Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko, except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

翻译：过去几十年来，在解析组合学与计算机代数领域的推动下，在通常实践中成立的假设条件下，确定满足多项式系数线性递推关系的序列渐近行为已基本成为常规操作。相关算法通常以递推关系和初始项编码的序列为输入，输出渐近展开式中直至大O误差项的领先项。然而，针对渐近误差项给出显式界的有效技术却鲜有研究。这类显式界通常允许用户通过展示正主导渐近行为支配所有误差项的指标，自动证明序列的正性（枚举代数组合学中的一个活跃领域）。本文提出一种实用算法，在序列生成级数为具有正则（Fuchs型）主导奇点的微分方程解的假设下，计算带有严格误差界的渐近近似。我们的算法大致遵循Flajolet和Odlyzko的奇点分析方法，但将渐近展开推导中涉及的所有大O项替换为显式误差项。误差项的计算结合了文献中的解析界与严谨数值计算及计算机代数的有效技术。我们在SageMath计算机代数系统中实现该算法，并通过多种应用实例展示其用途（包括最初的动机性问题——亏格一生物膜Canham模型中解的唯一性）。