正则p -递归序列渐近展开式的误差界计算

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2023-08-16 DOI:10.1090/mcom/3888
Ruiwen Dong, Stephen Melczer, Marc Mezzarobba
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引用次数: 6

摘要

在过去的几十年里,分析组合学和计算机代数领域的进步使得在实践中经常成立的假设下,确定满足多项式系数线性递归关系的序列的渐近行为在很大程度上是一种常规问题。所涉及的算法通常采用由递归关系和初始项编码的序列,并在渐近展开中返回到大0误差项的前导项。然而,研究较少的是给出渐近误差项的显式界的有效技术。除其他事项外,这种显式界限通常允许用户自动证明序列正性(枚举学和代数组合学的活跃领域),当正渐近行为主导任何错误项时,通过显示索引。在本文中,我们给出了一种实用的算法来计算这种具有严格误差界的渐近逼近,假设序列的生成级数是具有正则(Fuchsian)优势奇点的微分方程的解。我们的算法近似地遵循Flajolet和Odlyzko的奇异分析方法,只是在渐近展开的推导中涉及的所有大o项都被显式误差项所取代。误差项的计算结合了文献中的解析界与严格数值和计算机代数的有效技术。我们在SageMath计算机代数系统中实现了我们的算法,并展示了它在各种应用中的使用(包括我们最初的激励示例,属1生物膜形状的Canham模型的解唯一性)。
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Computing error bounds for asymptotic expansions of regular P-recursive sequences
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).
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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