Generalized averaged Gauss quadrature rules for the approximation of matrix functionals
type: Item
instance of: scholarly article
title: Generalized averaged Gauss quadrature rules for the approximation of matrix functionals
zbMATH Open document ID: 1352.65089
author: Miodrag M. Spalević
author: Tunan Tang
author: Lothar Reichel
published in: BIT
publication date: 2016-10-21T00:00:00Z
review text: Let \(A\) be a large square complex matrix and \(u\), \(v\) complex vectors. Further let \(f\) be a function. In many applications, one has to compute the matrix functional \(u^{\ast}f(A)v\). Usually one reduces \(A\) to a small matrix by a few steps of a Lanczos process and then one evaluates the reduced problem. In this paper, the authors discuss the application of generalized averaged Gauss quadrature rules to the approximation of the matrix functional. First matrix functionals \(u^{\ast}f(A)u\) with a Hermitian matrix \(A\) are approximated by generalized averaged Gauss quadrature rule introduced by \textit{M. M. Spalević} [Math. Comput. 76, No. 259, 1483--1492 (2007; Zbl 1113.65025)]. If \(A\) is non-Hermitian, then \(A\) can be reduced to a small non-Hermitian tridiagonal matrix by a few steps of the non-Hermitian Lanczos process. Then the non-Hermitian tridiagonal matrix defines a Gauss-type quadrature rule, and the authors introduce new generalized averaged Gauss quadrature rules, because this yields higher accuracy and requires almost the same computational effort. In numerical examples, the functions \(f(x) = (1 + x^2)^{-1}\), \(e^x\), and \(\log (x+ c)\) with \(c>0\) are considered. Examples from network analysis are given too.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 65F60
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 6641941
zbMATH Keywords: matrix functional
zbMATH Keywords: Gauss quadrature
zbMATH Keywords: generalized averaged Gauss quadrature rule
zbMATH Keywords: non-Hermitian Lanczos process
zbMATH Keywords: higher accuracy
zbMATH Keywords: numerical example
describes a project that uses: blgaussexp
describes a project that uses: sgcen
describes a project that uses: OPQ
describes a project that uses: mftoolbox
describes a project that uses: Pajek datasets
describes a project that uses: Pajek
MaRDI profile type: Publication
full work available at URL: s10543-015-0592-7
OpenAlex ID: W2278113833
cites work: Computation of Gauss-Kronrod quadrature rules with non-positive weights
cites work: Some large-scale matrix computation problems
cites work: Q3043782
cites work: Quadrature rule-based bounds for functions of adjacency matrices
cites work: Q4296886
cites work: An adaptive Chebyshev iterative method for nonsymmetric linear systems based on modified moments
cites work: Computation of Gauss-Kronrod quadrature rules
cites work: Q4938054
cites work: A method for numerical integration on an automatic computer
cites work: On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas
cites work: Q3100186
cites work: Network Properties Revealed through Matrix Functions
cites work: Network analysis via partial spectral factorization and Gauss quadrature
cites work: Block Gauss and Anti-Gauss Quadrature with Application to Networks
cites work: On Generating Orthogonal Polynomials
cites work: Q4820343
cites work: Q4288554
cites work: Q5851422
cites work: Calculation of Gauss Quadrature Rules
cites work: Matrix interpretations and applications of the continued fraction algorithm
cites work: Functions of Matrices
cites work: Calculation of Gauss-Kronrod quadrature rules
cites work: Minimization of functionals on the solution of a large-scale discrete ill-posed problem
cites work: An overview of the computational aspects of Kronrod quadrature rules
cites work: Positive quadrature formulas III: asymptotics of weights
cites work: Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator
cites work: Q4414854
cites work: On generalized averaged Gaussian formulas
cites work: A note on generalized averaged Gaussian formulas
cites work: A Breakdown-Free Variation of the Nonsymmetric Lanczos Algorithms
DOI: 10.1007/S10543-015-0592-7
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description: scientific article; zbMATH DE number 6641941