Bi-CGSTAB often converges about as fast as CGS,
sometimes faster and
sometimes not. CGS can be viewed as a method in which the BiCG
``contraction'' operator is applied twice. Bi-CGSTAB can be
interpreted as the product of BiCG and repeatedly applied GMRES(1). At
least locally, a residual vector is minimized , which leads to a
considerably smoother convergence behavior. On the
other hand, if the
local GMRES(1) step stagnates, then the Krylov subspace
is not
expanded, and Bi-CGSTAB will break down . This is a breakdown situation
that can occur in addition to the other breakdown possibilities in the
underlying BiCG algorithm. This type of breakdown may be avoided by
combining BiCG with other methods, *i.e.*, by selecting other
values for (see the algorithm). One such alternative is
Bi-CGSTAB2 (see Gutknecht [115]); more general
approaches are suggested by Sleijpen and Fokkema in [190].

Note that Bi-CGSTAB has two stopping tests: if the method has already converged at the first test on the norm of , the subsequent update would be numerically questionable. Additionally, stopping on the first test saves a few unnecessary operations, but this is of minor importance.

Mon Nov 20 08:52:54 EST 1995