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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Communication dans un congrès
ACM International Symposium on Symbolic and Algebraic Computations, Jul 2018, New York, United States
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Contributeur : Jean-Guillaume Dumas <>
Soumis le : lundi 26 février 2018 - 09:52:58
Dernière modification le : vendredi 31 août 2018 - 15:36:34
Document(s) archivé(s) le : samedi 5 mai 2018 - 00:26:38

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  • HAL Id : hal-01704793, version 1
  • ARXIV : 1802.10453

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Jean-Guillaume Dumas, Clement Pernet. Symmetric indefinite triangular factorization revealing the rank profile matrix. ACM International Symposium on Symbolic and Algebraic Computations, Jul 2018, New York, United States. 〈hal-01704793〉

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