We investigate the convergence of GMRES for an n by n Jordan block J . For each k that divides n we derive the exact form of the kth ideal GMRES polynomial and prove the equality max ‖v‖=1 min p∈πk ‖p(J)v‖ = min p∈πk max ‖v‖=1 ‖p(J)v‖, where πk denotes the set of polynomials of degree at most k and with value one at the origin, and ‖ · ‖ denotes the Euclidean norm. In other words, we show that ...

The convergence of the restarted GMRES method can be significantly improved, for some problems, by using a weighted inner product that changes at each restart. How does this weighting affect convergence, and when is it useful? We show that weighted inner products can help in two distinct ways: when the coefficient matrix has localized eigenvectors, weighting can allow restarted GMRES to focus o...

We consider the solution of a linear system of equations using the GMRES iterative method. In [3], a strategy to relax the accuracy of the matrix-vector product is proposed for general systems and illustrated on a large set of numerical experiments. This work is based on some heuristic considerations and proposes a strategy that often enables a convergence of the GMRES iterates xk within a rela...

Work on generalizing the deflated, restarted GMRES algorithm, useful in lattice studies using stochastic noise methods, is reported. We first show how the multi-mass extension of deflated GMRES can be implemented. We then give a deflated GMRES method that can be used on multiple right-hand sides of Ax = b in an efficient manner. We also discuss and give numerical results on the possibilty of co...

Through the research of the parallel computational model based on the principal and subordinate mode and the basic theory of Gmres Algorithm in Krylov subspace, this essay raises a improvement parallel Predict-Correct Gmres(m) algorithm which posses Predict-Correct pattern, and shows the computing examples for linear equations. After the comparison with the result from the new parallel Predict-...

We study the convergence of GMRES/FOM and QMR/BiCG methods for solving nonsymmetric Ax = b. We prove that given the results of a BiCG computation on Ax = b, we can obtain a matrix B with the same eigenvalues as A and a vector c such that the residual norms generated by a FOM computation on Bx = c are identical to those generated by the BiCG computations. Using a unitary equivalence for each of ...

The GMRES method is a popular iterative method for the solution of linear systems of equations with a large nonsymmetric nonsingular matrix. However, little is known about the performance of the GMRES method when the matrix of the linear system is of ill-determined rank, i.e., when the matrix has many singular values of different orders of magnitude close to the origin. Linear systems with such...

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, exibility and eeciency the GMRES solvers have been implemented using the reverse communication mechanism for the matrix-vector product, the preconditioning and the...