How residual bounds for restarted GMRES describe the real behaviour

Abstract

Let us consider the non-Hermitian and non-singular system Ax = b, A ∈ C n×n , x, b ∈ C n . The following notational conventions will be observed throughout this paper; The notation x 0 will be used for the initial approximation, and r 0 = b − Ax 0 = 0 will denote the corresponding residual vector. For the matrix A and a vector u ∈ C n , we define the KrylovThe letter v will be used for the quotient v = r 0 / r 0 , S n is the unit sphere in C n , · is the Euclidean norm, P m is the set of all polynomials of degree m, P 0 m is the set of all polynomials from P m which evaluate to zero at zero. The restarted GMRES algorithm (GMRES(m)), proposed by Y. Saad and M. H. Schultz [1] and described for various situations and modifications by many authors, is the most popular Krylov method for the solution of Ax = b.We start with the estimates for r m in the first restart. The GMRES algorithm constructs the new approximation x m in the affine space (A) . In the paper [2], three bounds for the p(A), and hence for r m , are derived and their quality discussed. This paper provides the motivation for the following discussion. Elman (see [3] v. Calculation of r m yields for every q ∈ P 0 m the inequalitieswhere H q and iS q denote the Hermitian and skew-Hermitian part of the matrix q(A) respectively and i 2 = −1.is the orthogonal projector for the space span{AK m (r 0 )} and therefore.• If we substitute the estimate σ 2Another upper bounds for the quotient r m 2 / r 0 2 can be found in the papers of Strakoš, Rozložník and Liesen.To obtain non-stagnation conditions for restarted GMRES, we will concentrate on estimates of the form r m 2 / r 0 2 ≤ 1 − ρ i , wherefor α ∈ C m and 1 − ρ 3 = 1 − min The restarted version GMRES(m)Let