Lanczos Methods for the Solution of Nonsymmetric Systems of Linear Equations

The Lanczos or biconjugate gradient method is often an effective means for solving nonsymmetric systems of linear equations. However, the method sometimes experiences breakdown, a near division by zero which may hinder or preclude convergence. In this paper we present some theoretical results on the nature and likelihood of the phenomenon of breakdown. We also define several new algorithms that substantially mitigate the problem of breakdown. Numerical comparisons of the new algorithms and the standard algorithms are given.