We describe and analyze a novel symmetric triangular factorization algorithm. The algorithm is essentially a block version of Aasenâ€™s triangular tridiagonalization. It factors a dense symmetric matrix A as the product A = PLTL TPT 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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