Open Access
October 2009 Algorithms for Accurate, Validated and Fast Polynomial Evaluation
Stef Graillat, Philippe Langlois, Nicolas Louvet
Japan J. Indust. Appl. Math. 26(2-3): 191-214 (October 2009).


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 positive integer. We prove this accuracy property with an a priori 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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Stef Graillat. Philippe Langlois. Nicolas Louvet. "Algorithms for Accurate, Validated and Fast Polynomial Evaluation." Japan J. Indust. Appl. Math. 26 (2-3) 191 - 214, October 2009.


Published: October 2009
First available in Project Euclid: 1 February 2010

zbMATH: 1184.65029
MathSciNet: MR2589472

Keywords: compensated algorithm , floating-point arithmetic , IEEE-754 , polynomial evaluation

Rights: Copyright © 2009 The Japan Society for Industrial and Applied Mathematics

Vol.26 • No. 2-3 • October 2009
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