Handbook of Computational Methods for Integration
Purchasing Options
Features
Summary
During the past 20 years, there has been enormous productivity in theoretical as well as computational integration. Some attempts have been made to find an optimal or best numerical method and related computer code to put to rest the problem of numerical integration, but the research is continuously ongoing, as this problem is still very much open-ended. The importance of numerical integration in so many areas of science and technology has made a practical, up-to-date reference on this subject long overdue.
The Handbook of Computational Methods for Integration discusses quadrature rules for finite and infinite range integrals and their applications in differential and integral equations, Fourier integrals and transforms, Hartley transforms, fast Fourier and Hartley transforms, Laplace transforms and wavelets. The practical, applied perspective of this book makes it unique among the many theoretical books on numerical integration and quadrature. It will be a welcomed addition to the libraries of applied mathematicians, scientists, and engineers in virtually every discipline.
Table of Contents
Preface
Notation
Preliminaries
Notation and Definitions
Orthogonal Polynomials
Finite and Divided Differences
Interpolation
Semi-Infinite Interval
Convergence Accelerators
Polynomial Splines
Interpolatory Quadrature
Riemann Integration
Euler-Maclaurin Expansion
Interpolatory Quadrature Rules
Newton-Cotes Formulas
Basic Quadrature Rules
Repeated Quadrature Rules
Romberg’s Scheme
Gregory’s Correction Scheme
Interpolatory Product Integration
Iterative and Adaptive Schemes
Test Integrals
Gaussian Quadrature
Gaussian Rules
Extended Gaussian Rules
Other Extended Rules
Analytic Functions
Bessel’s Rule
Gaussian Rules for the Moments
Finite Oscillatory Integrals
Noninterpolatory Product Integration
Test Integrals
Improper Integrals
Infinite Range Integrals
Improper Integrals
Slowly Convergent Integrals
Oscillatory Integrals
Product Integration
Singular Integrals
Quadrature Rules
Product Integration
Acceleration Methods
Singular and Hypersingular Integrals
Computer-Aided Derivations
Fourier Integrals and Transforms
Fourier Transforms
Interpolatory Rules for Fourier Integrals
Interpolatory Rules by Rational Functions
Trigonometric Integrals
Finite Fourier Transforms
Discrete Fourier Transforms
Hartley Transform
Inversion of Laplace Transforms
Use of Orthogonal Polynomials
Interpolatory Methods
Use of Gaussian Quadrature Rules
Use of Fourier Series
Use of Bromwich Contours
Inversion by the Riemann Sum
New Exact Laplace Inverse Transforms
Wavelets
Orthogonal Systems
Trigonometric System
Haar System
Other Wavelet Systems
Daubechies’ System
Fast Daubechies Transforms
Integral Equations
Nyström System
Integral Equations of the First Kind
Integral Equations of the Second Kind
Singular Integral Equations
Weakly Singular Equations
Cauchy Singular Equations of the First Kind
Cauchy Singular Equations of the Second Kind
Canonical Equation
Finite-Part Singular Equations
Integral Equations Over a Contour
Appendix A: Quadrature Tables
Cotesian Numbers, Tabulated for k£n/2, n=1(1)11
Weights for a Single Trapezoidal Rule and Repeated Simpson’s Rule
Weights for Repeated Simpson’s Rule and a Single Trapezoidal Rule
Weights for a Single 3/8-Rule and Repeated Simpson’s Rule
Weights for Repeated Simpson’s Rule and a Single 3/8-Rule
Gauss-Legendre Quadrature
Gauss-Laguerre Quadrature
Gauss-Hermite Quadrature
Gauss-Radau Quadrature
Gauss-Lobatto Quadrature
Nodes of Equal-Weight Chebyshev Rule
Gauss-Log Quadrature
Gauss-Kronrod Quadrature Rule
Patterson’s Quadrature Rule
Filon’s Quadrature Formula
Gauss-Cos Quadrature on [π/2, π/2]
Gauss-Cos Quadrature on [0, π/2]
Coefficients in (5.1.15) with w(x)=ln(1/x), 0
Editorial Reviews
“… excellent in form and ideas which it contains, is devoted to the very often occurred problem of computation of integrals in one variable. …is not only the overview of rules but it explains the ideas for which the given formula were previously derived and studied. …The reviewer is persuaded that the book is a very useful source for researchers in many fields of mathematics and for graduate students.”
—EMS Newsletter, March 2006
