复合Bessel函数零点数值计算方法及分布规律
Numerical calculation method and distribution law of zero points of the compound Bessel function

摘要： 为了解决复合Bessel函数零点计算问题,提出修正粒子群与量子粒子群优化算法。修正后的算法能够找到复合Bessel函数有限区间内绝大部分函数零点。为了进一步提高函数零点的搜索能力,结合前两种算法特点,借鉴交叉算子操作,对带交叉算子量子粒子群算法进行修正,修正后的算法能找到复合Bessel函数有限区间内所有函数零点,修正后带交叉算子量子粒子群算法收敛速度快,零点计算精度高。计算结果表明:同一参数复合Bessel函数除去前3个零点,后续零点与零点次序在双对数坐标系中符合直线关系。对不同参数计算的零点与零点次序进行直线拟合,相关程度达到99.99%,零点拟合的相对误差小于0.5%,能够满足工程计算的需求。
Abstract: In order to solve the problem of computing the zeros of compound Bessel functions, a modified optimization algorithm of the particle swarm and the quantum-behaved particle swarm were proposed. Most of the zero points of the compound Bessel function could be calculated by modified algorithm in the finite interval. In order to improve the searching ability of the zero points, the quantum-behaved particle swarm optimization algorithm with crossover operator was modified by using cross operator operation in combination with the characteristics of the two former algorithms. All the zero points of the compound Bessel function in the finite interval were found with the modified version of algorithm. The modified version of algorithm was faster with convergence rate and higher with zero points calculation accuracy. The calculation results showed that, except for the former three zero points, the following zero points showed linear relationship with their sequence on double logarithmic coordinate axis under the same parameters of the compound Bessel function. The straight-line fitting of the zero points and their sequence from different parameters was calculated. The results showed that the correlation coefficient was 99.99% and the relative error of zero point fitting was less than 0.5%, which could full fit the requirement of engineering calculation