(untitled) (Journal of Computational Mathematics, 2021)
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 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.