An Exchange Algorithm for Optimizing both Approximation and Finite-Precision Evaluation Errors in Polynomial Approximations - Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes
Pré-Publication, Document De Travail Année : 2024

An Exchange Algorithm for Optimizing both Approximation and Finite-Precision Evaluation Errors in Polynomial Approximations

Un algorithme d'échange pour l'optimisation simultanée des erreur d'approximation et d'évaluation en précision finie de polynômes d'approximation

Résumé

The finite precision implementation of mathematical functions frequently depends on polynomial approximations. A key characteristic of this approach is that rounding errors occur both when representing the coefficients of the polynomial on a finite number of bits, and when evaluating it in finite precision arithmetic. Hence, to find a best polynomial, for a given fixed degree, norm and interval, it is necessary to account for both the approximation error and the floating-point evaluation error. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad-hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is a best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez exchange algorithm. An open-source C implementation using the Sollya library is presented and tested on several examples, which are then analyzed and compared against state-of-the-art Sollya routines.
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Dates et versions

hal-04709615 , version 1 (25-09-2024)

Identifiants

  • HAL Id : hal-04709615 , version 1

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Denis Arzelier, Florent Bréhard, Tom Hubrecht, Mioara Joldes. An Exchange Algorithm for Optimizing both Approximation and Finite-Precision Evaluation Errors in Polynomial Approximations. 2024. ⟨hal-04709615⟩
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