超几何函数实零点的数值算法
Numerical algorithms for the real zeros of hypergeometric functions
AI总结：
本文提出基于全局不动点迭代和差分微分方程组合的超几何函数实零点数值算法，通过分析不同DDE组的收敛性，实现了均匀高效的零点计算。
摘要：
本文描述了用于计算超几何函数实零点的算法，这些超几何函数是二阶常微分方程的解。这些算法基于全局不动点迭代，适用于满足具有连续系数的一阶线性差分微分方程的函数族。为了计算超几何方程的任意解的零点，我们拥有几组不同的差分微分方程（DDE）。我们分析了这些不同组在相关不动点迭代收敛速度方面的表现。结果表明，根据参数范围和自变量的不同，组合使用不同的DDE组能够产生高效的计算零点的方法，且每个零点的收敛速度相当均匀。
英文摘要：
Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.
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