Year: 2012
Numerical Mathematics: Theory, Methods and Applications, Vol. 5 (2012), Iss. 3 : pp. 447–492
Abstract
With the aid of index functions, we re-derive the ML($n$)BiCGStab algorithm in [Yeung and Chan, SIAM J. Sci. Comput., 21 (1999), pp. 1263-1290] systematically. There are $n$ ways to define the ML($n$)BiCGStab residual vector. Each definition leads 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 is a bridge connecting BiCG and FOM. We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML($n$)BiCGStab. Implementation issues are also addressed.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2012.m1035
Numerical Mathematics: Theory, Methods and Applications, Vol. 5 (2012), Iss. 3 : pp. 447–492
Published online: 2012-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 46
Keywords: CGS BiCGStab ML($n$)BiCGStab multiple starting Lanczos Krylov subspace iterative methods linear systems.
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