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基于Prony-like方法的第一类贝塞尔函数逼近

纪宇 何一璇 吴国群 吴敏

纪宇, 何一璇, 吴国群, 吴敏. 基于Prony-like方法的第一类贝塞尔函数逼近[J]. 华东师范大学学报(自然科学版), 2019, (6): 42-60. doi: 10.3969/j.issn.1000-5641.2019.06.006
引用本文: 纪宇, 何一璇, 吴国群, 吴敏. 基于Prony-like方法的第一类贝塞尔函数逼近[J]. 华东师范大学学报(自然科学版), 2019, (6): 42-60. doi: 10.3969/j.issn.1000-5641.2019.06.006
JI Yu, HE Yi-xuan, WU Guo-qun, WU Min. On evaluation of Bessel functions of the first kind via Prony-like methods[J]. Journal of East China Normal University (Natural Sciences), 2019, (6): 42-60. doi: 10.3969/j.issn.1000-5641.2019.06.006
Citation: JI Yu, HE Yi-xuan, WU Guo-qun, WU Min. On evaluation of Bessel functions of the first kind via Prony-like methods[J]. Journal of East China Normal University (Natural Sciences), 2019, (6): 42-60. doi: 10.3969/j.issn.1000-5641.2019.06.006

基于Prony-like方法的第一类贝塞尔函数逼近

doi: 10.3969/j.issn.1000-5641.2019.06.006
基金项目: 

国家自然科学基金 11371143

详细信息
    作者简介:

    纪宇, 女, 硕士研究生, 研究方向为数值计算、符号计算.E-mail:15526476747@163.com

    通讯作者:

    吴敏, 女, 副教授, 主要研究方向为符号计算、数值计算.E-mail:mwu@sei.ecnu.edu.cn

  • 中图分类号: TP311.5

On evaluation of Bessel functions of the first kind via Prony-like methods

  • 摘要: 贝塞尔函数的数值逼近既有重要的理论意义,又在数学、物理学、工程等各个领域有着广泛的应用.研究整数阶第一类贝塞尔函数的数值逼近,基于Prony方法,采用不同三角函数(正弦、余弦)形式的Prony-like方法进行逼近.通过在符号计算软件Maple中对函数进行数值实验,分析不同整数阶的第一类贝塞尔函数在不同自变量区间上的数值逼近,将Prony-like方法的实验结果与基于傅里叶级数展开的方法进行对比,发现Prony-like方法的逼近效果远优于基于傅里叶级数的方法.
  • 图  1  J0 (x), J1 (x)和J2 (x)的图象

    Fig.  1  Graphs of J0 (x), J1 (x) and J2 (x)

    图  2  J0(y; b)、J1y; b)和J2(y; b)的图像

    Fig.  2  Graphs of J0(y; b), J1(y; b)and J2(y; b)

    图  3  Prony-like方法和基于傅里叶级数的函数${\rm J}_0 (y;1)$逼近图象

    Fig.  3  Logarithmic relative error of approximations via Fourier-based method and Prony-like method of J0(y; 1)

    图  4  Prony-like方法和基于傅里叶级数的函数${J}_0(y;5)$逼近图象

    Fig.  4  Logarithmic relative error of approximations via Fourier-based method and Prony-like method of ${J}_0 (y;5)$

    图  5  Prony-like方法和基于傅里叶级数的函数$J_1(y;1)$逼近图象

    Fig.  5  Logarithmic relative error of approximations via Fourier-based method and Prony-like method of ${J}_1 (y;1)$

    图  6  Prony-like方法和基于傅里叶级数的函数$J_1(y;5)$逼近图象

    Fig.  6  Logarithmic relative error of approximations via Fourier-based method and Prony-like method of ${J}_1 (y;5)$

    图  7  Prony-like方法和基于傅里叶级数的函数$J_2(y;1)$逼近图象

    Fig.  7  Logarithmic relative error of approximations via Fourier-based method and Prony-like method of $J_2 (y;1)$

    图  8  Prony-like方法和基于傅里叶级数的函数$J_2(y;5)$逼近图象

    Fig.  8  Logarithmic relative error of approximations via Fourier-based method and Prony-like method of $J_2 (y;5)$

    表  1  Prony-like方法和基于傅里叶级数逼近${J}_0(y;1)$的最大相对误差

    Tab.  1  Maximal logarithmic relative error of $m$-th approximants of ${J}_0(y;1)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -2.037~260~79 -27.701~183~5
    10 -2.316~961~97 -66.140~269~8
    25 -2.701~969~41 -202.538~626
    50 -2.998~691~42 -463.105~636
    75 -3.173~362~18 -746.232~874
    100 -3.297~603~19 -1~044.084~34
    下载: 导出CSV

    表  2  Prony-like方法和基于傅里叶级数逼近${J}_0(y;5) $的最大相对误差

    Tab.  2  Maximal logarithmic relative error of $m$-th approximants of ${J}_0(y;5)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -0.659 608 29 -13.434 112 0
    10 -0.924 782 74 -37.878 176 8
    25 -1.304 994 47 -132.322 355
    50 -1.600 954 77 -322.979 580
    75 -1.775 461 29 -536.214 948
    100 -1.899 628 47 -764.148 583
    下载: 导出CSV

