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.
Type de document :
Communication dans un congrès
ISSAC'18, Jul 2018, New York, United States. ACM, pp.151-158, Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computations. 〈10.1145/3208976.3209019〉
Liste complète des métadonnées

Littérature citée [15 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-01704793
Contributeur : Jean-Guillaume Dumas <>
Soumis le : lundi 26 février 2018 - 09:52:58
Dernière modification le : lundi 18 février 2019 - 13:24:07
Document(s) archivé(s) le : samedi 5 mai 2018 - 00:26:38

Fichiers

ldlt.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

Collections

Citation

Jean-Guillaume Dumas, Clément Pernet. Symmetric indefinite triangular factorization revealing the rank profile matrix. ISSAC'18, Jul 2018, New York, United States. ACM, pp.151-158, Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computations. 〈10.1145/3208976.3209019〉. 〈hal-01704793〉

Partager

Métriques

Consultations de la notice

241

Téléchargements de fichiers

98