The convergence behavior of QMR is typically much smoother than for BiCG. Freund and Nachtigal [102] present quite general error bounds which show that QMR may be expected to converge about as fast as GMRES. From a relation between the residuals in BiCG and QMR (Freund and Nachtigal [relation (5.10)]FrNa:qmr) one may deduce that at phases in the iteration process where BiCG makes significant progress, QMR has arrived at about the same approximation for . On the other hand, when BiCG makes no progress at all , QMR may still show slow convergence.

The look-ahead steps in the version of the QMR method discussed in [102] prevents breakdown in all cases but the so-called ``incurable breakdown'', where no practical number of look-ahead steps would yield a next iterate.

Mon Nov 20 08:52:54 EST 1995