Analytical and Numerical Methods For Slowly Convergent Integrals and Applications

缓慢收敛积分的分析和数值方法及其应用

• 批准号: RGPIN-2016-04317
• 负责人: Safouhi, Hassan
• 金额: $ 1.6万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635592
• 关键词: Analytical; Numerical; Methods; Slowly; Convergent

Slowly convergent integrals play a major role in science and engineering. They arise when solving the simplest to the most complicated problems through iterative, discretization or other expansion methods. Slowly convergent integrals must be evaluated in Fourier series to obtain a function approximation scheme that converges faster than the standard Taylor series. Many special functions, which play an extremely important role in applied mathematics and physics, have slowly convergent integral representations. Traditional quadrature rules have failed to provide accurate approximations to slowly convergent integrals, therefore new techniques are highly sought after and desired. The most challenging application concerns the computation of molecular multi-centre integrals needed for the computation of energy components, such as nuclear attraction energy. The computation of these integrals takes as much as 40% of the wall time of molecular structure calculations. Accordingly, any reduction in the calculation time will significantly impact the performance of any software used for molecular structure calculations.

缓慢收敛积分在科学和工程中发挥着重要作用。当通过迭代、离散化或其他扩展方法解决最简单到最复杂的问题时，就会出现这种情况。必须在傅里叶级数中评估缓慢收敛的积分，以获得比标准泰勒级数收敛得更快的函数逼近方案。许多在应用数学和物理学中起着极其重要作用的特殊函数都具有缓慢收敛的积分表示。传统的求积规则无法为缓慢收敛的积分提供精确的近似，因此新技术受到高度追捧和渴望。最具挑战性的应用涉及计算能量成分（例如核吸引力能）所需的分子多中心积分的计算。这些积分的计算占用了分子结构计算的 40% 的时间。因此，计算时间的任何减少都会显着影响用于分子结构计算的任何软件的性能。