Few theoretical results are known about the convergence of BiCG. For symmetric positive definite systems the method delivers the same results as CG, but at twice the cost per iteration. For nonsymmetric matrices it has been shown that in phases of the process where there is significant reduction of the norm of the residual, the method is more or less comparable to full GMRES (in terms of numbers of iterations) (see Freund and Nachtigal [102]). In practice this is often confirmed, but it is also observed that the convergence behavior may be quite irregular , and the method may even break down . The breakdown situation due to the possible event that can be circumvented by so-called look-ahead strategies (see Parlett, Taylor and Liu [172]). This leads to complicated codes and is beyond the scope of this book. The other breakdown situation, , occurs when the -decomposition fails (see the theory subsection of §), and can be repaired by using another decomposition. This is done in the version of QMR that we adopted (see §).

Sometimes, breakdown or near-breakdown situations can be satisfactorily avoided by a restart at the iteration step immediately before the (near-) breakdown step. Another possibility is to switch to a more robust (but possibly more expensive) method, like GMRES.

Mon Nov 20 08:52:54 EST 1995