Bounds for the decay of the entries in inverses and Cauchy–Stieltjes functions of certain sparse, normal matrices. Issue 4 (24th November 2017)

• Record Type: Journal Article
• Authors: Frommer, Andreas; Schimmel, Claudia; Schweitzer, Marcel
• Other Names: Bai Zhong‐Zhi guestEditor.; Neytcheva Maya G. guestEditor.; Reichel Lothar guestEditor.
• Is Part Of: Numerical linear algebra with applications. Volume 25:Issue 4(2018)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2017-11-24
• Subjects: banded matrices -- Chebyshev polynomials -- matrix function -- normal matrices -- off‐diagonal decay; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2131
• Languages: English
• ISSNs: 1070-5325

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.