Communication-avoiding symmetric-indefinite factorization

Grey Ballard, Dulceneia Becker, James Demmel, Jack Dongarra, Alex Druinsky, Inon Peled, Oded Schwartz, Sivan Toledo, Ichitaro Yamazaki

Research output: Contribution to journalArticlepeer-review


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 = PLTLT PT , 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. The current paper describes the algorithm, proves that it is numerically stable, and proves that it is communication optimal.

Original languageEnglish
Pages (from-to)1364-1406
Number of pages43
JournalSIAM Journal on Matrix Analysis and Applications
Issue number4
StatePublished - 2014


  • Aasen's factorization
  • Communication-avoiding algorithms
  • Symmetric-indefinite matrices


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