Marcel Schweitzer
S. 438 - 454
doi:
10.1553/etna_vol55s438
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol55s438
Abstract:
For many functions of matrices $f(A)$, it is known that their entries exhibit a rapid—often exponential or even superexponential—decay away from the sparsity pattern of the matrix $A$. In this paper, we specifically focus on the class of Bernstein functions, which contains the fractional powers $A^\alpha$, $\alpha \in (0,1)$, as an important special case, and derive new decay bounds by exploiting known results for the matrix exponential in conjunction with the Lévy-Khintchine integral representation. As a particular special case, we find a result concerning the power law decay of the strength of connection in nonlocal network dynamics described by the fractional graph Laplacian, which improves upon known results from the literature by doubling the exponent in the power law.
matrix functions, Bernstein functions, off-diagonal decay, graph Laplacian, fractional powers, nonlocal dynamics
Published Online:
2022/05/09 11:53:23
Object Identifier:
0xc1aa5576 0x003d6a9f
Rights:All rights reserved.For questions regarding copyright and copies please contact us by email.
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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