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

Zhihao Cao

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

Published online: 1998-12

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  • Abstract

This paper extends 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 perturbations 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 ${\rm method}^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoretical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given.

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@Article{JCM-16-539, author = {Cao , Zhihao}, 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 extends 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 perturbations 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 ${\rm method}^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoretical 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 AU - Cao , Zhihao 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, Iterative methods, Backward error. AB -

This paper extends 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 perturbations 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 ${\rm method}^{[12]}$. The minimization problem amounts to computing the smallest singular value and the corresponding right singular vector of a low-order upper-Hessenberg matrix. Theoretical properties of the algorithm are discussed and practical implementation issues are considered. The numerical examples are also given.

Zhihao 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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