BOUNDS OF EIGENVALUES OF PRECONDITIONED MATRICES

Some methods to bound individual eigenvalues of a generalized eigenvalue problem lambda-Cx = Ax are presented, both for general positive semidefinite matrices and for the special case where C is an incomplete factorization of A. This provides accurate estimates of the rate of convergence of preconditioned conjugate gradient methods to solve linear systems with A. In particular, methods are presented to actually numerically compute bounds of the extreme eigenvalues. The estimates enable us to compare modified and unmodified incomplete factorization methods.