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
2 ELIAUS - Electronique, Informatique, Automatique et Systèmes
3 DALI - Digits, Architectures et Logiciels Informatiques
4 ARENAIRE - Computer arithmetic

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.

Mots clés: Compensated algorithms; Accurate polynomial evaluation; IEEE-754 floating point arithmetic

Domaines: Logiciel mathématique [cs.MS]

Stef Graillat, Philippe Langlois, Nicolas Louvet. Algorithms for Accurate, Validated and Fast Polynomial Evaluation. Japan Journal of Industrial and Applied Mathematics, 2009, 26 (2-3), pp.191-214. ⟨10.1007/BF03186531⟩. ⟨hal-00285603⟩