Algorithms for Accurate, Validated and Fast Polynomial Evaluation

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 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.