Abstract
The Conjugate Gradient Squared (CGS) method has been proposed for solving linear systems with non-Hermitian matrix. This method is known to be unstable because the residual often increases rapidly due to near-breakdown. In this paper, we analyze two causes of near-breakdown in the CGS method and derive a condition of detecting it. Moreover, we propose a method for avoiding near-breakdown in the CGS method by restarting Lanczos process. Effects of the CGS method with restart have been illustrated by some numerical experiments.
