Bounds for the decay of the entries in inverses and Cauchy–Stieltjes functions of certain sparse, normal matrices

Abstract

Summary It is known that in many functions of banded and, more generally, sparse Hermitian positive definite matrices, the entries exhibit a rapid decay away from the sparsity pattern. This is particularly true for the inverse, and based on results for the inverse, bounds for Cauchy–Stieltjes functions of Hermitian positive definite matrices have recently been obtained. We add to the known results by considering certain types of normal matrices, for which fewer and typically less satisfactory results exist so far. Starting from a very general estimate based on approximation properties of Chebyshev polynomials on ellipses, we obtain as special cases insightful decay bounds for various classes of normal matrices, including (shifted) skew‐Hermitian and Hermitian indefinite matrices. In addition, some of our results improve over known bounds when applied to the Hermitian positive definite case.

Non-Toeplitz decay bounds for inverses of Hermitian positive definite tridiagonal matrices

“…where "≥" and "| • |" are understood componentwise. Similarly, all the results on decay bounds for banded matrices from [2,5,6,14,15,24,25,26] also result in a Toeplitz-structured matrix Q of bounds. See also [12] for Toeplitz type bounds for block-tridiagonal matrices.…”

“…Due to its importance in many applications, there are in particular many publications dealing with the matrix exponential [4,6,18,22] or general analytic functions [3,27]. Recently, also other classes of functions like, e.g., Stieltjes functions, have been studied with respect to their decay behavior; see, e.g., [2,5,6,15].…”

“…To use the same idea, we cannot use the bounds of Theorem 1.1 but have to resort to a slightly different approach: Instead of using a polynomial approximation for the inverse directly, we use shifted and normalized Chebyshev polynomials similarly to what was done in [15] for general line segments in the complex plane. In the case of a Hermitian positive definite matrix, this gives the same decay rate as that of Theorem 1.1 with a new constant C which fulfills C ≤ 2c, i.e., the new constant can be slightly worse than the constant c in (1.2).…”

Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices

“…In all these last articles, and also in our approach, bounds on the decay pattern of banded non-Hermitian matrices are derived that avoid the explicit reference to the possibly large condition number of the eigenvector matrix. Specialized off-diagonal decay results have been obtained for certain normal matrices, see, e.g., [20,11,23], and for analytic functions of banded matrices over C * -algebras [3].…”

Editorial: Novel methods and theories in numerical algebra with interdisciplinary applications

“…Frommer et al look at bounds for the entries of functions of matrices. For the matrix inverse, and more generally for Cauchy–Stieltjes functions, and for certain normal matrices A including shifted Hermitian and skew‐Hermitian matrices, they develop quite sharp bounds for these entries.…”