Stability of the notion of approximating class of sequences and applications
Journal of Computational and Applied Mathematics
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Starting from a set X= {Xn}n of Hermitian positive definite $n\times n$ matrices, we constructively define a class ${\cal LL}(X)$ 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 define 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 semi-elliptic differential equations defined on multidimensional domains as well as matrices coming from optimization problems connected with graph theory.