基于幂级数展开的基本初等函数的高精度快速计算

FAST CALCULATION AND HIGH PRECISION OF BASIC ELEMENTARY FUNCTION BASED ON POWER SERIES EXPANSIONS

本文考虑了基本初等函数的高精度快速算法问题.首先讨论与Bernoulli数B_(2n)或Euler数E_(2n)相关的基本初等函数(如tanx、secx、tanhx等)的幂级数展开问题,并给出相应的幂级数展开式的快速算法.然后,对于基本初等函数、双曲函数和反双曲函数,在复数域上给出基于幂级数展开的任意精度的快速算法.由于指数、对数函数可以用幂级数表示,本文设计的算法适用于所有初等函数的计算.算法的特点是编程简单、容易实现,可以自成计算初等函数的体系.

In this paper, we consider a fast and high-precision algorithm of the basic elementary functions. Firstly, we discuss the power series expansion of functions which are related to the Bernoulli number B2n or Euler number E2n, such as tan x, sec x, tanh x and so on, and study the corresponding fast computation. For the basic elementary functions, hyperbolic functions and inverse hyperbolic functions, we derive a fast and arbitrarily accurate algorithm based on the power series expansion in the complex domain. The algorithm proposed in this paper is suitable for all elementary functions due to that the exponential and logarithm functions can be expressed by the power series. The feature of this algorithm is that it can be easily coded and it is self-contained for the computation of the elementary functions.