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**Algorithms for accurate, validated and fast polynomial evaluation.**
*(English)*
Zbl 1184.65029

Summary: 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.

### MSC:

65D20 | Computation of special functions and constants, construction of tables |

26A09 | Elementary functions |

26C05 | Real polynomials: analytic properties, etc. |

12Y05 | Computational aspects of field theory and polynomials (MSC2010) |

### Keywords:

polynomial evaluation; compensated algorithm; floating-point arithmetic; IEEE-754; error-free transformation; Horner algorithm; numerical experiments
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\textit{S. Graillat} et al., Japan J. Ind. Appl. Math. 26, No. 2--3, 191--214 (2009; Zbl 1184.65029)

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