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

摘要这项研究使用了一种新的方法来计算合流超几何函数，包括实数和复数的各种论点。该方法包括Kummer级数的直接求和和扩展计算机精度以控制溢出。这种直接的方法对于精确计算特例函数的整个子集似乎很有希望：贝塞尔函数、库仑函数、拉盖尔多项式等。研究的目的是：确定方法的绝对精度；使函数求值器尽可能健壮；确保求值器的可移植性；并对函数求值器算法进行矢量化，以获得与现有求值器相当的计算时间。因为现在可以进行各种复杂的参数，所以这个函数求值器将是一个准确的、通用的和便携的软件包，适用于科学和工程中的各种问题。此外，用来控制复杂算术中溢出的方法可能适用于更广泛的一类需要幂函数求和的问题。

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.