Methods of Numerical Integration

Preface to First EditionPreface to Second EditionChapter 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 EquationsChapter 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 IntegrationChapter 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 InversionChapter 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 EstimationChapter 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 ArtChapter 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 RemarksAppendix 1 On the Practical Evaluation of IntegralsAppendix 2 Fortran ProgramsAppendix 3 Bibliography of Algol, Fortran, and PL/I ProceduresAppendix 4 Bibliography of TablesAppendix 5 Bibliography of Books and ArticlesIndex