Handbook of Computational Methods for Integration: 1st Edition (Hardback) book cover

Handbook of Computational Methods for Integration

1st Edition

By Prem K. Kythe, Michael R. Schäferkotter

Chapman and Hall/CRC

624 pages | 3 B/W Illus.

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Hardback: 9781584884286
pub: 2004-12-20
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Description

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.

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

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

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT021000
MATHEMATICS / Number Systems
TEC007000
TECHNOLOGY & ENGINEERING / Electrical