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).