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 -

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.