ML(n)BiCGStab: Reformulation, Analysis and Implementation

Abstract

With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in a paper by Yeung and Chan in 1999 in a more systematic way. It turns out that there are n ways to define the ML(n)BiCGStab residual vector. Each definition will lead to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG a bridge connecting BiCG and FOM. We also analyze the breakdown situations from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed.

arXiv commentThis paper is dedicated to the memory of Prof. Gene Golub. Most part of the paper was presented in Gene Golub Memorial Conference, Feb. 29-Mar. 1, 2008, University of Massachusetts, Dartmouth, U.S.A