Communication-Avoiding Symmetric-Indefinite Factorization

TitleCommunication-Avoiding Symmetric-Indefinite Factorization
Publication TypeJournal Article
Year of Publication2014
AuthorsBallard, G., D. Becker, J. Demmel, J. Dongarra, A. Druinsky, I. Peled, O. Schwartz, S. Toledo, and I. Yamazaki
JournalSIAM Journal on Matrix Analysis and Application
Volume35
Issue4
Pagination1364-1406
Date Published07-2014
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 = P LT L T P T 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.