TY - JOUR
T1 - ML($n$)BiCGStab: Reformulation, Analysis and Implementation
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 - <p style="text-align: justify;">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.</p>