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 perforned inK times the working precision, forK 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 theseK-times compensated algorithms are competitive forK up to 4, i.e., up to 212 mantissa bits.
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Graillat, S., Langlois, P. & Louvet, N. Algorithms for accurate, validated and fast polynomial evaluation. Japan J. Indust. Appl. Math. 26, 191–214 (2009). https://doi.org/10.1007/BF03186531
- polynomial evaluation
- compensated algorithm
- floating-point arithmetic