Rigorous Polynomial Approximations and Applications

Abstract : For purposes of evaluation and manipulation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, integration, ordinary differential equations (ODE) solving. For that, a wide range of numerical methods exists. We consider the application of such methods in the context of rigorous computing, where we need guarantees on the accuracy of the result, with respect to both the truncation and rounding errors. A rigorous polynomial approximation (RPA) for a function f defined over an interval [a,b] is a couple (P, Delta) where P is a polynomial and Delta is an interval such that f(x)-P(x) belongs to Delta, for all x in [a,b]. In this work we analyse and bring forth several ways of obtaining RPAs for univariate functions. Firstly, we analyse and refine an existing approach based on Taylor expansions. Secondly, we replace them with better approximations such as minimax approximations, Chebyshev truncated series or interpolation polynomials. Several applications are presented: one from standard functions implementation in mathematical libraries (libm), another regarding the computation of Chebyshev series expansions solutions of linear ODEs with polynomial coefficients, and finally an automatic process for function evaluation with guaranteed accuracy in reconfigurable hardware.
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https://tel.archives-ouvertes.fr/tel-00657843
Contributor : Mioara Joldes <>
Submitted on : Monday, January 9, 2012 - 1:39:35 PM
Last modification on : Friday, April 20, 2018 - 3:44:24 PM
Long-term archiving on : Tuesday, April 10, 2012 - 2:31:05 AM

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  • HAL Id : tel-00657843, version 1

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Mioara Joldes. Rigorous Polynomial Approximations and Applications. Computer Arithmetic. Ecole normale supérieure de lyon - ENS LYON, 2011. English. ⟨tel-00657843v1⟩

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