When solving a linear algebraic system Ax = b with GMRES, the relative residual norm at each step is bounded from above by the so-called ideal GMRES approximation. This worstcase bound is sharp (i.e. it is attainable by the relative GMRES residual norm) in case of a normal matrix A, but it need not characterize the worst-case GMRES behavior if A is nonnormal. Characterizing the tightness of thi...

This note describes the usage of the GMRES solver using reverse communication protocol. The GMRES control flow is outlined, and an example calling sequence explained.

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869] is an iterative method for approximately solving linear systems , with initial guess residual . employs the Arnoldi process to generate Krylov basis vectors (the columns ). It well known that this can be viewed as a factorization matrix at each iteration. Despite loss orthogonality, unit roundoff conditi...

In the past few years new methods have been proposed that can be seen as combinations of standard Krylov subspave methods, such as Bi{ CG and GMRES. One of the rst hybrid schemes of this type is CGS, actually the Bi{CG squared method. Other such hybrid schemes include BiCGSTAB (a combination of Bi{CG and GMRES(1)), QMRS, TFQMR, Hybrid GMRES (polynomial preconditioned GMRES) and the nested GMRES...

Restarting GMRES, a linear solver frequently used in numerical schemes, is known to suffer from stagnation. In this paper, a simple strategy is proposed to detect and avoid stagnation, without modifying the standard GMRES code. Numerical tests with the modified GMRES(m), GMRESH(m) procedure, alone and as part of an inexact Newton procedure with several choices for the forcing term, demonstrate ...

In the convergence analysis of the GMRES method for a given matrix A, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step k, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for A and k. We show that the worst case behavior of GMRES for the matrices A and A is the same, and we analyze proper...

The most popular iterative linear solvers in Computational Fluid Dynamics (CFD) calculations are restarted GMRES and BiCGStab. At the beginning of most incompressible flow calculations, the computation time and the number of iterations to converge for the pressure Poisson equation are quite high, since the initial guess is far from the solution. In this case, the BiCGStab algorithm, with relati...

We consider the solution of large linear systems with multiple right-hand sides using a block GMRES approach. We introduce a new algorithm that effectively handles the situation of almost rank deficient block generated by the block Arnoldi procedure and that enables the recycling of spectral information at restart. The first feature is inherited from an algorithm introduced by Robbé and Sadkane...

The Generalized Minimum Residual (GMRES) method is a popular Krylov subspace projection method for solving a nonsymmetric linear system of equations. On modern computers, communication is becoming increasingly expensive compared to arithmetic operations, and a communication-avoiding variant (CA-GMRES) may improve the performance of GMRES. To further enhance the performance of CAGMRES, in this p...