Two-frequency-dependent Gauss quadrature rules
type: Item
instance of: scholarly article
title: Two-frequency-dependent Gauss quadrature rules
zbMATH Open document ID: 1064.65022
author: Kyung Joong Kim
published in: Journal of Computational and Applied Mathematics
publication date: 2005-02-23T00:00:00Z
review text: The article presents Gauss quadrature rules to evaluate the integral of oscillatory functions of the form \(y(x)=h_1(x)\cos(\omega_1x)+h_2(x)\sin(\omega_1 x)+h_3(x)\cos(\omega_2x)+h_4(x)\sin(\omega_2 x)\), where \(\omega_1\) and \(\omega_2\) are frequencies and \(h_i\) are smooth functions. The problem of solving a nonlinear system of equations to calculate the quadrature coefficients \(w_k\) and the nodes \(x_k\) is posed according to the exponentially fitted approach by \textit{L. Gr. Ixaru} [Comput. Phys. Commun. 105, 1--19 (1997; Zbl 0930.65150)]. In the case of \(\omega_2\to \omega_1\) or \(\omega_1,\omega_2\to 0\) a regularization procedure is provided to avoid the singularity of the Jacobian matrix of the system of equations. In the section devoted to error analysis a theorem is given to describe the first non-vanishing term of the quadrature error. Numerical tests illustrate the power of the two-frequency-dependent quadrature rule in comparison with the classical Gauss rule and one-frequency-dependent rules.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 2137441
zbMATH Keywords: quadrature formula
zbMATH Keywords: oscillatory integrand
zbMATH Keywords: Gauss quadrature formula
zbMATH Keywords: numerical examples
zbMATH Keywords: regularization
zbMATH Keywords: error analysis
reviewed by: J. Illán
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: j.cam.2004.03.020
OpenAlex ID: W1996731649
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DOI: 10.1016/J.CAM.2004.03.020
description: scientific article; zbMATH DE number 2137441