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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 possiblities 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 ); more general approaches are suggested by Sleijpen and Fokkema in .