Methods of Numerical Integration

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.

Inhalt

Preface to First Edition
Preface to Second Edition
Chapter 1 Introduction
1.1 Why Numerical Integration?
1.2 Formal Differentiation and Integration on Computers
1.3 Numerical Integration and Its Appeal in Mathematics
1.4 Limitations of Numerical Integration
1.5 The Riemann Integral
1.6 Improper Integrals
1.7 The Riemann Integral in Higher Dimensions
1.8 More General Integrals
1.9 The Smoothness of Functions and Approximate Integration
1.10 Weight Functions
1.11 Some Useful Formulas
1.12 Orthogonal Polynomials
1.13 Short Guide to the Orthogonal Polynomials
1.14 Some Sets of Polynomials Orthogonal Over Figures in the Complex Plane
1.15 Extrapolation and Speed-Up
1.16 Numerical Integration and the Numerical Solution of Integral Equations
Chapter 2 Approximate Integration Over a Finite Interval
2.1 Primitive Rules
2.2 Simpson's Rule
2.3 Nonequally Spaced Abscissas
2.4 Compound Rules
2.5 Integration Formulas of Interpolatory Type
2.6 Integration Formulas of Open Type
2.7 Integration Rules of Gauss Type
2.8 Integration Rules Using Derivative Data
2.9 Integration of Periodic Functions
2.10 Integration of Rapidly Oscillatory Functions
2.11 Contour Integrals
2.12 Improper Integrals (Finite Interval)
2.13 Indefinite Integration
Chapter 3 Approximate Integration Over Infinite Intervals
3.1 Change of Variable
3.2 Proceeding to the Limit
3.3 Truncation of the Infinite Interval
3.4 Primitive Rules for the Infinite Interval
3.5 Formulas of Interpolatory Type
3.6 Gaussian Formulas for the Infinite Interval
3.7 Convergence of Formulas of Gauss Type for Singly and Doubly Infinite Intervals
3.8 Oscillatory Integrands
3.9 The Fourier Transform
3.10 The Laplace Transform and Its Numerical Inversion
Chapter 4 Error Analysis
4.1 Types of Errors
4.2 Roundoff Error for a Fixed Integration Rule
4.3 Truncation Error
4.4 Special Devices
4.5 Error Estimates through Differences
4.6 Error Estimates through the Theory of Analytic Functions
4.7 Application of Functional Analysis to Numerical Integration
4.8 Errors for Integrands with Low Continuity
4.9 Practical Error Estimation
Chapter 5 Approximate Integration in Two or More Dimensions
5.1 Introduction
5.2 Some Elementary Multiple Integrals Over Standard Regions
5.3 Change of Order of Integration
5.4 Change of Variables
5.5 Decomposition into Elementary Regions
5.6 Cartesian Products and Product Rules
5.7 Rules Exact for Monomials
5.8 Compound Rules
5.9 Multiple Integration by Sampling
5.10 The Present State of the Art
Chapter 6 Automatic Integration
6.1 The Goals of Automatic Integration
6.2 Some Automatic Integrators
6.3 Romberg Integration
6.4 Automatic Integration Using Tschebyscheff Polynomials
6.5 Automatic Integration in Several Variables
6.6 Concluding Remarks
Appendix 1 On the Practical Evaluation of Integrals
Appendix 2 Fortran Programs
Appendix 3 Bibliography of Algol, Fortran, and PL/I Procedures
Appendix 4 Bibliography of Tables
Appendix 5 Bibliography of Books and Articles
Index