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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



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Samuel I. Daitch
Daniel A. Spielman

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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

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

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