分数阶微积分的高精度递推算法

设计了一种计算分数阶微积分的高精度数值算法，提出了一种构造生成函数的简便方法.分析了基于快速Fourier变换的算法，该算法误差较大的原因是应用了不准确的生成函数的系数，而且没有考虑原函数的非零初值条件对计算精度的影响.新算法应用递推公式计算生成函数的系数，并将原函数分解成零初值条件和非零初值条件两部分，分别计算它们的分数阶微分和积分，这样可以减小计算误差.误差分析和计算实例证明新算法具有很高的计算精度.

High Precision Recursive Algorithm for Computing Fractional-Order Derivative and Integral

A high precision numerical algorithm was designed to compute fractional-order derivative and integral， and a simple method was proposed to construct the generating function. An algorithm based on fast Fourier transform was analyzed. It could be concluded that the reasons of its large computation error were using the inaccurate coefficient of the generating function and no considering the effect of nonzero initial condition of the original function on calculation precision. The recursive formula was used to compute the coefficient of the generating function in the new algorithm， what’s more， the original function was decomposed into two parts， i.e.， zero initial condition and nonzero initial condition， and their fractional-order derivative and integral were computed to decrease the computation error. The error analysis and the illustrative numerical examples showed that the computation accuracy of the new algorithm was very high.