Communication-Avoiding Symmetric-Indefinite Factorization

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

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 $A=PLTL^{T}P^{T},$ 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.