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Symmetric indefinite triangular factorization revealing the rank profile matrix

Abstract : We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization P T AP = LDL T where P is a permutation matrix, L is lower triangular with a unit diagonal and D is symmetric block diagonal with 1×1 and 2×2 antidiagonal blocks. The novel algorithm requires O(n 2 r ω−2) arithmetic operations. Furthermore , experimental results demonstrate that our algorithm can even be slightly more than twice as fast as the state of the art unsymmetric Gaus-sian elimination in most cases, that is it achieves approximately the same computational speed. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. There is an obstruction in characteristic 2 for revealing the rank profile matrix which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard PLDL T P T decomposition at a negligible cost.
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https://hal.archives-ouvertes.fr/hal-01704793
Contributor : Jean-Guillaume Dumas Connect in order to contact the contributor
Submitted on : Monday, February 26, 2018 - 9:52:58 AM
Last modification on : Wednesday, October 27, 2021 - 5:30:14 AM
Long-term archiving on: : Saturday, May 5, 2018 - 12:26:38 AM

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Jean-Guillaume Dumas, Clément Pernet. Symmetric indefinite triangular factorization revealing the rank profile matrix. ISSAC'18, Jul 2018, New York, United States. pp.151-158, ⟨10.1145/3208976.3209019⟩. ⟨hal-01704793⟩

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