Krylov sequences of maximal length and convergence of GMRES

Abstract

In most practical cases, the convergence of GMRES method applied to a linear algebraic system $Ax = b$ is determined by the distribution of eigenvalues of $A$. In theory, however, the information about the eigenvalues alone is not sufficient for determining the convergence. In this paper our previous work is extended in the following direction. It is given a complete parametrization of the set of all pairs $\{ A,b \}$ for which GMRES$(A,b)$ generates the prescribed convergence curve while the matrix $A$ has the prescribed eigenvalues. Moreover, a characterization of the right hand sides $b$ for which the GMRES$(A,b)$ converges exactly in $m$ steps, where $m$ is the degree of the minimal polynomial of $A$, is given. EMAIL:: ian@microian.ian.pv.cnr.it KEYWORDS:: Minimal polynomial, Krylov sequences, GMRES method, Convergence