Starting from the residual estimates for GMRES formulated by many authors, usually in terms of the quotient of the Hermitian part and the norm of a matrix or by using the field of values of a matrix, we present more general estimates which hold also for restarted and augmented GMRES. Sufficient conditions for convergence are formulated. All estimates are independent on the choice of an initial ...

In this paper we analyze the numerical behavior of several minimum residual methods, which are mathematically equivalent to the GMRES method. Two main approaches are compared: the one that computes the approximate solution (similar to GMRES) in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and the one where the approximate solutions are updated with a...

In this report we describe the implementations of the GMRES algorithm for both real and complex, single and double precision arithmetics suitable for serial, shared memory and distributed memory computers. For the sake of simplicity, flexibility and efficiency the GMRES solvers have been implemented using the reverse communication mechanism for the matrixvector product, the preconditioning and ...

Restarted GMRES methods augmented with approximate eigenvectors are widely used for solving large sparse linear systems. Recently a new scheme of augmenting with error approximations is proposed. The main aim of this paper is to develop a restarted GMRES method augmented with the combination of harmonic Ritz vectors and error approximations. We demonstrate that the resulted combination method c...

In the present work, we propose a new projection method for solving the matrix equation AXB = F . For implementing our new method, generalized forms of block Krylov subspace and global Arnoldi process are presented. The new method can be considered as an extended form of the well-known global generalized minimum residual (Gl-GMRES) method for solving multiple linear systems and it will be calle...

GMRES and CGS are well-known iterative methods for the solution of certain sparse linear systems with a non-symmetric matrix. These methods have been compared experimentally in many studies and speciic observations on their convergence behaviour have been reported. A new iterative method to solve a non-symmetric system is proposed by Eirola and Nevanlinna. The purpose of this paper is to invest...

In this paper, a two-stage iterative algorithm is proposed to improve the convergence of Krylov based iterative methods, typically those of GMRES variants. The principle of the proposed approach is to build an external iteration over the Krylov method, and to frequently store its current residual (at each GMRES restart for instance). After a given number of outer iterations, a least-squares min...