Bounds of Eigenvalues of Preconditioned Matrices
Owe Axelsson · SIAM Journal on Matrix Analysis and Applications · 1992
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.