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