Strassen's algorithm is not optimally accurate
Document type :
Pré-publication ou Document de travail
Title :
Strassen's algorithm is not optimally accurate
Author(s) :
Dumas, Jean-Guillaume [Auteur]
Calcul Algébrique et Symbolique, Sécurité, Systèmes Complexes, Codes et Cryptologie [CASC]
Pernet, Clément [Auteur]
Calcul Algébrique et Symbolique, Sécurité, Systèmes Complexes, Codes et Cryptologie [CASC]
Sedoglavic, Alexandre [Auteur]
Université de Lille
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Calcul Algébrique et Symbolique, Sécurité, Systèmes Complexes, Codes et Cryptologie [CASC]
Pernet, Clément [Auteur]
Calcul Algébrique et Symbolique, Sécurité, Systèmes Complexes, Codes et Cryptologie [CASC]
Sedoglavic, Alexandre [Auteur]
Université de Lille
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
HAL domain(s) :
Informatique [cs]/Analyse numérique [cs.NA]
Informatique [cs]/Calcul formel [cs.SC]
Informatique [cs]/Calcul formel [cs.SC]
English abstract : [en]
We propose a non-commutative algorithm for multiplying 2×2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and ...
Show more >We propose a non-commutative algorithm for multiplying 2×2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2×2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy.Show less >
Show more >We propose a non-commutative algorithm for multiplying 2×2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2×2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy.Show less >
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Anglais
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