SYMMLQ-like procedure for Ax = b where A is a special normal matrix

Abstract

We propose a method of Lanczos type for solving a linear system with a normal matrix whose spectrum is contained in a second-degree curve. This is a broader class of matrices than that of the (l,m)-normal matrices introduced in a recent paper by Barth and Manteuffel. Our approach is similar to that of Huhtanen in the sense that both use the condensed form of normal matrices discovered by Elsner and Ikramov. However, there are a number of differences, among which are: (i) our method is modeled after the SYMMLQ algorithm of Paige and Saunders; (ii) it uses only one matrix-vector product per step; (iii) we provide effective means for monitoring the size of the residual during the process. Numerical experiments are presented. Keywords: generalized Krylov sequence, Rayleigh-Ritz projection, SYMMLQ algorithm, GMRES algorithm, normal matrix