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.
Document type :
Journal articles
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-00285603
Contributor : Philippe Langlois <>
Submitted on : Friday, June 6, 2008 - 12:55:56 AM
Last modification on : Wednesday, May 15, 2019 - 3:41:12 AM
Long-term archiving on : Friday, September 28, 2012 - 3:35:49 PM

File

jjiam.pdf
Files produced by the author(s)

Identifiers

Citation

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⟩

Share

Metrics

Record views

436

Files downloads

281