Autour de l'évaluation numérique des fonctions D-finies

Marc Mezzarobba 1
1 ALGORITHMS - Algorithms
Inria Paris-Rocquencourt
Abstract : D-Finite functions of one variable (also known as holonomic functions) are the solutions of linear ordinary differential equations with polynomial coefficients. Developing a unified treatment of these functions has proven fruitful in Computer Algebra for the last twenty years. In line with these ideas, the present thesis is concerned with the numeric evaluation of D-finite functions and related issues. We focus on three directions. First, we describe an algorithm to compute upper bounds on the coefficients of power series expansions of D-finite functions, starting from the associated differential equations. The main originality of our approach is that we state a tightness criterion with respect to which the computed bounds are optimal. Our second focus is the multiple precision analytic continuation of D-finite functions. Here, we put into practice the "bit burst" method of Chudnovsky and Chudnovsky by implementing it, eliminating several redundancies and suggesting various other improvements. Both the bound computation algorithm and the numeric analytic continuation cover the case of generalized series expansions at regular singular points of differential equations. Finally, we develop a method for the approximation of D‑finite functions by polynomials of prescribed degree over a segment, based on their Chebyshev expansions. All these questions are addressed with the triple goal of mathematical rigor (all approximations include error bounds), generality (the entire class of D-finite functions is treated in a uniform way) and efficiency. In almost all cases, our algorithms are accompanied by publicly available implementations.
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Theses
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https://pastel.archives-ouvertes.fr/pastel-00663017
Contributor : Marc Mezzarobba <>
Submitted on : Wednesday, January 25, 2012 - 6:30:43 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
Long-term archiving on : Wednesday, December 14, 2016 - 1:58:27 AM

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Marc Mezzarobba. Autour de l'évaluation numérique des fonctions D-finies. Calcul formel [cs.SC]. Ecole Polytechnique X, 2011. Français. ⟨pastel-00663017⟩

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