Extrapolation methods and sequence transformations for computing slowly convergent integrals

众所周知，在应用数学和科学与工程问题的数值处理中，缓慢收敛的积分和级数经常出现。它们是由近似过程产生的，取决于参数、摄动技术和由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不能提供这些积分和级数的精确近似。解决这些问题的有效方法是使用外推法和加速收敛。这些方法在改善无穷级数和无穷区间积分的收敛方面非常有用，但在处理复杂积分时会遇到许多挑战。 在这个研究项目中，我们将介绍一种新的方法，该方法基于对非常强大的S的推广。这种推广将消除原始S变换所需的边界条件，并将适用范围扩展到球面贝塞尔积分之外。这种新方法用一个发散级数作为边界项来表示一个缓慢收敛的积分，并且变换后的积分比初始积分渐近地更有利。对于边界项的实现，我们提出使用序列变换来求散度级数的和。在变换积分的情况下，我们将使用外推方法、非线性变换以及计算振荡积分的最新技术，即数值最陡下降法、Filon型和Levin型方法。 这项研究计划的一部分涉及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。最具挑战性的应用将涉及到所谓的分子多中心积分和核磁共振参数积分的计算。这些核磁共振积分的解析处理，也是研究程序的一部分，将使用傅立叶变换获得。

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.