RUI: A Numerical Evaluation of the Confluent HypergeometricFunction with Large, Complex Arguments

摘要 这项调查采用了一种新的方法来评估 具有广泛自变量的合流超几何函数 既真实的又复杂的。 该方法包括直接 库默级数求和与计算推广 机器精度控制溢出。 这种直接的方法 8似乎有希望的准确评估整个子集 特殊情况下的函数：贝塞尔函数，库仑函数， 拉盖尔多项式等。研究的目标是： 确定方法的绝对精度;使函数 评估器尽可能健壮;确保 求值器;以及向量化函数求值器算法，以 实现与现有评估器竞争的计算时间。 由于现在可以进行广泛的复杂论证， 函数评估器将是一个准确的，多功能的，和便携式的 软件包适用于广泛的科学问题， 工程. 此外，用于控制溢出的方法 在复杂的算术中可能适用于更广泛的类 求幂级数和的问题。

ABSTRACT This investigation utilizes a new approach for evaluating the confluent hypergeometric function for a wide range of arguments both real and complex. The approach consist of the direct summation of Kummer's series and extension of the computing machine precision to control overflow. This direct approach 8seems promising for the accurate evaluation of the entire subset of special case functions: Bessel functions, Coulomb functions, Laguerre polynomials, etc. The goals of the research are to: determine the absolute accuracy of the method; make the function evaluator as robust as possible; ensure the portability of the evaluator; and vectorize the function evaluator algorithm to achieve computation times competitive with existing evaluators. Because a wide range of complex arguments are now possible, this function evaluator will be an accurate, versatile, and portable package applicable to a wide spectrum of problems in science and engineering. In addition, the approach used to control overflow in the complex arithematic may be applicable to a broader class of problems requiring the summation of a power series.