Always Chebyshev Interpolation For Elementary Function Computations
08.10.07 - 07:50
French-Algerian Cycle of Seminars
Following an initiative of Marc Daumas (LIRMM & ELIAUS) and Professeur Mohamed Mezghiche, we had in the summer of 2007 a cycle of seminars between the University of Perpignan Via Domitia and the University M’hamed Bougara of Boumerdès in Algeria. The first two seminars were broadcasted live and the last two ones were recorded.
Speaker: Ren-Cang Li
Department of Mathematics, University of Texas at Arlington
Ren-Cang Li has been an Invited Professor of the University of Perpignan Via Domitia in June 2007 funded by the French Région Languedoc-Roussillon. He participated to the FP2 library available from NetLib repository. Report of his work is available from HAL.
Abstract
A common practice for computing an elementary transcendental function in a libm implementation nowadays has two phases : reductions of input arguments to fall into a tiny interval and polynomial approximations for the function within the interval. Typically the interval is made tiny enough so that polynomials of very high degrees aren’t required for accurate approximations. Often approximating polynomials as such are taken to be the best polynomials or any others such as the Chebyshev interpolating polynomials. The best polynomial of degree n has the property that the biggest difference between it and the function is smallest among all possible polynomials of degrees no higher than n. Thus it is natural to choose the best polynomials over others. In this talk, we’ll show that the best polynomial can only be more accurate by at most a fractional bit than the Chebyshev interpolating polynomial of the same degree in computing elementary functions, or in other words the Chebyshev interpolating polynomials will do just as well as the best polynomials.
Tags: mathématiques, polynomes
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