在科学计算中特殊函数和基本函数的高精度快速算法研究

In engineering and natural science, a large number of scientific computing problems are involved. People urgently need reliable computing methods and softwares with high precision as well as high computation speed. It is worthy to notice that many existing algorithms are excellent in theory, while actual calculation results by them are dissatisfactory. In numerical computation, besides the algorithm designing in particular problems, one need frequently utilize some special functions and elementary functions in mathematics, and the computation precision and speed for these functions appears to be very important. The main purpose of this research project is to give high-precision (containing effective numbers of one hundred or thousand) fast algorithms for special functions and elementary functions, especially for some special functions composed by integrals and series, and for some related constants. The research will be fulfilled in the following three methods.1) To express the functions and constants in the form of series with fast convergence, and each item of the series approachs Rc as far as possible, where Rc is the set of rational numbers which can be expressed accurately by computer, and the precision for Rc depends on the accuracy of the computer itself. 2) To establish a recursive formula with the number of operations as few as possible in calculating special functions and mathematical physics related to generalized constants. 3) For the above-mentioned special function and generalized integral given its closed form. By the existing high precision algorithms for these closed forms, the high precision fast algorithms for the special functions and generalized integral can be established.

在工程和自然科学中涉及到大量的科学计算问题，人们迫切需要计算精度高、计算速度快且可靠的方法和计算软件。值得注意的是，许多理论上很好的算法，实际计算效果却不尽人意。在数值计算中，除了需要设计算法以外，常常需要调用数学中的特殊函数和基本函数，这些函数的计算精度以及速度变得至关重要。本项目研究主要目的是, 对特殊函数和基本函数尤其是由积分和级数组成的特殊函数，给出高精度(有效数字为百位甚至千位)快速算法。具体的做法有三种：1) 将所求函数和常数表示成快速收敛的级数，且级数的每项尽可能接近Rc, 其中Rc是能够由计算机精确表示的有理数构成的集合,是由计算机本身的精度决定的；2) 建立所求特殊函数和数学物理涉及到的广义积分的运算次数尽可能少的递推公式；3) 建立这些特殊函数和广义积分的闭形式，由于这些闭形式已经有了高精度快速算法，因而对所求特殊函数和广义积分也就有了高精度快速算法。

本项研究主要是初等函数和一些特殊函数高精度快速计算的算法研究，随着现代科学计算的发展，这些问题是迫切要考虑的问题。经过四年的努力，对(不完全)Gamma 函数类和(不完全)Beta 函数类以及相关函数类，比如Riemann zeta 函数、Digamma 函数多重对数函数、Gauss 超几何函数和Nielsen广义多重对数函数,我们较系统地建立了高精度快速算法。对于这些特殊函数的高精度快速算法都做了不同程度的讨论和研究。对(不完全)Gamma 函数类，具体的结果是对Gamma函数、不完全Gamma函数和一些广义Gamma函数的导数建立了一些递推公式和快速算法。对于Beta函数和不完全Beta函数的偏导数建立了一些递推公式和快速算法。对于初等函数的任意精度的快速计算进行较系统的讨论。对于（Hurwitz）Riemann zeta函数以及(偏)导数、Nielsen广义多重对数函数、相关函数的积分以及分数部分的多重积分和非线性Euler和都做了较深入的研究，发表SCI论文6篇，EI论文6篇，核心期刊和外文期刊4篇，一般论文3篇。