A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive unper and lower estimates of individual eigenvalues and of __ condition number. This includes a discussion that the condition number of preconditioned second matrices is O(h-‘). Some of the methods are applied to compute certain parameters involved in preconditioner.