By combining element-by-element estimates for the field of values of a preconditioned matrix with GMRES-convergence estimates it is possible to derive an easily computable upper bound on the GMRES-residual norm. This method can be applied to general finite element systems, but the preconditioner has to be Hermitian and positive definite. The resulting upper bound for the GMRES-residual norm can...

The ILU factorization is one of the most popular preconditioners for the Krylov subspace method, alongside the GMRES. Properties of the preconditioner derived from the ILU factorization are relayed onto the dropping rules. Recently, Zhang et al. [Numer. Linear. Algebra. Appl., Vol. 19, pp. 555–569, 2011] proposed a Flexible incomplete Cholesky (IC) factorization for symmetric linear systems. Th...

SUMMARY Recently the GMRESR inner-outer iteration scheme for the solution of linear systems of equations has been proposed by Van der Vorst and Vuik. Similar methods have been proposed by Axelsson and Vassilevski 1] and Saad (FGMRES) 10]. The outer iteration is GCR, which minimizes the residual over a given set of direction vectors. The inner iteration is GMRES, which at each step computes a ne...

Abstract. We analyze the residuals of GMRES [9], when the method is applied to tridiagonal Toeplitz matrices. We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks. This problem has been studied previously by Ipsen [5], Eiermann and Ernst [2], but we formulate and prove our results in a different way. We then extend the (lower) bidiagona...

We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and non-Hermitian. Some Krylov iterative methods s...