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

Authors: Samuel I. Daitch and Daniel A. Spielman

Published in: Dagstuhl Seminar Proceedings, Volume 9061, Combinatorial Scientific Computing (2009)


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

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)


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@InProceedings{daitch_et_al:DagSemProc.09061.3,
  author =	{Daitch, Samuel I. and Spielman, Daniel A.},
  title =	{{A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix}},
  booktitle =	{Combinatorial Scientific Computing},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9061},
  editor =	{Uwe Naumann and Olaf Schenk and Horst D. Simon and Sivan Toledo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09061.3},
  URN =		{urn:nbn:de:0030-drops-20803},
  doi =		{10.4230/DagSemProc.09061.3},
  annote =	{Keywords: M-matrix, diagonally-dominant matrix, linear system solver, iterative algorithm, randomized algorithm, network flow, gain graph}
}
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