    表  3  Prony-like方法和基于傅里叶级数逼近 ${J}_0(y;20)$ 的最大相对误差

    Tab.  3  Maximal logarithmic relative error of $m$-th approximants of ${J}_0(y;20)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -0.272 889 33 -2.573 217 82
    10 -0.333 740 91 -14.544 573 7
    25 -0.249 573 16 -72.519 517 0
    50 -0.436 180 51 -202.846 437
    75 -0.589 057 60 -355.833 679
    100 -0.705 527 47 -523.539 798
    下载: 导出CSV

    表  4  Prony-like方法和基于傅里叶级数逼近${J}_1(y;1)$的最大相对误差

    Tab.  4  Maximal logarithmic relative error of $m$-th approximants of ${J}_1(y;1)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -2.514 252 75 -28.212 755 5
    10 -2.794 047 38 -66.653 094 0
    25 -3.179 084 59 -203.051 977
    50 -3.475 811 15 -463.619 129
    75 -3.650 482 78 -746.746 445
    100 -3.774 724 09 -1 044.597 91
    下载: 导出CSV

    表  5  Prony-like方法和基于傅里叶级数逼近$J_1(y;5)$的最大相对误差

    Tab.  5  Maximal logarithmic relative error of $m$-th approximants of $J_1(y;5)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -0.796 908 90 -13.224 032 1
    10 -1.068 388 19 -37.671 450 6
    25 -1.450 689 03 -132.126 135
    50 -1.746 972 06 -322.788 583
    75 -1.921 539 77 -536.026 002
    100 -2.045 728 54 -763.960 797
    下载: 导出CSV

    表  6  Prony-like方法和基于傅里叶级数逼近$J_1(y;20)$的最大相对误差

    Tab.  6  Maximal logarithmic relative error of $m$-th approximants of $J_1(y;20)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -0.067 765 28 -2.114 145 13
    10 -0.040 342 38 -14.080 735 0
    25 0.280 248 722 -72.064 701 9
    50 -0.207 786 81 -202.393 867
    75 -0.400 047 88 -355.381 787
    100 -0.638 124 17 -523.088 228
    下载: 导出CSV

    表  7  Prony-like方法和基于傅里叶级数逼近$J_2(y;1)$的最大相对误差

    Tab.  7  Maximal logarithmic relative error of $m$-th approximants of $J_2(y;1)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -1.476 830 29 -30.312 522 8
    10 -1.614 937 91 -69.033 081 8
    25 -1.807 076 01 -205.818 481
    50 -1.956 252 46 -466.682 748
    75 -2.044 453 72 -749.983 714
    100 -2.107 369 26 -1 047.960 29
    下载: 导出CSV

    表  8  Prony-like方法和基于傅里叶级数逼近$J_2(y;5)$的最大相对误差

    Tab.  8  Maximal logarithmic relative error of $m$-th approximants of $J_2(y;5)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -0.140 752 61 -14.992 899
    10 -0.245 484 98 -39.695 214
    25 -0.418 134 08 -135.549 74
    50 -0.560 357 82 -325.464 58
    75 -0.645 798 60 -538.873 75
    100 -0.707 046 03 -768.597 60
    下载: 导出CSV

    表  9  Prony-like方法和基于傅里叶级数逼近$J_2(y;20)$的最大相对误差

    Tab.  9  Maximal logarithmic relative error of $m$-th approximants of $J_2(y;20)$ via Fourier-based method and Prony-like method

    项数 方法
    傅里叶级数展开 Prony-cos
    5 -0.116 706 38 -2.627 610 0
    10 -0.038 715 01 -14.892 997
    25 0.488 410 80 -73.260 577
    50 0.525 586 10 -203.882 33
    75 0.486 685 31 -357.043 00
    100 0.446 139 68 -524.872 56
    下载: 导出CSV

    表  10  Prony-like方法和基于傅里叶级数的计算时间对比(时间单位: s)

    Tab.  10  Calculation times of Fourier-based method and Prony-like method

    逼近函数 方法 [0, 1] [0, 5]
    5 25 50 5 25 50
    $J_0(y;b)$ 傅里叶方法 0.001 0.005 0.006 0.001 0.003 0.005
    Prony-cos 0.156 9.689 86.956 0.168 9.765 86.491
    $J_1(y;b)$ 傅里叶方法 0.001 0.007 0.008 0.001 0.005 0.006
    Prony-cos 0.159 9.942 91.406 0.198 8.795 89.188
    $J_2(y;b)$ 傅里叶方法 0.001 0.006 0.008 0.001 0.007 0.008
    Prony-sin 0.193 9.865 105.718 0.154 9.781 117.422
    下载: 导出CSV
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  • 收稿日期:  2018-11-30
  • 刊出日期:  2019-11-25

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