Numerical Stability Of GMRES

. The Generalized Minimal Residual Method (GMRES) is one of the significant methods for solving linear algebraic systems with nonsymmetric matrices. It minimizes the norm of the residual on the linear variety determined by the initial residual and the n-th Krylov residual subspace and is therefore optimal, with respect to the size of the residual, in the class of Krylov subspace methods. One possible way of computing the GMRES approximations is based on constructing the orthonormal basis of the Krylov subspaces (Arnoldi basis) and then solving the transformed least squares problem. This paper studies the numerical stability of such formulations of GMRES. Our approach is based on the Arnoldi recurrence for the actually, i.e. in finite precision arithmetic, computed quantities. We consider the Householder (HHA), iterated modified GramSchmidt (IMGSA), and iterated classical Gram-Schmidt (ICGSA) implementations. Under the obvious assumption on the numerical nonsingularity of the system m..