Some theoretical aspects of generalised quadrature methods.
type: Item
instance of: scholarly article
title: Some theoretical aspects of generalised quadrature methods.
zbMATH Open document ID: 1035.65024
DOI: 10.1016/S0885-064X(03)00004-9
author: D. Massart
published in: Journal of Complexity
publication date: 2003-08-19T00:00:00Z
review text: Generalized quadrature methods for the numerical approximation of integrals \[ Q(f)=\int_a^bw(x)f(x)dx \] with a given irregular oscillatory weight function \(w\), satisfying \(Lw=0\) for a differential operator \(L\), are constructed using Lagrange's identity. Forcing exactness for a set of basis functions leads to a set of linear equations for the weights \(a_k\) of the quadrature formula \(Q(f) \approx \sum a_kf(x_k)\). These equations are solved numerically. Troubles due to ill-conditioning can occur. The paper is mainly concerned with the question, of how the numerical errors in the computed weights \(a_k\) affect the accuracy of the approximation to the integral \(Q(f)\). Four examples illustrate the theoretical predictions.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 1967208
zbMATH Keywords: quadrature formula
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: error analysis
zbMATH Keywords: Lagrange's identity
zbMATH Keywords: numerical examples
zbMATH Keywords: ill-conditioning
reviewed by: Rudolf Wegmann
MaRDI profile type: Publication
cites work: Comparison of some methods for evaluating infinite range oscillatory integrals
cites work: Far Field Asymptotics of the Two-Dimensional Linearised Sloping Beach Problem
cites work: A method to generate generalized quadrature rules for oscillatory integrals
cites work: A high order, progressive method for the evaluation of irregular oscillatory integrals
cites work: Q5734756
cites work: Q5829329
cites work: On high precision methods for the evaluation of Fourier integrals with finite and infinite limits
cites work: Symmetry reduction of Fourier kernels
full work available at URL: s0885-064x(03)00004-9
OpenAlex ID: W2073377075
description: scientific article; zbMATH DE number 1967208