Variants Of The Residual Minimizing Krylov Space Methods

Several variants of the GMRES method for solving systems of linear algebraic equations are described. These variants differ in building up different sets of orthonormalized vectors used for the construction of the approximate solution. A new A T A variant of GMRES is proposed and optimal implementations of the algorithms are thoroughly discussed. It is shown that the described implementations are superior to widely used schemes ORTHODIR, ORTHOMIN and their relatives. 1 Introduction Let Ax = b be a system of linear algebraic equations, where A is a real nonsingular N by N matrix and b an N-dimensional real vector. Many iterative methods for solving this system start with an initial guess x 0 for the solution and seek the n-th approximate solution x n in the linear variety x n 2 x 0 +K n (A; r 0 ) (1. 1) where r 0 = b \Gamma Ax 0 is the initial residual and K n (A; r 0 ) is the n-th Krylov subspace generated by A; r 0 , K n (A; r 0 ) = spanfr 0 ; Ar 0 ; : :..

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..