Truncation Bounds for Differentially Finite Series

Abstract

We describe a flexible symbolic-numeric algorithm for computing bounds on the tails of series solutions of linear differential equations with polynomial coefficients. Such bounds are useful in rigorous numerics, in particular in rigorous versions of the Taylor method of numerical integration of ODEs and related algorithms. The focus of this work is on obtaining tight bounds in practice at an acceptable computational cost, even for equations of high order with coefficients of large degree. Our algorithm fully covers the case of generalized series expansions at regular singular points. We provide a complete implementation in SageMath and use it to validate the method in practice. Résumé.-Nous décrivons un algorithme symbolique-numérique souple pour le calcul de bornes sur les restes de solutions séries d'équations différentielles linéaires à coefficients polynomiaux. Ces bornes sont destinées au calcul numérique rigoureux, et utiles notamment dans des versions rigoureuses de la méthode de Taylor d'intégration des EDO ou d'autres algorithmes apparentés. L'objectif principal de ce travail est d'obtenir des bornes fines en pratique pour un coût de calcul acceptable, y compris dans le cas d'équations d'ordre élevé à coefficients de grand degré. Notre algorithme couvre entièrement le cas des développement en séries généralisées au voisinage de points singuliers réguliers. Nous présentons une implémentation complète de la méthode en SageMath, et nous l'utilisons pour valider son bon comportement en pratique.

Sequence positivity through numeric analytic continuation: uniqueness of the Canham model for biomembranes

“…Proof. The bounds are computed using the implementation in ore algebra of the algorithm described in [Mez19], and full details can be found in the accompanying Sage notebook. We write the singular expansion of f at ρ in the form…”

Rounding Error Analysis of Linear Recurrences Using Generating Series

“…This means that the algorithm can be modified to simultaneously bound the truncation and rounding error, most of the steps involved being shared between both bounds. See [26] and the references therein for more on computing tight bounds on truncation errors. (ũ r−s , .…”

“…11 As a first step in this direction, we have implemented a variant of Algorithm 1 based on the more flexible framework of [26], in the ore algebra package [23,25] for SageMath. While very effective in some cases, it does not at this stage run consistently faster than naive interval summation, due both to overestimation issues and to the computational overhead of computing good bounds.…”

Guessing Gr{ö}bner Bases of Structured Ideals of Relations of Sequences

“…Computing or guessing linear recurrence relations satisfied by a table is a fundamental problem in coding theory for cyclic codes [8,25] of dimension n ≥ 1, combinatorics and computer algebra for solving sparse linear systems, performing sparse polynomial interpolation, polynomial least-square approximation and Gröbner bases changes of orderings in n ≥ 1 variables [20,21]. Furthermore, computing these relations with polynomial coefficients in the indices allows us to predict the growth of its terms, to classify the differential nature of their generating series or to evaluate said generating series [29].…”