On transformations of graded matrices, with applications to stiff ODE's
type: Item
instance of: scholarly article
title: On transformations of graded matrices, with applications to stiff ODE's
zbMATH Open document ID: 0599.65020
DOI: 10.1007/BF01389585
author: Germund Dahlquist
published in: Numerische Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133039
review text: The author presents an iterative block LR refining algorithm for graded matrices together with its application to the approximate solution of singularly perturbed systems of ordinary differential equations with several parameters. A block matrix \(A=[A_{ij}]\), \(i,j=1,2,...,m\), where \(A_{ij}\) are \((n_ i\times n_ j)\) matrices, is called a graded matrix if there is a diagonal matrix \(E=[\epsilon_ iI_{ii}]\) with parameters \(0\leq \epsilon_ 1<\epsilon_ 2<...<\epsilon_ m\) such that \(A=E^{-1}B\) and B is a matrix whose elements are small compared to the number \(\epsilon =\max (\epsilon_ i/\epsilon_{i+1})\). Assume that A may be decomposed into a block LR product \(A=L^{(1)}R^{(1)}\) where \(L^{(1)}\) is block lower diagonal and \(R^{(1)}\) is block upper diagonal with \(ER^{(1)}\), \(EL^{(1)}E^{-1}\) bounded. The iterative refining algorithm produces a new product \(A=LR\) from \(L^{(1)}\), \(R^{(1)}\) with the property that the diagonal blocks of L are unit matrices and \(L=I+O(\epsilon)\). A proof of convergence of the algorithm is presented based on criteria involving norms of the matrices \(R_{ij}^{(1)}\), \(L_{ij}^{(1)}\), \((R_{ij}^{(1)})^{-1}\).
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 3967795
zbMATH Keywords: stiff equations
zbMATH Keywords: iterative block LR refining algorithm
zbMATH Keywords: graded matrices
zbMATH Keywords: singularly perturbed systems
zbMATH Keywords: convergence
reviewed by: John B. Butler jun.
describes a project that uses: NSDTST
describes a project that uses: STDTST
describes a project that uses: Matlab
MaRDI profile type: Publication
cites work: Q4053029
cites work: On matrix majorants and minorants, with applications to differential equations
cites work: Q3319710
cites work: Two FORTRAN packages for assessing initial value methods
cites work: Q3333992
cites work: Q3939134
cites work: Q5735511
cites work: Q3967010
cites work: On Logarithmic Norms
cites work: Q5624527
description: scientific article; zbMATH DE number 3967795