GMRES and the Arioli, Pták, and Strakoš parametrization

Abstract

In this paper we study the parametrization proposed by Arioli, Ptak and Strakos (BIT Numerical Mathematics, v 38, 1998) for the class of matrices having the same GMRES residual norm convergence curve. We give expressions for the Hessenberg matrix and the orthonormal basis vectors constructed by GMRES as well as for iterates and error vectors. The iterates do not depend on the eigenvalues in the sense that changing the coefficients of the characteristic polynomial in the parametrization does not change the GMRES iterates as well as the residuals. However, the error vectors do depend on these coefficients.