Communication efficient gaussian elimination with partial pivoting using a shape morphing data layout

Grey Ballard, James Demmel, Benjamin Lipshitz, Oded Schwartz, Sivan Toledo

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations

Abstract

High performance for numerical linear algebra often comes at the expense of stability. Computing the LU decomposition of a matrix via Gaussian Elimination can be organized so that the computation involves regular and efficient data access. However, maintaining numerical stability via partial pivoting involves row interchanges that lead to inefficient data access patterns. To optimize communication efficiency throughout the memory hierarchy we confront two seemingly contradictory requirements: partial pivoting is efficient with column-major layout, whereas a block-recursive layout is optimal for the rest of the computation. We resolve this by introducing a shape morphing procedure that dynamically matches the layout to the computation throughout the algorithm, and show that Gaussian Elimination with partial pivoting can be performed in a communication efficient and cache-oblivious way. Our technique extends to QR decomposition, where computing Householder vectors prefers a different data layout than the rest of the computation.

Original languageEnglish
Title of host publicationSPAA 2013 - Proceedings of the 25th ACM Symposium on Parallelism in Algorithms and Architectures
PublisherAssociation for Computing Machinery
Pages232-240
Number of pages9
ISBN (Print)9781450315722
DOIs
StatePublished - 2013
Event25th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2013 - Montreal, QC, Canada
Duration: 23 Jul 201325 Jul 2013

Publication series

NameAnnual ACM Symposium on Parallelism in Algorithms and Architectures

Conference

Conference25th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2013
Country/TerritoryCanada
CityMontreal, QC
Period23/07/1325/07/13

Keywords

  • Cache oblivious algorithms
  • Communication-avoiding algorithms
  • Matrix data layouts
  • Matrix factorization

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