| Identifiant de l'article : |
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ensl-00335918, version 1 |
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| Identifiant arXiv : |
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arXiv:0810.5647 |
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| Domaine : |
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| Titre : |
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Kaltofen's division-free determinant algorithm differentiated for matrix adjoint computation |
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| Auteur(s) : |
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Gilles Villard1, 2 |
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| Laboratoire : |
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| 1 : |
LIP - Laboratoire de l'Informatique du Parallélisme |
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| 2 : |
Inria Grenoble Rhône-Alpes / LIP Laboratoire de l'Informatique du Parallélisme - ARENAIRE |
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| Équipe de recherche : |
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[ARENAIRE - Arithmétique des ordinateurs] |
| Résumé : |
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Kaltofen has proposed a new approach in 1992 for computing matrix determinants without divisions. The algorithm is based on a baby steps/giant steps construction of Krylov subspaces, and computes the determinant as the constant term of a characteristic polynomial. For matrices over an abstract ring, by the results of Baur and Strassen, the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix. However, the latter adjoint algorithm is obtained by the reverse mode of automatic differentiation, hence somehow is not ''explicit''. We present an alternative (still closely related) algorithm for the adjoint thatcan be implemented directly, we mean without resorting to an automatic transformation. The algorithm is deduced by applying program differentiation techniques ''by hand'' to Kaltofen's method, and is completely decribed. As subproblem, we study the differentiation of programs that compute minimum polynomials of lineraly generated sequences, and we use a lazy polynomial evaluation mechanism for reducing the cost of Strassen's avoidance of divisions in our case. |
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Langue du texte
intégral : |
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Anglais |
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| Mots-clés : |
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matrix determinant – matrix adjoint – matrix inverse – characteristic polynomial – exact algorithm – division-free complexity – Wiedemann algorithm – automatic differentiation |
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