Methods Of Numerical Integration 2ed Davis Rabinowitz
METHODS
OF NUMERICAL
INTEGRATION
SECOND EDITION
Philip J. Davis
APPLIED MATHEMATICS DIVISION
BROWN UNIVERSITY
PROVIDENCE, RHODE ISLAND
Philip Rabinowitz
DEPARTMENT OF APPLIED MATHEMATICS
THE WEIZMANN INSTITUTE OF SCIENCE
REHOVOT, ISRAEL
@
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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COPYRIGHT @ 1984, BY ACADEMIC PRESS, INC.
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United Kingdom Edition published by
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Library of Congress Cataloging in Publication Data
Davis, Philip J, Date
Methods of numerical integration.
(Computer science and applied mathematics)
Includes bibliographies and index.
1. Numerical integration. I. Rabinowitz, Philip.
II. Title. III. Series.
QA299.3.D28 1983 515'.624 83-13522
ISBN 0-12-206360-0 (alk. paper)
PRINTED IN THE UNITED STATES OF AMERICA
88 89 9 8 7 6 5 4 :\ 2
10
Contents
Preface to First Edition
Preface to Second Edition
CHAPTER 1 INTRODUCTION
1 . 1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1. 11
1.12
1.13
1.14
1.15
1.16
Why Numerical Integration?
Formal Differentiation and Integration on Computers
Numerical Integration and Its Appeal in Mathematics
Limitations of Numerical Integration
The Riemann Integral
Improper Integrals
The Riemann Integral in Higher Dimensions
More General Integrals
The Smoothness of Functions and Approximate
Integration
Weight Functions
Some Useful Formulas
Orthogonal Polynomials
Short Guide to the Orthogonal Polynomials
Some Sets of Polynomials Orthogonal over Figures
in the Complex Plane
Extrapolation and Speed-Up
Numerical Integration and the Numerical Solution
of Integral Equations
vii
Xl
XIll
I
3
4
5
7
10
17
20
20
21
22
28
33
42
43
48
viii
CONTENTS
CHAPTER 2 APPROXIMATE INTEGRATION OVER A FINITE
INTER VAL
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
Primitive Rules
Simpson's Rule
Nonequally Spaced Abscissas
Compound Rules
Integration Formulas of Interpolatory Type
Integration Formulas of Open Type
Integration Rules of Gauss Type
Integration Rules Using Derivative Data
Integration of Periodic Functions
Integration of Rapidly Oscillatory Functions
Contour Integrals
Improper Integrals (Finite Interval)
Indefinite Integration
51
57
60
70
74
92
95
132
134
146
168
172
190
CHAPTER 3 APPROXIMATE INTEGRATION OVER INFINITE
INTER V ALS
3.1 Change of Variable 199
3.2 Proceeding to the Limit 202
3.3 Truncation of the Infinite Interval 205
3.4 Primitive Rules for the Infinite Interval 207
3.5 Formulas of Interpolatory Type 219
3.6 Gaussian Formulas for the Infinite Interval 222
3.7 Convergence of Formulas of Gauss Type for Singly
and Doubly Infinite Intervals 227
3.8 Oscillatory Integrands 230
3.9 The Fourier Transform 236
3.10 The Laplace Transform and Its Numerical Inversion 264
CHAPTER 4 ERROR ANALYSIS
4.1 Types of Errors 271
4.2 Roundoff Error for a Fixed Integration Rule 272
4.3 Truncation Error 285
4.4 Special Devices 295
4.5 Error Estimates through Differences 297
4.6 Error Estimates through the Theory of Analytic
Functions 300
4.7 Application of Functional Analysis to Numerical
Integration 317
4.8 Errors for Integrands with Low Continuity 332
4.9 Practical Error Estimation 336
CONTENTS ix
CHAPTER 5 APPROXIMATE INTEGRATION IN TWO OR MORE
DIMENSIONS
5. I Introduction 344
5.2 Some Elementary Multiple Integrals over Standard
Regions 346
5.3 Change of Order of Integration 348
5.4 Change of Variables 348
5.5 Decomposition into Elementary Regions 350
5.6 Cartesian Products and Product Rules 354
5.7 Rules Exact for Monomials 363
5.8 Compound Rules 379
5.9 Multiple Integration by Sampling 384
5. 10 The Present State of the Art 415
CHAPTER 6 AUTOMATIC INTEGRATION
6.1 The Goals of Automatic Integration 418
6.2 Some Automatic Integrators 425
6.3 Romberg Integration 434
6.4 Automatic Integration Using Tschebyscheff
Polynomials 446
6.5 Automatic Integration in Several Variables 450
6.6 Concluding Remarks 461
APPENDIX ION THE PRACTICAL EVALUATION
OF INTEGRALS,
Milton Abramowitz 463
APPENDIX 2FORTRAN PROGRAMS 480
APPENDIX 3 BIBLIOGRAPHY OF ALGOL, FORTRAN,
AND PL/I PROCEDURES 509
APPENDIX 4 BIBLIOGRAPHY OF TABLES 518
APPENDIX 5 BIBLIOGRAPHY OF BOOKS AND ARTICLES 524
Index 605
好像有人求这本书
It is a good book.
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