research-article

Partial factorization of a dense symmetric indefinite matrix

Authors:

Jennifer A. ScottAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 38, Issue 2

Article No.: 10, Pages 1 - 19

https://doi.org/10.1145/2049673.2049674

Published: 05 January 2012 Publication History

Abstract

At the heart of a frontal or multifrontal solver for the solution of sparse symmetric sets of linear equations, there is the need to partially factorize dense matrices (the frontal matrices) and to be able to use their factorizations in subsequent forward and backward substitutions. For a large problem, packing (holding only the lower or upper triangular part) is important to save memory. It has long been recognized that blocking is the key to efficiency and this has become particularly relevant on modern hardware. For stability in the indefinite case, the use of interchanges and 2 × 2 pivots as well as 1 × 1 pivots is equally well established. In this article, the challenge of using these three ideas (packing, blocking, and pivoting) together is addressed to achieve stable factorizations of large real-world symmetric indefinite problems with good execution speed.

The ideas are not restricted to frontal and multifrontal solvers and are applicable whenever partial or complete factorizations of dense symmetric indefinite matrices are needed.

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Cited By

Grubišić LTambača J(2019)Direct solution method for the equilibrium problem for elastic stentsNumerical Linear Algebra with Applications10.1002/nla.223126:3Online publication date: 21-Jan-2019
https://doi.org/10.1002/nla.2231
Scott JTůma M(2017)Improving the stability and robustness of incomplete symmetric indefinite factorization preconditionersNumerical Linear Algebra with Applications10.1002/nla.209924:5Online publication date: 30-Mar-2017
https://doi.org/10.1002/nla.2099
Hogg JOvtchinnikov EScott J(2016)A Sparse Symmetric Indefinite Direct Solver for GPU ArchitecturesACM Transactions on Mathematical Software10.1145/275654842:1(1-25)Online publication date: 29-Jan-2016
https://dl.acm.org/doi/10.1145/2756548
Show More Cited By

Index Terms

Partial factorization of a dense symmetric indefinite matrix
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Numerical differentiation
  2. Mathematical software
2. Theory of computation
  1. Design and analysis of algorithms

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 38, Issue 2

December 2011

136 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2049673

Issue’s Table of Contents

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 05 January 2012

Accepted: 01 May 2011

Revised: 01 April 2011

Received: 01 November 2010

Published in TOMS Volume 38, Issue 2

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Cited By

Grubišić LTambača J(2019)Direct solution method for the equilibrium problem for elastic stentsNumerical Linear Algebra with Applications10.1002/nla.223126:3Online publication date: 21-Jan-2019
https://doi.org/10.1002/nla.2231
Scott JTůma M(2017)Improving the stability and robustness of incomplete symmetric indefinite factorization preconditionersNumerical Linear Algebra with Applications10.1002/nla.209924:5Online publication date: 30-Mar-2017
https://doi.org/10.1002/nla.2099
Hogg JOvtchinnikov EScott J(2016)A Sparse Symmetric Indefinite Direct Solver for GPU ArchitecturesACM Transactions on Mathematical Software10.1145/275654842:1(1-25)Online publication date: 29-Jan-2016
https://dl.acm.org/doi/10.1145/2756548
Texter J(2015)A Kinetic Model for Exfoliation Kinetics of Layered MaterialsAngewandte Chemie International Edition10.1002/anie.20150469354:35(10258-10262)Online publication date: 14-Jul-2015
https://doi.org/10.1002/anie.201504693
Texter J(2015)A Kinetic Model for Exfoliation Kinetics of Layered MaterialsAngewandte Chemie10.1002/ange.201504693127:35(10396-10400)Online publication date: 14-Jul-2015
https://doi.org/10.1002/ange.201504693
Ballard GBecker DDemmel JDongarra JDruinsky APeled ISchwartz OToledo SYamazaki I(2014)Communication-Avoiding Symmetric-Indefinite FactorizationSIAM Journal on Matrix Analysis and Applications10.1137/13092906035:4(1364-1406)Online publication date: Jan-2014
https://doi.org/10.1137/130929060
Hogg JScott J(2014)Compressed Threshold Pivoting for Sparse Symmetric Indefinite SystemsSIAM Journal on Matrix Analysis and Applications10.1137/13092062935:2(783-817)Online publication date: Jan-2014
https://doi.org/10.1137/130920629
Hogg JScott J(2013)New Parallel Sparse Direct Solvers for Multicore ArchitecturesAlgorithms10.3390/a60407026:4(702-725)Online publication date: 1-Nov-2013
https://doi.org/10.3390/a6040702
Hogg JScott J(2013)Pivoting strategies for tough sparse indefinite systemsACM Transactions on Mathematical Software10.1145/2513109.251311340:1(1-19)Online publication date: 3-Oct-2013
https://dl.acm.org/doi/10.1145/2513109.2513113

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