A Method for Avoiding Breakdown in Product-Type Iterative Methods and its Behavior for Toeplitz Linear Systems

In this paper, we propose product-type iterative methods with restart for avoiding breakdown. The residual vectors of product-type iterative methods are denoted as the product of a polynomial and the residual vector of BiCG. In particular, the residual polynomial of BiCG is called the Lanczos polynomial. In product-type iterative methods, breakdown may occur in the recurrence relations of the Lanczos polynomial due to division by zero. We investigated the behavior of the proposed method through experiments using Toeplitz linear systems with various parameters. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)