A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix

Daitch, Samuel I.; Spielman, Daniel A.

doi:10.4230/DagSemProc.09061.3

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A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix

Authors Samuel I. Daitch, Daniel A. Spielman

Part of: Volume: Dagstuhl Seminar Proceedings, Volume 9061
Part of: Series: Dagstuhl Seminar Proceedings (DagSemProc)
License: Creative Commons Attribution 4.0 International license
Publication Date: 2009-07-24

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Document Identifiers

DOI: 10.4230/DagSemProc.09061.3
URN: urn:nbn:de:0030-drops-20803

Subject Classification

Keywords

M-matrix
diagonally-dominant matrix
linear system solver
iterative algorithm
randomized algorithm
network flow
gain graph

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Abstract

We present an algorithm for solving a linear system in a symmetric M-matrix.
In particular, for $n times n$ symmetric M-matrix $M$, we show how to find a diagonal matrix $D$ such that
$DMD$ is diagonally-dominant. To compute $D$, the algorithm must solve $O{log n}$ linear systems in diagonally-dominant matrices. If we solve these diagonally-dominant systems approximately using the Spielman-Teng
nearly-linear time solver, then we obtain an algorithm for approximately solving linear systems in symmetric M-matrices, for which the expected running time is also nearly-linear.

Cite As Get BibTex

Samuel I. Daitch and Daniel A. Spielman. A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix. In Combinatorial Scientific Computing. Dagstuhl Seminar Proceedings, Volume 9061, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009) https://doi.org/10.4230/DagSemProc.09061.3

Author Details

Samuel I. Daitch

Daniel A. Spielman

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