Matrices structurées et matrices de Toeplitz par blocs de Toeplitz en calcul numérique et formel

Houssam Khalil 1, 2
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : Several problems in applied mathematics require the solving of linear systems with very large sizes, and sometimes these systems must be solved multiple times. In such cases, the standard algorithms based on the Gauss elimination require O (n ^ 3) arithmetic operations to solve a system of size n, and it will be a handicap for the computation. That is why we look for using of the structure to reduce the computation's time.

The structure of Toeplitz, Hankel, Cauchy, Vandermonde and other more general structure are very well expoiled to reduce the complexity of solving a linear system to O (n log n ^ 2) arithmetic operations.

The two levels structured matrices and especially Toeplitz block Toeplitz (TBT) matrices appear in many applications. The purpose of this work is to find fast algorithms for TBT systems.

In this thesis, we describe the difficulties of this problem. We give three fast algorithms, of complexity O (n ^ 3 / 2) operations, for Toeplitz band block Toeplitz band systems. we also give a new method of solving Toeplitz systems by giving a relationship between the solution of a Toeplutz system and syzygies of polynomials in one variable. We generalize this method for TBT matrices and we give a relationship between the solution of such linear system and syzygies of polynomials in two variables.
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Theses
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https://tel.archives-ouvertes.fr/tel-00306987
Contributor : Houssam Khalil <>
Submitted on : Monday, July 28, 2008 - 4:53:46 PM
Last modification on : Wednesday, December 12, 2018 - 3:17:41 PM
Long-term archiving on : Friday, October 5, 2012 - 11:27:09 AM

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  • HAL Id : tel-00306987, version 1

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Houssam Khalil. Matrices structurées et matrices de Toeplitz par blocs de Toeplitz en calcul numérique et formel. Mathématiques [math]. Université Claude Bernard - Lyon I, 2008. Français. ⟨tel-00306987⟩

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