Simplified anti-Gauss quadrature rules with applications in linear algebra
type: Item
instance of: scholarly article
title: Simplified anti-Gauss quadrature rules with applications in linear algebra
zbMATH Open document ID: 1390.41036
author: Hessah Alqahtani
author: Lothar Reichel
published in: Numerical Algorithms
publication date: 2018-02-09T00:00:00Z
review text: The authors develop inexpensive techniques for the efficient computation of upper and lower bounds for expressions of the form \(F(A) := w^{\mathrm T} f(A) v\) where \(A\) is a large square \(n\)-by-\(n\) matrix, \(f\) is a function, and \(v, w \in \mathbb R^n\). In contrast to earlier works, there are no highly restrictive assumptions such as the symmetry of \(A\), \(v=w\), or the requirement that certain derivatives of \(f\) do not change sign. Also, the methods developed here do not require any knowledge about the spectrum of \(A\). The approach is based on rewriting \(F(A)\) in the form of a Stieltjes integral and on using a Gaussian quadrature rule associated to its distribution function in combination with a second quadrature rule to obtain the desired upper and lower bounds. The second rule is essentially an anti-Gaussian rule, slightly modified in order to significantly reduce the computational cost of its construction. The authors provide a set of relatively mild conditions under which one can prove that one of the formulas yields a lower bound for the required value and the other one gives an upper bound.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 6836929
zbMATH Keywords: matrix functions
zbMATH Keywords: quadrature
zbMATH Keywords: error estimates
describes a project that uses: Pajek
describes a project that uses: Pajek datasets
describes a project that uses: mftoolbox
describes a project that uses: ABLE
describes a project that uses: sgcen
describes a project that uses: hubauth
describes a project that uses: blgaussexp
describes a project that uses: OPQ
MaRDI profile type: Publication
full work available at URL: s11075-017-0329-6
OpenAlex ID: W2609830030
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DOI: 10.1007/S11075-017-0329-6
description: scientific article; zbMATH DE number 6836929