整数阶复宗量贝塞尔函数的计算程序研究
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摘要
鉴于目前的算法程序集中没有现成的计算复宗量贝塞尔函数的程序,本文基于贝塞尔函数的逆向递推关系编写了计算整数阶复宗量第一类贝塞尔函数的Fortran程序源代码与Matlab软件的计算结果比较,两者至少有12位有效数字一致.接着运用此程序,分析了徐士良的《FORTRAN常用算法程序集》中的纯虚宗量的贝塞尔函数,即变形贝塞尔函数程序的准确度,发现其准确度为6位有效数字.最后,对基于实宗量贝塞尔函数和纯虚宗量贝塞尔函数相乘然后用无限求和来计算复宗量贝塞尔函数值的方法的准确性进行了探讨.证明其仅能对有限的贝塞尔函数进行准确计算.这是由于当求和项中有远大于最终的求和项时,会导致求和结果的有效数字减少甚至完全错误.
Fortran source code for calculating the integer order of Bessel functions of the first kind with complex arguments is presented. The method is based on the backward recurrence relation of Bessel functions. Values of the Bessel function generated by our program are in-agreement with the values generated by Matlab to at least 12 significant digits. We use the program to calculate the integer order of Bessel functions of the first kind with pure imaginary arguments, provided by Xu Shiliang's Fortran algorithm assembly.The results show that the first 6 significant digits are accurate. We also analyze the algorithm for calculating Bessel functions with complex arguments, which use the infinite sum of the product of the real arguments of the Bessel function and the pure imaginary arguments of the Bessel function, provided by Xu Shiliang's algorithm assembly. The results show that this algorithm does not always get accurate values for Bessel functions with complex arguments. The reason lies with the fact that the term in the sum larger than the function value causes the loss of significant digits.
Fortran source code for calculating the integer order of Bessel functions of the first kind with complex arguments is presented. The method is based on the backward recurrence relation of Bessel functions. Values of the Bessel function generated by our program are in-agreement with the values generated by Matlab to at least 12 significant digits. We use the program to calculate the integer order of Bessel functions of the first kind with pure imaginary arguments, provided by Xu Shiliang's Fortran algorithm assembly.The results show that the first 6 significant digits are accurate. We also analyze the algorithm for calculating Bessel functions with complex arguments, which use the infinite sum of the product of the real arguments of the Bessel function and the pure imaginary arguments of the Bessel function, provided by Xu Shiliang's algorithm assembly. The results show that this algorithm does not always get accurate values for Bessel functions with complex arguments. The reason lies with the fact that the term in the sum larger than the function value causes the loss of significant digits.
引文
[1] LEWENSTEIN M, BALCOU P, IVANOV M Y, et al. Theory of high harmonic generation by low frequency laser fields[J]. Physical Review A, 1994, 49(3):2117-2132.
[2] GUO Y, FU P, YAN Z C, et al. Imaging the geometrical structure of the H_2~+molecular ion by high order above-threshold ionization in an intense laser field[J]. Physical Review A, 2009, 80(6):3694-3697.
[3] JIA X Y, LI W D, FAN J, et al. Suppression effect in the nonsequential double ionization of molecules by an intense laser field[J]. Physical review A, 2008, 77(6):3195-3199.
[4]徐士良.FORTRAN常用算法程序集[M]. 2版.北京:清华大学出版社,2012.
[5] PRESS W H, TEUKOLSKY S A, VETTERLING W T, et al. Numerical Recipes:The Art of Scientific Computing[M]. 3rd ed. Cambridge:Cambridge University Press, 2007.
[6] DU TOIT C F. The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments[J]. IEEE Transactions on Antennas and Propagation, 1990, 38(9):1341-1349.
[7]魏彦玉,宫玉彬,王文样.任意阶复宗量贝塞尔函数的数值计算[J].电子科技大学学报,1998, 27(2):171-176.
[8]张爽,郭欣,宋立军.利用贝塞尔函数的级数形式进行数值计算的误差分析[J].长春大学学报,2004, 14(2):57-59.
[9]张善杰,唐汉.任意实数阶复宗量第一类和第二类bessel函数的精确计算[J].电子学报,1996, 24(3):77-81.
[2] GUO Y, FU P, YAN Z C, et al. Imaging the geometrical structure of the H_2~+molecular ion by high order above-threshold ionization in an intense laser field[J]. Physical Review A, 2009, 80(6):3694-3697.
[3] JIA X Y, LI W D, FAN J, et al. Suppression effect in the nonsequential double ionization of molecules by an intense laser field[J]. Physical review A, 2008, 77(6):3195-3199.
[4]徐士良.FORTRAN常用算法程序集[M]. 2版.北京:清华大学出版社,2012.
[5] PRESS W H, TEUKOLSKY S A, VETTERLING W T, et al. Numerical Recipes:The Art of Scientific Computing[M]. 3rd ed. Cambridge:Cambridge University Press, 2007.
[6] DU TOIT C F. The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments[J]. IEEE Transactions on Antennas and Propagation, 1990, 38(9):1341-1349.
[7]魏彦玉,宫玉彬,王文样.任意阶复宗量贝塞尔函数的数值计算[J].电子科技大学学报,1998, 27(2):171-176.
[8]张爽,郭欣,宋立军.利用贝塞尔函数的级数形式进行数值计算的误差分析[J].长春大学学报,2004, 14(2):57-59.
[9]张善杰,唐汉.任意实数阶复宗量第一类和第二类bessel函数的精确计算[J].电子学报,1996, 24(3):77-81.