Program for calculating the integer order of Bessel functions with complex arguments
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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.
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关键词:
- 复宗量贝塞尔函数 /
- Fortran源程序 /
- 逆递推计算
Abstract: 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. -
表 1 J$_n(z)$的计算举例
Tab. 1 Numerical examples of J$_n(z)$
$n$ $z$ 本文结果 [1]中基于逆向递推的结果 5 18+0i $-$0.155370098779049 $-$0.155370098779049 5 0+18i i3.05782771756610$×$10$^{6}$ i3.05782771756610$×$10$^{6}$ 50 $-$180+100i 3.235775172706374$×$10$^{40}$$-$i2.054938087958093$×$10$^{40}$ 3.23577517270637$×$10$^{40}$$-$i2.05493808795809$×$10$^{40}$ 600 $-$4000$-$300i $-$2.767691224061821$×$10$^{126}$+i3.172834828322395$×$10$^{126}$ $-$2.76769122406189$×$10$^{126}$+i3.172834828322395$×$10$^{126}$ 1000 600$-$200i 5.373339700916123$×$10$^{-104}$+i1.658289459267489$×$10$^{-103}$ 5.37333970091597$×$10$^{-104}$+i1.65828945926745$×$10$^{-103}$ 0 10000000+333i $-$1.810334213201858$×$10$^{140}$$-$i4.938482320281676$×$10$^{140}$ Not computable with this algorithm 4090 10000000+333i 4.878566679363567$×$10$^{140}$$-$i1.965697327877443$×$10$^{140}$ Not computable with this algorithm 3300 3000+10i $-$3.031921315712407$×$10$^{-42}$$-$i2.477259163766169$×$10$^{-41}$ $-$3.03192131571261$×$10$^{-42}$$-$i2.47725916376618$×$10$^{-41}$ $n$ $z$ [1]中基于幂级数展开截断的结果 Matlab的结果 5 18+0i $-$0.155370098779049 $-$0.155370098779049 5 0+18i $-$3.38362801581971$×$10$^{-8}$+i3.05782771756610$×$10$^{6}$ 1.872379463330692$×$10$^{-9}$+i3.057827717566103$×$10$^{6}$ 50 $-$180+100i 3.23577517271025$×$10$^{40}$$-$i2.05493808795888$×$10$^{40}$ 3.235775172706450$×$10$^{40}$$-$i2.054938087958026$×$10$^{40}$ 600 $-$4000$-$300i $-$2.76769122367143$×$10$^{126}$+i3.172834828322395$×$10$^{126}$ $-$2.767691224059726$×$10$^{126}$+i3.172834828323653$×$10$^{126}$ 1000 600$-$200i Not computable with this algorithm 5.373339700915873$×$10$^{-104}$ +i1.658289459267151$×$10$^{-103}$ 0 10000000+333i $-$1.81033421565001$×$10$^{140}$$-$i4.93848231938389$×$10$^{140}$ $-$1.810334213201893$×$10$^{140}$$-$i4.938482320281608$×$10$^{140}$ 4090 10000000+333i 4.878566815077401$×$10$^{140}$$-$i1.96569732864925$×$10$^{140}$ 4.878566679363453$×$10$^{140}$$-$i1.965697327877465$×$10$^{140}$ 3300 3000+10i Not computable with this algorithm $-$3.031921315712416$×$10$^{-42}$$-$i2.477259163766451$×$10$^{-41}$ 
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表 2 ${\rm I}_0(y)$的计算举例
Tab. 2 Numerical examples of ${\rm I}_0(y)$
$y$ 本文第1节的结果 程序集[4]中的结果 1 1.266065877752008 1.266065848034260 71 3.243091297746851E+029 3.243089889079756E+029 141 5.783741126182957E+059 5.783738566300272E+059 211 1.188946782454909E+090 1.188946347893052E+090 281 2.591193569443138E+120 2.591192781417313E+120 351 5.831422925433985E+150 5.831421416694420E+150 421 1.339291152932047E+181 1.339290852152675E+181 491 3.119396717782502E+211 3.119396099566693E+211 561 7.340570892169514E+241 7.340569591071896E+241 631 1.741003196211549E+272 1.741002917182085E+272 701 4.154898016095451E+302 4.154897408481159E+302
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表 3 运用程序集[4]中的现有程序结合公式(13)能准确计算${\rm J}_{n}(x+{\rm i}y)$的$n$的范围
Tab. 3 The range of the orders of ${\rm J}_n(x+{\rm i}y)$ which can be accurately
($x, y$) $n$的范围 ($x, y$) $n$的范围 ($x, y$) $n$的范围 ($x, y$) $n$的范围 (1, 1) ($-\infty$, $+\infty$) (40, 1) ($-\infty$, $+\infty$) (100, 1) ($-\infty$, $+\infty$) (500, 1) ($-\infty$, $+\infty$) (1, 10) ($-\infty$, $+\infty$) (40, 10) ($-$131, 131) (100, 10) ($-$51, 51) (500, 10) ($-\infty$, $+\infty$) (1, 50) ($-\infty$, $+\infty$) (40, 50) ($-$41, 41) (100, 50) ($-$21, 21) (500, 50) ($-246$, 246) (1, 80) ($-\infty$, $+\infty$) (40, 80) ($-$46, 46) (100, 80) ($-$26, 26) (500, 80) ($-56$, 56) (1, 120) ($-\infty$, $+\infty$) (40, 120) ($-$56, 56) (100, 120) ($-$41, 41) (500, 120) ($-56$, 56) (1, 300) ($-\infty$, $+\infty$) (40, 300) ($-$131, 131) (100, 300) ($-$61, 61) (500, 300) ($-61$, 61)
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