Solving Eigenproblems: From Arnoldi via Jacobi-Davidson to the Riccati Method

Abstract

The formulation of eigenproblems as generalized algebraic Riccati equations removes the non-uniqueness problem of eigenvectors. This basic idea gave birth to the Jacobi-Davidson (JD) method of Sleijpen and Van der Vorst (1996). JD converges quadratically when the current iterate is close enough to the solution that one targets for. Unfortunately, it may take quite some effort to get close enough to this solution. In this paper we present a remedy for this. Instead of linearizing the Riccati equation (which is done in JD) and replacing the linearization by a low-dimensional linear system, we propose to replace the Riccati equation by a low-dimensional Riccati equation and to solve it exactly. The performance of the resulting Riccati algorithm compares extremely favorable to JD while the extra costs per iteration compared to JD are in fact negligible.