Solving linear equations with a stabilized GPBiCG method

Abstract

Any residual polynomial of hybrid Bi-Conjugate Gradient (Bi-CG) methods, as Bi-CG STABilized (Bi-CGSTAB), BiCGstab(ℓ), Generalized Product-type Bi-CG (GPBiCG), and BiCG×MR2, can be expressed as the product of a Lanczos polynomial and a so-called stabilizing polynomial. The stabilizing polynomials of GPBiCG have originally been built by coupled two-term recurrences, but, as in BiCG×MR2, they can also be constructed by a three-term recurrence similar to the one for the Lanczos polynomials. In this paper, we propose to use this three-term recurrence and to combine it with a slightly modified version of the coupled two-term recurrences for Bi-CG. The modifications appear to lead to more accurate Bi-CG coefficients. We consider two combinations. The recurrences of the resulting two algorithms are different from those of the original GPBiCG, BiCG×MR2, and other variants in literature. Specifically in cases where the convergence has a long stagnation phase, the convergence seems to rely on the underlying Bi-CG process. We therefore also propose a “stabilization” strategy that allows the Bi-CG coefficients in our variants to be more accurately computed. Numerical experiments show that our two new variants are less affected by rounding errors, and a GPBiCG method with the stabilization strategy is more effective.