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Algorithms for Accurate, Validated and Fast Polynomial Evaluation

Stef Graillat 1 Philippe Langlois 2, 3 Nicolas Louvet 4
1 PEQUAN - Performance et Qualité des Algorithmes Numériques
LIP6 - Laboratoire d'Informatique de Paris 6
2 ELIAUS - Electronique, Informatique, Automatique et Systèmes
PROMES - Procédés, Matériaux et Energie Solaire
3 DALI - Digits, Architectures et Logiciels Informatiques
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier, UPVD - Université de Perpignan Via Domitia
4 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We survey a class of algorithms to evaluate polynomials with floating point coefficients and for computation performed with IEEE-754 floating point arithmetic. The principle is to apply, once or recursively, an error-free transformation of the polynomial evaluation with the Horner algorithm and to accurately sum the final decomposition. These compensated algorithms are as accurate as the Horner algorithm performed in K times the working precision, for K an arbitrary integer. We prove this accuracy property with an \apriori error analysis. We also provide validated dynamic bounds and apply these results to compute a faithfully rounded evaluation. These compensated algorithms are fast. We illustrate their practical efficiency with numerical experiments on significant environments. Comparing to existing alternatives these K-times compensated algorithms are competitive for K up to 4, i.e., up to 212 mantissa bits.
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Submitted on : Friday, June 6, 2008 - 12:55:56 AM
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Stef Graillat, Philippe Langlois, Nicolas Louvet. Algorithms for Accurate, Validated and Fast Polynomial Evaluation. Japan Journal of Industrial and Applied Mathematics, Kinokuniya Company, 2009, 26 (2-3), pp.191-214. ⟨10.1007/BF03186531⟩. ⟨hal-00285603⟩



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