利用切比雪夫展开式严格逼近D-有限函数
Rigorous uniform approximation of D-finite functions using Chebyshev expansions
AI总结：
本文研究了利用切比雪夫展开式在保证精度的前提下逼近D-有限函数的方法，结合递推关系与验证方法，提出新的计算框架并分析算法复杂度。
摘要：
现有大量数值方法用于基于切比雪夫级数展开或切比雪夫插值多项式计算常微分方程解的多项式近似。本文考虑在需要结果准确性的严格计算中应用此类方法，并从复杂度角度出发。已知在给定区间上切比雪夫展开的阶数n截断是该函数在该区间上的近最佳均匀多项式近似。对于系数为多项式的线性微分方程解，展开系数服从具有多项式系数的线性递推关系。不幸的是，这些递推关系无法直接递推计算系数，因为缺乏初始条件。本文展示如何利用这些递推关系，作为验证过程的一部分，计算D-有限函数的良好均匀近似并提供严格误差界，并研究由此产生的算法复杂度。本文方法基于Clenshaw经典数值方法的新观点，结合功能围挡方法。
英文摘要：
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions. We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.
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