Michele Benzi,
Paraskevi Fika,
Marilena Mitrouli
S. 439 - 458
doi:
10.1553/etna_vol53s439
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol53s439
Abstract:
We consider the computation of generalized inverses of the graph Laplacian for both undirected and directed graphs, with a focus on the group inverse and the closely related absorption inverse. These generalized inverses encode valuable information about the underlying graph as well as the regular Markov process generated by the graph Laplacian. In [Benzi et al., Linear Algebra Appl., 574 (2019), pp. 123–152], both direct and iterative numerical methods have been developed for the efficient computation of the absorption inverse and related quantities. In this work, we present two direct algorithms for the computation of the group inverse and the absorption inverse. The first is based on the Gauss-Jordan elimination and the reduced row echelon form of the Laplacian matrix and the second on the bottleneck matrix, the inverse of a submatrix of the Laplacian matrix. These algorithms can be faster than the direct algorithms proposed in [Benzi et al., Linear Algebra Appl., 574 (2019), pp. 123–152].
Laplacian matrix, group inverse, absorption inverse, Gauss-Jordan method
Published Online:
2020/06/09 14:37:45
Object Identifier:
0xc1aa5576 0x003b932d
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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