Methods of Numerical Integration Second Edition

by Davis, Philip J.; Rabinowitz, Philip

• ISBN13: 9780486453392
• ISBN10: 0486453391
• Edition: 2nd
• Format: Paperback
• Copyright: 2007-10-25
• Publisher: Dover Publications

Summary

Useful to programmers and stimulating for theoreticians, this text offers a balanced presentation accessible to those with a background in calculus. Topics include approximate integration over finite and infinite intervals, error analysis, approximate integration in two or more dimensions, and automatic integration. Includes five helpful appendixes. 1984 edition.

Table of Contents

Preface to First Edition	p. ix
Preface to Second Edition	p. xi
Introduction
Why Numerical Integration?	p. 1
Formal Differentiation and Integration on Computers	p. 3
Numerical Integration and Its Appeal in Mathematics	p. 4
Limitations of Numerical Integration	p. 5
The Riemann Integral	p. 7
Improper Integrals	p. 10
The Riemann Integral in Higher Dimensions	p. 17
More General Integrals	p. 20
The Smoothness of Functions and Approximate Integration	p. 20
Weight Functions	p. 21
Some Useful Formulas	p. 22
Orthogonal Polynomials	p. 28
Short Guide to the Orthogonal Polynomials	p. 33
Some Sets of Polynomials Orthogonal over Figures in the Complex Plane	p. 42
Extrapolation and Speed-Up	p. 43
Numerical Integration and the Numerical Solution of Integral Equations	p. 48
Approximate Integration Over a Finite Interval
Primitive Rules	p. 51
Simpson's Rule	p. 57
Nonequally Spaced Abscissas	p. 60
Compound Rules	p. 70
Integration Formulas of Interpolatory Type	p. 74
Integration Formulas of Open Type	p. 92
Integration Rules of Gauss Type	p. 95
Integration Rules Using Derivative Data	p. 132
Integration of Periodic Functions	p. 134
Integration of Rapidly Oscillatory Functions	p. 146
Contour Integrals	p. 168
Improper Integrals (Finite Interval)	p. 172
Indefinite Integration	p. 190
Approximate Integration Over Infinite Intervals
Change of Variable	p. 199
Proceeding to the Limit	p. 202
Truncation of the Infinite Interval	p. 205
Primitive Rules for the Infinite Interval	p. 207
Formulas of Interpolatory Type	p. 219
Gaussian Formulas for the Infinite Interval	p. 222
Convergence of Formulas of Gauss Type for Singly and Doubly Infinite Intervals	p. 227
Oscillatory Integrands	p. 230
The Fourier Transform	p. 236
The Laplace Transform and Its Numerical Inversion	p. 264
Error Analysis
Types of Errors	p. 271
Roundoff Error for a Fixed Integration Rule	p. 272
Truncation Error	p. 285
Special Devices	p. 295
Error Estimates through Differences	p. 297
Error Estimates through the Theory of Analytic Functions	p. 300
Application of Functional Analysis to Numerical Integration	p. 317
Errors for Integrands with Low Continuity	p. 332
Practical Error Estimation	p. 336
Approximate Integration in Two or More Dimensions
Introduction	p. 344
Some Elementary Multiple Integrals over Standard Regions	p. 346
Change of Order of Integration	p. 348
Change of Variables	p. 348
Decomposition into Elementary Regions	p. 350
Cartesian Products and Product Rules	p. 354
Rules Exact for Monomials	p. 363
Compound Rules	p. 379
Multiple Integration by Sampling	p. 384
The Present State of the Art	p. 415
Automatic Integration
The Goals of Automatic Integration	p. 418
Some Automatic Integrators	p. 425
Romberg Integration	p. 434
Automatic Integration Using Tschebyscheff Polynomials	p. 446
Automatic Integration in Several Variables	p. 450
Concluding Remarks	p. 461
On the Practical Evaluation of Integrals	p. 463
Fortran Programs	p. 480
Bibliography of Algol, Fortran, and PL/I Procedures	p. 509
Bibliography of Tables	p. 518
Bibliography of Books and Articles	p. 524
Index	p. 605
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