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Volume 5, Issue 3
ML($n$)BiCGStab: Reformulation, Analysis and Implementation

Man-Chung Yeung

DOI: 10.4208/nmtma.2012.m1035

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 447-492.

Published online: 2012-05

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  • 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.

  • Keywords

CGS, BiCGStab, ML($n$)BiCGStab, multiple starting Lanczos, Krylov subspace, iterative methods, linear systems.

  • AMS Subject Headings

65F10, 65F15, 65F25, 65F30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-447, author = {Man-Chung Yeung}, title = {ML($n$)BiCGStab: Reformulation, Analysis and Implementation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {3}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1035}, url = {http://global-sci.org/intro/article_detail/nmtma/5946.html} }
TY - JOUR T1 - ML($n$)BiCGStab: Reformulation, Analysis and Implementation AU - Man-Chung Yeung JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 447 EP - 492 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1035 UR - https://global-sci.org/intro/article_detail/nmtma/5946.html KW - CGS, BiCGStab, ML($n$)BiCGStab, multiple starting Lanczos, Krylov subspace, iterative methods, linear systems. AB -

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.

Man-Chung Yeung. (2012). ML($n$)BiCGStab: Reformulation, Analysis and Implementation. Numerical Mathematics: Theory, Methods and Applications. 5 (3). 447-492. doi:10.4208/nmtma.2012.m1035
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