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Title: Communication-avoiding symmetric-indefinite factorization

Abstract

We describe and analyze a novel symmetric triangular factorization algorithm. The algorithm is essentially a block version of Aasen's triangular tridiagonalization. It factors a dense symmetric matrix A as the product A=PLTLTPT where P is a permutation matrix, L is lower triangular, and T is block tridiagonal and banded. The algorithm is the first symmetric-indefinite communication-avoiding factorization: it performs an asymptotically optimal amount of communication in a two-level memory hierarchy for almost any cache-line size. Adaptations of the algorithm to parallel computers are likely to be communication efficient as well; one such adaptation has been recently published. As a result, the current paper describes the algorithm, proves that it is numerically stable, and proves that it is communication optimal.

Authors:
 [1];  [2];  [3];  [4];  [5];  [6];  [3];  [6];  [2]
+ Show Author Affiliations
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  2. Univ. of Tennessee, Knoxville, TN (United States)
  3. Univ. of California, Berkeley, CA (United States)
  4. Univ. of Tennessee, Knoxville, TN (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Univ. of Manchester (United Kingdom)
  5. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
  6. Tel Aviv Univ., Tel Aviv (Israel)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1237467
Report Number(s):
SAND-2015-1851J
Journal ID: ISSN 0895-4798; 579666
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Matrix Analysis and Applications
Additional Journal Information:
Journal Volume: 35; Journal Issue: 4; Journal ID: ISSN 0895-4798
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; symmetric-indefinite matrices; communication-avoiding algorithms; Aasen's factorization

Citation Formats

Ballard, Grey Malone, Becker, Dulcenia, Demmel, James, Dongarra, Jack, Druinsky, Alex, Peled, Inon, Schwartz, Oded, Toledo, Sivan, and Yamazaki, Ichitaro. Communication-avoiding symmetric-indefinite factorization. United States: N. p., 2014. Web. doi:10.1137/130929060.
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Ballard, Grey Malone, Becker, Dulcenia, Demmel, James, Dongarra, Jack, Druinsky, Alex, Peled, Inon, Schwartz, Oded, Toledo, Sivan, & Yamazaki, Ichitaro. Communication-avoiding symmetric-indefinite factorization. United States. https://doi.org/10.1137/130929060
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Ballard, Grey Malone, Becker, Dulcenia, Demmel, James, Dongarra, Jack, Druinsky, Alex, Peled, Inon, Schwartz, Oded, Toledo, Sivan, and Yamazaki, Ichitaro. Thu . "Communication-avoiding symmetric-indefinite factorization". United States. https://doi.org/10.1137/130929060. https://www.osti.gov/servlets/purl/1237467.
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@article{osti_1237467,
title = {Communication-avoiding symmetric-indefinite factorization},
author = {Ballard, Grey Malone and Becker, Dulcenia and Demmel, James and Dongarra, Jack and Druinsky, Alex and Peled, Inon and Schwartz, Oded and Toledo, Sivan and Yamazaki, Ichitaro},
abstractNote = {We describe and analyze a novel symmetric triangular factorization algorithm. The algorithm is essentially a block version of Aasen's triangular tridiagonalization. It factors a dense symmetric matrix A as the product A=PLTLTPT where P is a permutation matrix, L is lower triangular, and T is block tridiagonal and banded. The algorithm is the first symmetric-indefinite communication-avoiding factorization: it performs an asymptotically optimal amount of communication in a two-level memory hierarchy for almost any cache-line size. Adaptations of the algorithm to parallel computers are likely to be communication efficient as well; one such adaptation has been recently published. As a result, the current paper describes the algorithm, proves that it is numerically stable, and proves that it is communication optimal.},
doi = {10.1137/130929060},
journal = {SIAM Journal on Matrix Analysis and Applications},
number = 4,
volume = 35,
place = {United States},
year = {Thu Nov 13 00:00:00 EST 2014},
month = {Thu Nov 13 00:00:00 EST 2014}
}
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https://doi.org/10.1137/130929060
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Cited by: 11 works
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    Works referencing / citing this record:

    Non-GPU-resident symmetric indefinite factorization: Non-GPU-resident Dense Symmetric Indefinite Factorization
    journal, November 2016

    • Yamazaki, Ichitaro; Tomov, Stanimire; Dongarra, Jack
    • Concurrency and Computation: Practice and Experience, Vol. 29, Issue 5
    • DOI: 10.1002/cpe.4012
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