Analysis of a Minimum Perturbation Algorithm for Nonsymmetric Linear Systems
Ebrahim M. Kasenally, Valeria Simoncini · SIAM Journal on Numerical Analysis · 1997
This paper presents a theoretical analysis of a minimum perturbation Krylov subspace method for the solution of nonsymmetric linear systems $Ax=b$. At each step, the algorithm minimizes the joint backward perturbation norm $\|[\Delta_A, \Delta_b]\|_F$ of the matrix $[A,\, b]$ and computes an approximate solution $x_m$ satisfying $(A-\Delta_A)x_m=(b+\Delta_b)$. This process is a generalization of the Saad and Schultz generalized minimum residual (GMRES) algorithm [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869] and the generalized minimum backward error (GMBACK) process of Kasenally [ SIAM J. Sci. Comput., 16 (1995), pp. 698--719]. The minimum perturbation algorithm is used to analyze the performance of these Krylov subspace methods by gauging the proximity of their solutions to joint backward perturbation optimality. The relationships among these methods are thoroughly investigated and numerical results are provided to illustrate their key differences.