Introduction To Numerical Computation:

This text, designed for an introductory course in scientific computation at an advanced undergraduate level, is a revision and translation of a Swedish work which focuses on simplicity and readability in order to help students prepare for complex mathematical software. (Computer Books - Languages/Programming)

PREFACE

v

1. INTRODUCTION

1.1. Mathematical Models and Numerical Approximations

1

(5)

1.2. Numerical Computation

6

(2)

1.3. References

8

(1)

2. ERROR ANALYSIS AND COMPUTER ARITHMETIC

2.1. Sources of Error

9

(1)

2.2. Basic Concepts

10

(3)

2.3. Error Propagation

13

(7)

2.4. Number Representation

20

(4)

2.5. Rounding Errors in Floating Point

24

(2)

2.6. Arithmetic Operations in Floating Point

26

(4)

2.7. Accumulated Errors

30

(4)

2.8. IEEE Standard

34

(3)

Exercises

37

(4)

Computer Exercises

41

(1)

References

42

(3)

3. FUNCTION EVALUATION

3.1. Introduction

45

(1)

3.2. Remainder Term Estimates

46

(3)

3.3. Standard Functions

49

(4)

3.4. Range Reduction

53

(3)

3.5. Trigonometric Functions

56

(6)

Exercises

62

(1)

References

63

(2)

4. NONLINEAR EQUATIONS

4.1. Introduction

65

(1)

4.2. Crude Localization

66

(2)

4.3. Iteration Methods

68

(6)

4.4. Convergence Analysis

74

(4)

4.5. Error Estimation and Stopping Criteria

78

(6)

4.6. Algebraic Equations

84

(3)

4.7. Square Root

87

(3)

4.8. Nonlinear Systems

90

(6)

Exercises

96

(1)

Computer Exercises

96

(1)

References

97

(2)

5. INTERPOLATION

5.1. Introduction

99

(1)

5.2. Interpolation by Polynomials

100

(4)

5.3. Linear Interpolation

104

(3)

5.4. Newton's Interpolation Formula

107

(6)

5.5. Neville's Method

113

(2)

5.6. Lagrange Interpolation

115

(1)

5.7. Hermite Interpolation

116

(2)

5.8. Runge's Phenomenon

118

(1)

5.9. Splines

119

(2)

5.10. Linear Spline Functions

121

(3)

5.11. Cubic Splines

124

(10)

5.12. Cubic B-Splines

134

(3)

Exercises

137

(2)

Computer Exercises

139

(2)

References

141

(2)

6. DIFFERENTIATION AND RICHARDSON EXTRAPOLATION

6.1. Introduction

143

(1)

6.2. Difference Approximations

144

(1)

6.3. Difference Approximation Errors

145

(5)

6.4. Richardson Extrapolation

150

(10)

Exercises

160

(1)

Computer Exercises

161

(1)

References

162

(1)

7. INTEGRATION

7.1. Introduction

163

(1)

7.2. Trapezoidal Rule

163

(4)

7.3. Newton-Cotes Formulas

167

(4)

7.4. Romberg's Method

171

(5)

7.5. Difficulties with Numerical Integration

176

(4)

7.6. Adaptive Quadrature

180

(3)

Exercises

183

(1)

Computer Exercises

184

(1)

References

185

(2)

8. LINEAR SYSTEMS OF EQUATIONS

8.1. Introduction

187

(3)

8.2. Triangular Systems

190

(2)

8.3. Gaussian Elimination

192

(6)

8.4. Pivoting

198

(5)

8.5. Permutations, Gauss Transformations

203

(5)

8.6. LU Factorization

208

(6)

8.7. Symmetric, Positive Definite Matrices

214

(5)

8.8. Band Matrices

219

(4)

8.9. Inverse Matrix

223

(2)

8.10. Vector and Matrix Norms

225

(6)

8.11. Sensitivity Analysis

231

(5)

8.12. Rounding Errors

236

(5)

8.13. Estimation of Condition Number

241

(3)

8.14. Overdetermined Systems

244

(6)

8.15. QR Factorization

250

(7)

Exercises

257

(1)

Computer Exercises

258

(1)

References

259

(2)

9. APPROXIMATION

9.1. Introduction

261

(3)

9.2. Important Concepts

264

(5)

9.3. Least Squares Method

269

(5)

9.4. Orthogonal Functions

274

(2)

9.5. Orthogonal Polynomials

276

(5)

9.6. Legendre Polynomials

281

(3)

9.7. Chebyshev Polynomials

284

(4)

9.8. Discrete Cosine Transform

288

(8)

9.9. Minimax Approximation

296

(7)

Exercises

303

(2)

Computer Exercises

305

(2)

References

307

(2)

10. ORDINARY DIFFERENTIAL EQUATIONS

10.1. Introduction

309

(3)

10.2. Initial Value Problems

312

(4)

10.3. Local and Global Error

316

(3)

10.4. Runge-Kutta Methods

319

(4)

10.5. An Implicit Method

323

(2)

10.6. Stability

325

(9)

10.7. Adaptive Step Length Control

334

(3)

10.8. Boundary Value Problems

337

(2)

10.9. A Difference Method

339

(5)

10.10. A Finite Element Method

344

(8)

10.11. The Shooting Method

352

(3)

Exercises

355

(1)

Computer Exercises

356

(1)

References

357

(2)

SHORT BIOGRAPHIES

359

(5)

ANSWERS TO EXERCISES

364

(6)

INDEX

370

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Introduction To Numerical Computation: Analysis And MATLAB Illustrations

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