Mathieu函数傅里叶展开系数的高精度混合递推算法

High accuracy hybrid recursive algorithm for the computation of Fourier expansion coefficients of Mathieu functions

针对Mathieu函数傅里叶展开系数计算中的不稳定问题,提出了一种新的基于系数比率的向前-向后混合递推算法.首先,按照传统递推格式向前递推得到前n项系数;然后,令尾项系数的比率充分小并作为初值,利用所提出的后向递推公式反向计算尾项到第n项的比率因子;而后,根据比率与系数的关系计算第n+1项到尾项的系数;最后,将所得系数进行归一化便得Mathieu函数傅里叶展开系数.数值结果表明:基于系数比率的向前-向后混合递推算法稳定且高精度,与参考值间的误差可达到机器精度.对于Mathieu函数傅里叶展开系数的高精度计算可保证使用椭圆无反射人工边界条件和谱方法求解偏微分方程外问题的谱精度.

In order to solve the instability problem in calculating the Fourier expansion coefficients of the Mathieu function,a new forward-backward hybrid recursive algorithm based on the coefficient ratio is proposed.First,the first n coefficients are obtained by forward recursion via the traditional recursive format.Second,when the ratio of the tail coefficient is small enough and used as the initial value,the ratio factors from the last term to the nth term are calculated inversely according to the backward recursive formula proposed in this paper.Moreover,the relationship between the ratio and the coefficient is used to calculate the coefficients from the(n+1)th term to the tail.Finally,the Fourier expansion coefficients of the Mathieu functions are obtained after normalization.The numerical results show that the proposed algorithm is stable and highly accurate,and the error between the approximation and the reference values can reach the machine's accuracy.The high-precision calculation of the Fourier expansion coefficients of the Mathieu function can ensure the spectral accuracy of solving the external problems of partial differential equations using the elliptic non-reflection artificial boundary condition and the spectral method.