Among the Lanczos-type product methods, which are characterized by residual polynomials pntn that are the product of the Lanczos polynomial pn and another polynomial tn of exact degree n with tn(0) = 1, Zhang's algorithm GPBiCG has the feature that the polynomials tn 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 xn is guaranteed to be high and the cost is at most slightly increased. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.
Röllin, S., & Gutknecht, M. H. (2002). Variations of Zhang’s Lanczos-type product method. In Applied Numerical Mathematics (Vol. 41, pp. 119–133). https://doi.org/10.1016/S0168-9274(01)00114-3