A Survey of Numerical Mathematics, Volume I

Volume 2 of 2-volume set. Broad self-contained coverage of computer-oriented numerical algorithms for solving various types of mathematical problems in linear algebra, ordinary and partial, differential equations, much more. Includes exercises.

Chapter 1 Numerical Analysis as a Subject Area

1

(18)

1.1 Introduction

1

(1)

1.2 Some pitfalls in computation

2

(3)

1.3 Mathematical and computer aspects of an algorithm

5

(2)

1.4 Numerical instability of algorithms and ill-conditioned problems

7

(3)

1.5 Typical problems of interest to the numerical analyst

10

(3)

1.6 Iterative methods

13

(6)

Chapter 2 Elementary Operations with Automatic Digital Computers

19

(51)

2.1 Introduction

19

(1)

2.2 Binary arithmetic

19

(3)

2.3 Conversion from base D to base B representation

22

(6)

2.4 Representation of integers on a binary computer

28

(7)

2.5 Floating-point representations

35

(5)

2.6 Computer-representable numbers

40

(5)

2.7 Floating-point arithmetic operations

45

(11)

2.8 Fortran analysis of a floating-point number

56

(5)

2.9 Calculation of elementary functions

61

(9)

Chapter 3 Surveillance of Number Ranges

70

(23)

3.1 Introduction

70

(1)

3.2 Allowable number ranges

71

(3)

3.3 Basic real arithmetic operations

74

(3)

3.4 The quadratic equation

77

(6)

3.5 Complex arithmetic operations

83

(10)

Chapter 4 Solution of Equations

93

(83)

4.1 Introduction

93

(3)

4.2 Attainable accuracy

96

(7)

4.3 Graphical methods

103

(2)

4.4 The method of bisection

105

(7)

4.5 The method of false position

112

(6)

4.6 The secant method

118

(1)

4.7 General properties of iterative methods

119

(8)

4.8 Generation of iterative methods

127

(5)

4.9 The Newton method

132

(4)

4.10 Muller's method

136

(9)

4.11 Orders of convergence of iterative methods

145

(14)

4.12 Acceleration of the convergence

159

(1)

4.13 Systems of nonlinear equations

160

(16)

Chapter 5 Roots of Polynomial Equations

176

(70)

5.1 Introduction

176

(2)

5.2 General properties of polynomials

178

(11)

5.3 The Newton method and related methods

189

(6)

5.4 Muller's method and Cauchy's method

195

(1)

5.5 Location of the roots

196

(16)

5.6 Root acceptance and refinement

212

(7)

5.7 Matrix related methods: the modified Bernoulli method

219

(8)

5.8 Matrix related methods: the IP method

227

(8)

5.9 Polyalgorithms

235

(6)

5.10 Other methods

241

(5)

Chapter 6 Interpolation and Approximation

246

(98)

6.1 Introduction

246

(2)

6.2 Linear interpolation

248

(3)

6.3 Convergence and accuracy of linear interpolation

251

(7)

6.4 Lagrangian interpolation

258

(8)

6.5 Convergence and accuracy of Lagrangian interpolation

266

(4)

6.6 Interpolation with equal intervals

270

(18)

6.7 Hermite interpolation

288

(7)

6.8 Limitations on polynomial interpolation: smooth interpolation

295

(6)

6.9 Inverse interpolation

301

(7)

6.10 Approximation by polynomials

308

(10)

6.11 Least squares approximation by polynomials

318

(8)

6.12 Rational approximation

326

(3)

6.13 Trigonometric interpolation and approximation

329

(11)

6.14 Interpolation in two variables

340

(4)

Chapter 7 Numerical Differentiation and Quadrature

344

(78)

7.1 Introduction

344

(1)

7.2 The method of undetermined weights

345

(5)

7.3 Numerical differentiation

350

(11)

7.4 Numerical quadrature--equal intervals

361

(13)

7.5 The Euler-MacLaurin formula

374

(7)

7.6 Romberg integration

381

(4)

7.7 Error determination

385

(16)

7.8 Numerical quadrature--unequal intervals

401

(21)

Chapter 8 Ordinary Differential Equations

422

8.1 Introduction

422

(3)

8.2 Existence and uniqueness

425

(6)

8.3 Analytic methods

431

(5)

8.4 Integral equation formulation--the Picard method of successive approximations

436

(5)

8.5 The Euler method

441

(8)

8.6 Methods based on numerical quadrature

449

(11)

8.7 Error estimation for predictor-corrector methods

460

(9)

8.8 A numerical example

469

(4)

8.9 Runge-Kutta methods

473

(5)

8.10 Methods based on numerical differentiation

478

(4)

8.11 Higher-order equations and systems of first-order equations

482

(6)

8.12 The use of high-speed computers

488

Appendix A

A-1

(18)

Appendix B

A-19

(2)

Appendix C

A-21

Bibliography

B-1

Index

I-1

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A Survey of Numerical Mathematics, Volume I

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Table of Contents

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