Variations of Zhang's Lanczos-Type Product Method

Abstract

Among the Lanczos-type product methods, which are characterized by residual polynomials $p_n t_n$ that are the product of the Lanczos polynomial $p_n$ and another polynomial $t_n$ of exact degree $n$ with $t_n(0) = 1$, Zhang's algorithm \GPBICG\ has the feature that the polynomials $t_n$ are implicitly built up by a pair of coupled two-term recurrences whose coefficients are chosen so that the new residual is minimized in a 2-dimensional space. There are several ways to achieve this. We discuss here alternative algorithms that are mathematically equivalent (that is, produce in exact arithmetic the same results). The goal is to find one where the ultimate accuracy of the iterates $x_n$ is guaranteed to be high and the cost is at most slightly increased.