Title Theory and applications of Gaussian quadrature methods / Narayan Kovvali

Published Cham, Switzerland : Springer, ©2011

Online access available from: Synthesis Digital Library    View Resource Record

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Description 1 online resource (viii, 57 pages) : illustrations Series Synthesis lectures on algorithms and software in engineering, 1938-1735 ; #8 Synthesis lectures on algorithms and software in engineering ; #8. Contents 1. Introduction -- 1.1 Quadrature -- 1.2 Families of quadrature rules -- 1.3 Organization 2. Approximating with polynomials and related functions -- 2.1 Trigonometric and orthogonal polynomials -- 2.2 Prolate spheroidal wave functions -- 2.3 Approximation theory -- 2.3.1 Fourier series -- 2.3.2 Chebyshev series -- 2.3.3 Legendre series -- 2.3.4 Prolate series 3. Gaussian quadrature -- 3.1 Gaussian quadrature based on orthogonal polynomials -- 3.1.1 Computation of the quadrature nodes and weights -- 3.2 Gaussian quadrature based on prolate spheroidal wave functions -- 3.3 Gauss-Lobatto quadrature 4. Applications -- 4.1 Integration examples -- 4.2 Computations involving contour integration -- 4.2.1 Oscillatory integral -- 4.2.2 Matrix functions -- 4.3 Pseudospectral methods -- 4.3.1 Background -- 4.3.2 Legendre pseudospectral method for the Fresnel integrals A. Links to mathematical software -- Bibliography -- Author's biography Summary "Gaussian quadrature is a powerful technique for numerical integration that falls under the broad category of spectral methods. The purpose of this work is to provide an introduction to the theory and practice of Gaussian quadrature. We study the approximation theory of trigonometric and orthogonal polynomials and related functions and examine the analytical framework of Gaussian quadrature. We discuss Gaussian quadrature for bandlimited functions, a topic inspired by some recent developments in the analysis of prolate spheroidal wave functions. Algorithms for the computation of the quadrature nodes and weights are described. Several applications of Gaussian quadrature are given, ranging from the evaluation of special functions to pseudospectral methods for solving differential equations. Software realization of select algorithms is provided"--Provided by publisher Analysis approximation theory bandlimited functions Gaussian quadrature numerical integration prolate spheroidal wave functions spectral methods trigonometric and orthogonal polynomials Bibliography Includes bibliographical references (pages 53-55) Notes Print version record Subject Gaussian quadrature formulas. Approximation theory. MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. Approximation theory Gaussian quadrature formulas Form Electronic book ISBN 9781608457540 1608457540 9783031015175 3031015177

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