Ill-conditioned matrices and the integration of stiff ODEs
type: Item
instance of: scholarly article
title: Ill-conditioned matrices and the integration of stiff ODEs
zbMATH Open document ID: 0798.65081
DOI: 10.1016/0377-0427(93)90025-7
author: L. F. Shampine
published in: Journal of Computational and Applied Mathematics
publication date: 1994-03-27T00:00:00Z
review text: The article is concerned with the numerical integration of stiff initial value problems for systems of ordinary differential equations. It is shown that the matrices occurring in the linear algebraic systems, which have to be solved when implicit linear multistep methods are applied, are ``almost always'' very ill-conditioned. Nevertheless, multistep methods can be evaluated accurately by iteration, since (resp. if) each iteration step improves the solution by one or a few accurate digits. Even semi- implicit methods perform well although they do not involve iteration. This is explained by the fact that the right-hand sides occurring in the linear algebraic systems behave in a way which makes the quotient of the relative errors of solution and right-hand side much less than the matrix condition.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E13
zbMATH DE Number: 540485
zbMATH Keywords: ill-conditioned matrices
zbMATH Keywords: Rosenbrock-methods
zbMATH Keywords: stiff initial value problems
zbMATH Keywords: systems of ordinary differential equations
zbMATH Keywords: implicit linear multistep methods
zbMATH Keywords: semi-implicit methods
reviewed by: Michael Plum
describes a project that uses: RODAS
describes a project that uses: LINPACK
MaRDI profile type: Publication
full work available at URL: 0377-0427(93)90025-7
OpenAlex ID: W2029240763
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description: scientific article; zbMATH DE number 540485