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Efficient algorithms for verified scientific computing : Numerical linear algebra using interval arithmetic

Interval arithmetic is a means to compute verified results. However, a naive use of interval arithmetic does not provide accurate enclosures of the exact results. Moreover, interval arithmetic computations can be time-consuming. We propose several accurate algorithms and efficient implementations in verified linear algebra using interval arithmetic. Two fundamental problems are addressed, namely the multiplication of interval matrices and the verification of a floating-point solution of a linear system. For the first problem, we propose two algorithms which offer new tradeoffs between speed and accuracy. For the second problem, which is the verification of the solution of a linear system, our main contributions are twofold. First, we introduce a relaxation technique, which reduces drastically the execution time of the algorithm. Second, we propose to use extended precision for few, well-chosen parts of the computations, to gain accuracy without losing much in term of execution time.

Identiferoai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00680352
Date18 January 2011
CreatorsNguyen, Hong Diep
PublisherEcole normale supérieure de lyon - ENS LYON
Source SetsCCSD theses-EN-ligne, France
LanguageEnglish
Detected LanguageEnglish
TypePhD thesis

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