Locally X matrices, spectral distributions, preconditioning, and applications

Starting from a set X = {X-n}(n) of Hermitian positive definite n x n matrices, we constructively define a class LL(X) of "locally X" matrices which can be viewed as the range of a special sequence of linear normally positive operators. Regarding the spectra of these matrices and of the related preconditioned matrices, we prove some Szego-style ergodic formulas. These results allow one to de ne a very general procedure for devising optimal and superlinear preconditioners. As special cases, we can deal with matrices coming from the discretization of elliptic and semielliptic differential equations defined on multidimensional domains as well as matrices coming from optimization problems connected with graph theory.