Spectral behavior of matrix sequences and discretized boundary value problems

In this paper we provide theoretical tools for dealing with the spectral pr operties of general sequences of matrices of increasing dimension. More spe cifically, we give a unified treatment of notions such as distribution, equ al distribution, localization, equal localization, clustering and sub-clust ering. As a case study we consider the matrix sequences arising from the fi nite difference (FD) discretization of elliptic and semielliptic boundary v alue problems (BVPs). The spectral analysis is then extended to Toeplitz-ba sed preconditioned matrix sequences with special attention to the case wher e the coefficients of the differential operator are not regular (belong to L-1) and to the case of multidimensional problems. The related clustering p roperties allow the establishment of some ergodic formulas for the eigenval ues of the preconditioned matrices. (C) 2001 Published by Elsevier Science Inc.