An effective predictor-corrector process for large scale linear system of equations
type: Item
instance of: scholarly article
title: An effective predictor-corrector process for large scale linear system of equations
zbMATH Open document ID: 1105.65032
published in: Applied Mathematics and Computation
publication date: 2006-11-15T00:00:00Z
review text: The author treats the old problem of seriously ill-conditioned systems of linear algebraic equations \(A x = b, \enspace A \in \mathbb R^{m \times m}, \enspace x,b \in \mathbb R^m\), i.e. systems, whose condition number cond\((A) = \enspace \mid \mid A \mid \mid \enspace \mid \mid A^{-1} \mid \mid \) is very large. To improve the accuracy of the solution by direct methods commonly iterative refinement methods of \textit{R. S. Martin, G. Peters}, and \textit{J.-H. Wilkinson} [Numer. Math. 7, 362--383 (1965; Zbl 0135.37402)] are used. But e.g., for linear systems with a Hilbert matrix of order \(m \geq 12\) Wilkinson's refinement iteration gives solutions without any significant figures using double precision arithmetic. The Wilkinson refinement algorithm for ill-conditioned linear system is related to the explicit Euler method for the solution of stiff ordinary differential equations \[ \frac{dz}{dx} = A^{-1}(b - A z), \enspace z(0) = 0, \] where \(z = A^{-1}b\) is a unique stationary point. Starting from the knowledge that the explicit Euler method is not appropriate for stiff systems the author introduces a one-step second derivative extended backward differentiation formula of order four combined with Enright's second derivative linear method that computes the asymptotically stable equilibrium point of the dynamic system which is the solution of the linear system. The accuracy of the predictor-corrector process is validated by Hilbert matrix examples up to the order of 4000.
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F22
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 65L05
zbMATH DE Number: 5072700
zbMATH Keywords: ill-posedness
zbMATH Keywords: ill-conditioned linear system of algebraic equations
zbMATH Keywords: iterative improvement of solution
zbMATH Keywords: Hilbert matrix
zbMATH Keywords: stiff system of ordinary differential equations
zbMATH Keywords: equilibrium point
zbMATH Keywords: dynamic system
zbMATH Keywords: Wilkinson refinement algorithm
zbMATH Keywords: explicit Euler method
zbMATH Keywords: backward differentiation formula
author: Xinyuan Wu
MaRDI profile type: Publication
full work available at URL: j.amc.2005.11.131
OpenAlex ID: W2011273315
cites work: Iterative refinement implies numerical stability
cites work: Iterative Refinement Implies Numerical Stability for Gaussian Elimination
cites work: Iterative refinement for linear systems and LAPACK
cites work: Iterative refinement of solution with biparameter for solving ill-conditioned systems of linear algebraic equations.
cites work: Symmetric decomposition of a positive definite matrix
cites work: Iterative refinement of the solution of a positive definite system of equations
cites work: Q3779682
DOI: 10.1016/J.AMC.2005.11.131
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description: scientific article; zbMATH DE number 5072700