Davis / Rabinowitz / Rheinbolt Methods of Numerical Integration

2. Auflage 2014
ISBN: 978-1-4832-6428-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 626 Seiten, Web PDF

Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. A chapter describes the approximate integration in two or more dimensions. The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. This book will be of great value to theoreticians and computer programmers.

Autoren/Hrsg.

Davis, Philip J.

Rabinowitz, Philip

Rheinbolt, Werner

Inhaltsverzeichnis

1;Front Cover;1
2;Methods of Numerical Integration;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface to First Edition;12
7;Preface to Second Edition;14
8;CHAPTER 1. INTRODUCTION;16
8.1;1.1 Why Numerical Integration?;16
8.2;1.2 Formal Differentiation and Integration on Computers;18
8.3;1.3 Numerical Integration and Its Appeal in Mathematics;19
8.4;1.4 Limitations of Numerical Integration;20
8.5;1.5 The Riemann Integral;22
8.6;1.6 Improper Integrals;25
8.7;1.7 The Riemann Integral in Higher Dimensions;32
8.8;1.8 More General Integrals;35
8.9;1.9 The Smoothness of Functions and Approximate Integration;35
8.10;1.10 Weight Functions;36
8.11;1.11 Some Useful Formulas;37
8.12;1.12 Orthogonal Polynomials;43
8.13;1.13 Short Guide to the Orthogonal Polynomials;48
8.14;1.14 Some Sets of Polynomials Orthogonal over Figures in the Complex Plane;57
8.15;1.15 Extrapolation and Speed-Up;58
8.16;1.16 Numerical Integration and the Numerical Solution of Integral Equations;63
9;CHAPTER 2. APPROXIMATE INTEGRATION OVER A FINITE INTERVAL;66
9.1;2.1 Primitive Rules;66
9.2;2.2 Simpson's Rule;72
9.3;2.3 Nonequally Spaced Abscissas;75
9.4;2.4 Compound Rules;85
9.5;2.5 Integration Formulas of Interpolatory Type;89
9.6;2.6 Integration Formulas of Open Type;107
9.7;2.7 Integration Rules of Gauss Type;110
9.8;2.8 Integration Rules Using Derivative Data;147
9.9;2.9 Integration of Periodic Functions;149
9.10;2.10 Integration of Rapidly Oscillatory Functions;161
9.11;2.11 Contour Integrals;183
9.12;2.12 Improper Integrals (Finite Interval);187
9.13;2.13 Indefinite Integration;205
10;CHAPTER 3. APPROXIMATE INTEGRATION OVER INFINITE INTERVALS;214
10.1;4.1 Types of Errors;286
10.2;4.2 Roundoff Error for a Fixed Integration Rule;287
10.3;4.3 Truncation Error;300
10.4;4.4 Special Devices;310
10.5;4.5 Error Estimates through Differences;312
10.6;4.6 Error Estimates through the Theory of Analytic Functions;315
10.7;4.7 Application of Functional Analysis to Numerical Integration;332
10.8;4.8 Errors for Integrands with Low Continuity;347
10.9;4.9 Practical Error Estimation;351
10.10;3.1 Change of Variable;214
10.11;3.2 Proceeding to the Limit;217
10.12;3.3 Truncation of the Infinite Interval;220
10.13;3.4 Primitive Rules for the Infinite Interval;222
10.14;3.5 Formulas of Interpolatory Type;234
10.15;3.6 Gaussian Formulas for the Infinite Interval;237
10.16;3.7 Convergence of Formulas of Gauss Type for Singly and Doubly Infinite Intervals;242
10.17;3.8 Oscillatory Integrands;245
10.18;3.9 The Fourier Transform;251
10.19;3.10 The Laplace Transform and Its Numerical Inversion;279
11;CHAPTER 4. ERROR ANALYSIS;286
11.1;4.1 Types of Errors;286
11.2;4.2 Roundoff Error for a Fixed Integration Rule;287
11.3;4.3 Truncation Error;300
11.4;4.4 Special Devices;310
11.5;4.5 Error Estimates through Differences;312
11.6;4.6 Error Estimates through the Theory of Analytic Functions;315
11.7;4.7 Application of Functional Analysis to Numerical Integration;332
11.8;4.8 Errors for Integrands with Low Continuity;347
11.9;4.9 Practical Error Estimation;351
12;CHAPTER 5. APPROXIMATE INTEGRATION IN TWO OR MORE DIMENSIONS;359
12.1;5.1 Introduction;359
12.2;5.2 Some Elementary Multiple Integrals over Standard Regions;361
12.3;5.3 Change of Order of Integration;363
12.4;5.4 Change of Variables;363
12.5;5.5 Decomposition into Elementary Regions;365
12.6;5.6 Cartesian Products and Product Rules;369
12.7;5.7 Rules Exact for Monomials;378
12.8;5.8 Compound Rules;394
12.9;5.9 Multiple Integration by Sampling;399
12.10;5.10 The Present State of the Art;430
13;CHAPTER 6. AUTOMATIC INTEGRATION;433
13.1;6.1 The Goals of Automatic Integration;433
13.2;6.2 Some Automatic Integrators;440
13.3;6.3 Romberg Integration;449
13.4;6.4 Automatic Integration Using Tschebyscheff Polynomials;461
13.5;6.5 Automatic Integration in Several Variables;465
13.6;6.6 Concluding Remarks;476
14;APPENDIX 1: ON THE PRACTICAL EVALUATION OF INTEGRALS;478
15;APPENDIX 2: FORTRAN PROGRAMS;495
16;APPENDIX 3: BIBLIOGRAPHY OF ALGOL, FORTRAN, AND PL/I PROCEDURES;524
17;APPENDIX 4: BIBLIOGRAPHY OF TABLES;533
18;APPENDIX 5: BIBLIOGRAPHY OF BOOKS AND ARTICLES;539
19;Index;620