On the Implementation of Some Residual Minimizing Krylov Space Methods

. Several variants of the GMRES method for solving linear nonsingular systems of 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 the efficient implementation of the algorithm is discussed. 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 +Kn (A; r 0 ); (1) where r 0 = b \Gamma Ax 0 is the initial residual and Kn (A; r 0 ) is the n-th Krylov subspace generated by A; r 0 , Kn (A; r 0 ) = spanfr 0 ; Ar 0 ; : : : ; A n\Gamma1 r 0 g: (2) Among the broad variety of the Krylov space methods (surveys can be found, e.g., in [3], [7]..

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 ; : :..

On the Role of Orthogonality in the GMRES Method

. In the paper we deal with some computational aspects of the Generalized minimal residual method (GMRES) for solving systems of linear algebraic equations. The key question of the paper is the importance of the orthogonality of computed vectors and its influence on the rate of convergence, numerical stability and accuracy of different implementations of the method. Practical impact on the efficiency in the parallel computer environment is considered. 1 Introduction Scientific and engineering research is becoming increasingly dependent upon development and implementation of efficient parallel algorithms on modern highperformance computers. Numerical linear algebra is an important part of such research and numerical linear algebra algorithms represent the most widely used computational tools in science and engineering. Matrix computations, including the solution of systems of linear equations, least squares problems, and algebraic eigenvalue problems, govern the performance of many app..

Numerical Behaviour of the Modified Gram-Schmidt GMRES Implementation

. In [6] the Generalized Minimal Residual Method (GMRES) which constructs the Arnoldi basis and then solves the transformed least squares problem was studied. It was proved that GMRES with the Householder orthogonalization - based implementation of the Arnoldi process (HHA), see [9], is backward stable. In practical computations, however, the Householder orthogonalization is too expensive, and it is usually replaced by the modified Gram-Schmidt process (MGSA). Unlike the HHA case, in the MGSA implementation the orthogonality of the Arnoldi basis vectors is not preserved near the level of machine precision. Despite this, the MGSA GMRES performs surprisingly well, and its convergence behaviour and the ultimately attainable accuracy do not differ significantly from those of the HHA GMRES. As it was observed, but not explained, in [6], it is the linear independence of the Arnoldi basis, not the orthogonality near machine precision, that is important. Until the linear independence of the b..

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..