Fields of values and the ADI method for non-normal matrices
type: Item
instance of: scholarly article
title: Fields of values and the ADI method for non-normal matrices
zbMATH Open document ID: 0784.65026
DOI: 10.1016/0024-3795(93)90531-R
author: Gerhard Starke
published in: Linear Algebra and its Applications
publication date: 1994-03-29T00:00:00Z
review text: Generalizing the results of \textit{M. Eiermann} [Lin. Algebra Appl. 180, 167-197 (1993; Zbl 0784.65022)], the author obtains an upper bound for the error reduction of the ADI method in terms of the fields of values of the matrices which define splitting of the matrix in the iterations. Then it is shown that if \(A\) is an irreducible tridiagonal matrix and if \(B\) is the symmetric matrix similar to \(A\), then the field of values of \(B\) is included in that of \(A\), and the result is generalized to Kronecker sums of tridiagonal matrices. This result may be considered as an alternative to preconditioning for non-normal matrices. The estimate is tested on a model boundary value problem. Numerical results demonstrate the power of analysis based on the field of values for considered problems and indicate that the resulting estimates may be useful for an a priori prediction of the behaviour of the process in the finite stage of computation.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 179139
zbMATH Keywords: numerical results
zbMATH Keywords: error reduction
zbMATH Keywords: ADI method
zbMATH Keywords: fields of values
zbMATH Keywords: irreducible tridiagonal matrix
zbMATH Keywords: Kronecker sums of tridiagonal matrices
zbMATH Keywords: preconditioning
reviewed by: Z. Dostál
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 179139