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Algorithmique semi-numérique rapide des séries de Tchebychev

Alexandre Benoit 1 
1 ALGORITHMS - Algorithms
Inria Paris-Rocquencourt
Abstract : A Chebyshev series is an expansion in the basis of Chebyshev polynomials of the first kind. These series are important in approximation theory. Unlike Taylor series, their algorithmic aspects are not very developed in computer algebra. This thesis proposes new algorithms for these series. A first part gives fast algorithms that convert a truncated Chebyshev series into a truncated Taylor series and vice versa, and others that multiply or divide two truncated Chebyshev series. The rest of the thesis is devoted to Chebyshev series that are solutions of a linear differential equation with polynomial coefficients. In this class, the coefficients of series are solutions of a linear recurrence. This thesis show how to compute this recurrence efficiently, then how to use it for the efficient computation of approximated coefficients despite numerical instabilities. These algorithms lead to an efficient computation of the approximation by polynomials of fixed degree on a segment for solutions of linear differential equations. Finally, the computation of recurrences for the coefficients of series is generalized to the case of generalized Fourier series. The document is illustrated by examples using implementations developed during this thesis.
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Submitted on : Tuesday, September 18, 2012 - 8:12:59 PM
Last modification on : Thursday, February 3, 2022 - 11:17:23 AM
Long-term archiving on: : Friday, December 16, 2016 - 8:51:06 AM


  • HAL Id : pastel-00726487, version 1



Alexandre Benoit. Algorithmique semi-numérique rapide des séries de Tchebychev. Calcul formel [cs.SC]. Ecole Polytechnique X, 2012. Français. ⟨pastel-00726487⟩



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