Extrapolation methods and sequence transformations for computing slowly convergent integrals

用于计算缓慢收敛积分的外推方法和序列变换

• 批准号: 250223-2011
• 负责人: Safouhi, Hassan
• 金额: $ 1.09万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570851
• 关键词: Extrapolation; methods; sequence; transformations; computing

It is well known that in applied mathematics and in the numerical treatment of scientific and engineering problems, slowly convergent integrals and series occur very frequently. They are produced by approximation procedures depending on a parameter, perturbation techniques and reliable evaluation of functions that are defined by integrals. These slowly convergent integrals and series present severe numerical and computational difficulties. Traditional quadrature rules and summation techniques fail to provide accurate approximations to these integrals and series. An effective remedy for these problems is to use extrapolation methods and convergence acceleration. These methods have proved very useful for improving convergence of infinite series and infinite-range integrals but many challenges occur when dealing with complicated integrals. In this research program, we will introduce a new method based on a generalization of the extremely powerful S. This generalization will eliminate the boundary conditions required by the original S transformation and will expand applicability beyond spherical Bessel integrals. This new method will represent a slowly convergent integral by a divergent series as boundary terms and a transformed integral asymptotically more favorable than the initial one. For the implementation of the boundary terms, we propose the use of sequence transformations for the summation of divergent series. In the case of the transformed integral, we will use extrapolation methods, nonlinear transformations as well as the state of the arts techniques for computing oscillatory integrals, namely numerical steepest descent, Filon-type and Levin-type methods. A part of this research program is concerned with challenging applications of the developed methods and algorithms. These applications include the Sommerfield-type integrals and incomplete Bessel functions. The most challenging application will concern the computation of the so-called molecular multi-center integrals and integrals of nuclear magnetic resonance (NMR) parameters. The analytical treatment of these NMR integrals, which is also a part of the research program, will be obtained using the Fourier transformation.

众所周知，在应用数学以及科学和工程问题的数值处理中，缓慢收敛的积分和级数经常出现。它们是通过近似程序产生的，该近似程序取决于参数、扰动技术和对积分定义的函数的可靠评估。这些缓慢收敛的积分和级数带来了严重的数值和计算困难。传统的求积规则和求和技术无法为这些积分和级数提供准确的近似值。解决这些问题的有效方法是使用外推法和收敛加速。事实证明，这些方法对于提高无限级数和无限范围积分的收敛性非常有用，但在处理复杂积分时会出现许多挑战。 在本研究计划中，我们将引入一种基于极其强大的 S 的推广的新方法。这种推广将消除原始 S 变换所需的边界条件，并将适用性扩展到球面贝塞尔积分之外。这种新方法将用发散级数作为边界项来表示缓慢收敛的积分，并且变换后的积分渐近地比初始积分更有利。为了实现边界项，我们建议使用序列变换来对发散级数求和。对于变换积分，我们将使用外推法、非线性变换以及计算振荡积分的最先进技术，即数值最速下降法、Filon 型和 Levin 型方法。 该研究计划的一部分涉及所开发的方法和算法的挑战性应用。这些应用包括 Sommerfield 型积分和不完全贝塞尔函数。最具挑战性的应用涉及所谓的分子多中心积分和核磁共振（NMR）参数积分的计算。这些核磁共振积分的分析处理也是研究计划的一部分，将使用傅立叶变换来获得。