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 - <p style="text-align: justify;">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&#39;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.</p>