小变量情况下第一类整数阶Bessel函数的计算

Computation of the Integer Order Bessel Functions of First Kind with Small Arguments

在计算第一类整数阶Bessel函数时,后向递推算法稳定高效.然而,起始点的选取必须有足够高的阶数,并且需要进行归一化处理.本文对Taylor级数展开算法进行研究,并对其级数展开规律、计算精度,以及求和项与参数间的关系进行了讨论.此外,本文利用指数形式,极大扩展了该算法的可计算范围.与du Toit算法、MATLAB和Mathematica应用软件的计算结果比较显示,本文的算法具有较高的准确性和稳定性.

Algorithm based on the backward recurrence for computing the integer order Bessel functions of the first kind is stable and efficient. However, the orders of the starting points should be high enough and the normalization is re- quired. In this paper, we introduce an algorithm based on the Taylor series expansion （TSE） ,in which all the quantifies in- volved are expressed in the exponential format so as to expand the numeric range of calculation. Comparison against du Toit＇ s algorithm as well as MATLAB and Mathematica shows that our algorithm is stable and reliable.