being translated or has appeared in the following languages: Italian, German, French, Chinese and Spanish. Reviews for "Scientific Computing with MATLAB" - 1st edition: " ... Scientific Computing with MATLAB is written in a clear and concise style, figures, tables and formula boxes complement the explanations... The whole book is an invitation, if not a request, of the authors to the reader to play with MATLAB, apply its powerful menagerie of functions to solve the given (or own) problems - in brief, supervised learning by doing .... is a stimulating introductory textbook about numerical methods that successfully combines mathematical theory with programming experience..." Anselm A.C. Horn, Journal of Molecular Modeling 2004 "... An excellent addition to academic libraries and university bookstores, this book will be useful for self-study and as a complement to other MATLAB-based books. Highly recommended. Upper-division undergraduates through professionals." S.T. Karris, Choice 2003

1 What can't be ignored

1

(38)

1.1 Real numbers

2

(4)

1.1.1 How we represent them

2

(2)

1.1.2 How we operate with floating-point numbers

4

(2)

1.2 Complex numbers

6

(2)

1.3 Matrices

8

(7)

1.3.1 Vectors

14

(1)

1.4 Real functions

15

(8)

1.4.1 The zeros

16

(2)

1.4.2 Polynomials

18

(3)

1.4.3 Integration and differentiation

21

(2)

1.5 To err is not only human

23

(5)

1.5.1 Talking about costs

26

(2)

1.6 The MATLAB and Octave environments

28

(1)

1.7 The MATLAB language

29

(8)

1.7.1 MATLAB statements

31

(1)

1.7.2 Programming in MATLAB

32

(4)

1.7.3 Examples of differences between MATLAB and Octave languages

36

(1)

1.8 What we haven't told you

37

(1)

1.9 Exercises

37

(2)

2 Nonlinear equations

39

(32)

2.1 The bisection method

41

(4)

2.2 The Newton method

45

(6)

2.2.1 How to terminate Newton's iterations

47

(2)

2.2.2 The Newton method for systems of nonlinear equations

49

(2)

2.3 Fixed point iterations

51

(5)

2.3.1 How to terminate fixed point iterations

55

(1)

2.4 Acceleration using Aitken method

56

(4)

2.5 Algebraic polynomials

60

(5)

2.5.1 Homer's algorithm

61

(2)

2.5.2 The Newton-Horner method

63

(2)

2.6 What we haven't told you

65

(2)

2.7 Exercises

67

(4)

3 Approximation of functions and data

71

(30)

3.1 Interpolation

74

(12)

3.1.1 Lagrangian polynomial interpolation

75

(5)

3.1.2 Chebyshev interpolation

80

(1)

3.1.3 Trigonometric interpolation and FFT

81

(5)

3.2 Piecewise linear interpolation

86

(2)

3.3 Approximation by spline functions

88

(4)

3.4 The least-squares method

92

(5)

3.5 What we haven't told you

97

(1)

3.6 Exercises

98

(3)

4 Numerical differentiation and integration

101

(22)

4.1 Approximation of function derivatives

103

(2)

4.2 Numerical integration

105

(6)

4.2.1 Midpoint formula

106

(2)

4.2.2 Trapezoidal formula

108

(1)

4.2.3 Simpson formula

109

(2)

4.3 Interpolatory quadratures

111

(4)

4.4 Simpson adaptive formula

115

(4)

4.5 What we haven't told you

119

(1)

4.6 Exercises

120

(3)

5 Linear systems

123

(44)

5.1 The LU factorization method

126

(8)

5.2 The pivoting technique

134

(2)

5.3 How accurate is the LU factorization'?

136

(4)

5.4 How to solve a tridiagonal system

140

(1)

5.5 Overdetermined systems

141

(2)

5.6 What is hidden behind the command \

143

(1)

5.7 Iterative methods

144

(6)

5.7.1 How to construct an iterative method

146

(4)

5.8 Richardson and gradient methods

150

(3)

5.9 The conjugate gradient method

153

(3)

5.10 When should an iterative method be stopped?

156

(3)

5.11 To wrap-up: direct or iterative?

159

(5)

5.12 What we haven't told you

164

(1)

5.13 Exercises

164

(3)

6 Eigenvalues and eigenvectors

167

(20)

6.1 The power method

170

(4)

6.1.1 Convergence analysis

173

(1)

6.2 Generalization of the power method

174

(2)

6.3 How to compute the shift

176

(3)

6.4 Computation of all the eigenvalues

179

(4)

6.5 What we haven't told you

183

(1)

6.6 Exercises

183

(4)

7 Ordinary differential equations

187

(50)

7.1 The Cauchy problem

190

(1)

7.2 Euler methods

191

(6)

7.2.1 Convergence analysis

194

(3)

7.3 The Crank-Nicolson method

197

(2)

7.4 Zero-stability

199

(3)

7.5 Stability on unbounded intervals

202

(10)

7.5.1 The region of absolute stability

204

(1)

7.5.2 Absolute stability controls perturbations

205

(7)

7.6 High order methods

212

(4)

7.7 The predictor-corrector methods

216

(3)

7.8 Systems of differential equations

219

(6)

7.9 Some examples

225

(9)

7.9.1 The spherical pendulum

225

(3)

7.9.2 The three-body problem

228

(2)

7.9.3 Some stiff problems

230

(4)

7.10 What we haven't told you

234

(1)

7.11 Exercises

234

(3)

8 Numerical methods for (initial-)boundary-value problems

237

(30)

8.1 Approximation of boundary-value problems

240

(13)

8.1.1 Approximation by finite differences

241

(2)

8.1.2 Approximation by finite elements

243

(2)

8.1.3 Approximation by finite differences of two-dimensional problems

245

(6)

8.1.4 Consistency and convergence

251

(2)

8.2 Finite difference approximation of the heat equation

253

(4)

8.3 The wave equation

257

(6)

8.3.1 Approximation by finite differences

260

(3)

8.4 What we haven't told you

263

(1)

8.5 Exercises

264

(3)

9 Solutions of the exercises

267

(40)

9.1 Chapter 1

267

(3)

9.2 Chapter 2

270

(6)

9.3 Chapter 3

276

(4)

9.4 Chapter 4

280

(5)

9.5 Chapter 5

285

(4)

9.6 Chapter 6

289

(4)

9.7 Chapter 7

293

(8)

9.8 Chapter 8

301

(6)

References

307

(4)

Index

311

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