Communication-Avoiding Symmetric-Indefinite Factorization

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,...

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Vydané v:	SIAM journal on matrix analysis and applications Ročník 35; číslo 4; s. 1364 - 1406
Hlavní autori:	Ballard, Grey, Becker, Dulceneia, Demmel, James, Dongarra, Jack, Druinsky, Alex, Peled, Inon, Schwartz, Oded, Toledo, Sivan, Yamazaki, Ichitaro
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	United States SIAM 01.01.2014
Predmet:	Aasen's factorization Adaptation Algorithms Asymptotic properties Blocking communication-avoiding algorithms Factorization Hierarchies MATHEMATICS AND COMPUTING Optimization Parallel computers symmetric-indefinite matrices
ISSN:	0895-4798, 1095-7162
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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 $A=PLTLT}PT},$ 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.
AbstractList	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. 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.
Author	Yamazaki, Ichitaro Toledo, Sivan Druinsky, Alex Schwartz, Oded Becker, Dulceneia Dongarra, Jack Ballard, Grey Demmel, James Peled, Inon
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SubjectTerms	Aasen's factorization Adaptation Algorithms Asymptotic properties Blocking communication-avoiding algorithms Factorization Hierarchies MATHEMATICS AND COMPUTING Optimization Parallel computers symmetric-indefinite matrices
Title	Communication-Avoiding Symmetric-Indefinite Factorization
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