Decay patterns of matrix inverses have recently attracted considerable interest, due to their relevance in numerical analysis, and in applications requiring matrix function approximations. In this paper we analyze the decay pattern of the inverse of banded matrices in the form $S = M \oplus I_n + I_n \oplus M$, where M is tridiagonal, symmetric and positive definite, In is the identity matrix, and stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal. However, the entries in $S^{-1}$ show a non- monotonic decay, which is not caught by classical bounds. By using an alternative expression for $S^{-1}$, we derive computable upper bounds that closely capture the actual behavior of its entries. We also show that similar estimates can be obtained when M has a larger bandwidth, or when the sum of Kronecker products involves two different matrices. Numerical experiments illustrating the new bounds are also reported.

On the decay of the inverse of matrices that are sum of Kronecker products / Canuto, Claudio; V., Simoncini; M., Verani. - In: LINEAR ALGEBRA AND ITS APPLICATIONS. - ISSN 0024-3795. - 452:(2014), pp. 21-39. [10.1016/j.laa.2014.03.029]

On the decay of the inverse of matrices that are sum of Kronecker products

CANUTO, CLAUDIO;
2014

Abstract

Decay patterns of matrix inverses have recently attracted considerable interest, due to their relevance in numerical analysis, and in applications requiring matrix function approximations. In this paper we analyze the decay pattern of the inverse of banded matrices in the form $S = M \oplus I_n + I_n \oplus M$, where M is tridiagonal, symmetric and positive definite, In is the identity matrix, and stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal. However, the entries in $S^{-1}$ show a non- monotonic decay, which is not caught by classical bounds. By using an alternative expression for $S^{-1}$, we derive computable upper bounds that closely capture the actual behavior of its entries. We also show that similar estimates can be obtained when M has a larger bandwidth, or when the sum of Kronecker products involves two different matrices. Numerical experiments illustrating the new bounds are also reported.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2582343
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