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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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 35; no. 4; pp. 1364 - 1406
Main Authors: Ballard, Grey, Becker, Dulceneia, Demmel, James, Dongarra, Jack, Druinsky, Alex, Peled, Inon, Schwartz, Oded, Toledo, Sivan, Yamazaki, Ichitaro
Format: Journal Article
Language:English
Published: United States SIAM 01.01.2014
Subjects:
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
Online Access:Get full text
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Summary: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.
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AC04-94AL85000
USDOE National Nuclear Security Administration (NNSA)
SAND-2015-1851J
ISSN:0895-4798
1095-7162
DOI:10.1137/130929060