Decay rates for inverses of band matrices

Demko, Stephen; Moss, William F.; Smith, Philip W.

doi:10.1090/S0025-5718-1984-0758197-9

Decay rates for inverses of band matrices
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by Stephen Demko, William F. Moss and Philip W. Smith PDF

Math. Comp. 43 (1984), 491-499 Request permission

Abstract:

Spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices. The rate of decay of ${A^{ - 1}}$ can be bounded in terms of the (essential) spectrum of $A{A^\ast }$ for general A and in terms of the (essential) spectrum of A for positive definite A. In the positive definite case the bound can be attained. These results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices. We also establish decay rates for certain generalized inverses.

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