Volume 16, Issue 6
Total Generalized Minimum Backward Error Algorithm for Solving Nonsymmetric Linear Systems

Zhi-hao Cao

DOI:

J. Comp. Math., 16 (1998), pp. 539-550

Published online: 1998-12

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  • Abstract

This paper extendes the results by E.M. Kasenally$^{[7]}$ on a Generalized Minimum Backward Error Algorithm for nonsymmetric linear systems $Ax=b$ to the problem in which pertubations are simultaneously permitted on $A$ and $b$. The approach adopted by Kasenally has been to view the approximate solution as the exact solution to a perturbed linear system in which changes are permitted to the matrix $A$ only. The new method introduced in this paper is a Krylov subspace iterative method which minimizes the norm of the perturbations to both the observation vector $b$ and the data matrix $A$ and has better performance than the Kasenally?s method and the restarted GMRES method$^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoratical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given.

  • Keywords

Nonsymmetric linear systems Iterative methods Backward error

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@Article{JCM-16-539, author = {}, title = {Total Generalized Minimum Backward Error Algorithm for Solving Nonsymmetric Linear Systems}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {6}, pages = {539--550}, abstract = { This paper extendes the results by E.M. Kasenally$^{[7]}$ on a Generalized Minimum Backward Error Algorithm for nonsymmetric linear systems $Ax=b$ to the problem in which pertubations are simultaneously permitted on $A$ and $b$. The approach adopted by Kasenally has been to view the approximate solution as the exact solution to a perturbed linear system in which changes are permitted to the matrix $A$ only. The new method introduced in this paper is a Krylov subspace iterative method which minimizes the norm of the perturbations to both the observation vector $b$ and the data matrix $A$ and has better performance than the Kasenally?s method and the restarted GMRES method$^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoratical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9181.html} }
TY - JOUR T1 - Total Generalized Minimum Backward Error Algorithm for Solving Nonsymmetric Linear Systems JO - Journal of Computational Mathematics VL - 6 SP - 539 EP - 550 PY - 1998 DA - 1998/12 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9181.html KW - Nonsymmetric linear systems KW - Iterative methods KW - Backward error AB - This paper extendes the results by E.M. Kasenally$^{[7]}$ on a Generalized Minimum Backward Error Algorithm for nonsymmetric linear systems $Ax=b$ to the problem in which pertubations are simultaneously permitted on $A$ and $b$. The approach adopted by Kasenally has been to view the approximate solution as the exact solution to a perturbed linear system in which changes are permitted to the matrix $A$ only. The new method introduced in this paper is a Krylov subspace iterative method which minimizes the norm of the perturbations to both the observation vector $b$ and the data matrix $A$ and has better performance than the Kasenally?s method and the restarted GMRES method$^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoratical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given.
Zhi-hao Cao. (1970). Total Generalized Minimum Backward Error Algorithm for Solving Nonsymmetric Linear Systems. Journal of Computational Mathematics. 16 (6). 539-550. doi:
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