In the symmetric case (where and the preconditioner are both symmetric) for the Chebyshev Iteration we have the same upper bound as for the Conjugate Gradient method, provided and are computed from and (the extremal eigenvalues of the preconditioned matrix ).

There is a severe penalty for overestimating or underestimating the field of values. For example, if in the symmetric case is underestimated, then the method may diverge; if it is overestimated then the result may be very slow convergence. Similar statements can be made for the nonsymmetric case. This implies that one needs fairly accurate bounds on the spectrum of for the method to be effective (in comparison with CG or GMRES).

Mon Nov 20 08:52:54 EST 1995