On evaluation of Bessel functions of the first kind via Prony-like methods
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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.
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关键词:
- 贝塞尔函数 /
- Prony-like方法 /
- 三角函数 /
- 基于傅里叶级数展开 /
- 数值逼近
Abstract: Numerical approximations of Bessel functions are both of important theoretical significance and widely applied in mathematics, physics, engineering. In this work, we apply two variants of Prony's method on Bessel functions of the first kind of integer order. The Prony-like methods in cosine or sine version yield approximations as sums of sinusoidal functions of Bessel functions of the first kind of integer order. In the symbolic computation software Maple, we compute the approximations for different orders and over different intervals, and compare these approximations with those obtained through the Fourier method. Experiments show that Prony-like methods perform much better than the Fourier method. -
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 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 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 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 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 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 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 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 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 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 -
[1] ABRAMOWITZ M, STEGUN I A, ROMAIN J E. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[M]. New York:Dover, 1964. [2] LOZIER D W. NIST Digital library of mathematical functions[J]. Annals of Mathematics & Artificial Intelligence, 2003, 38(1/3):105-119. http://cn.bing.com/academic/profile?id=de8e2c4849cc966624702bd2ca0d95d2&encoded=0&v=paper_preview&mkt=zh-cn [3] 常晓阳.几类特殊函数的赋值分析研究[D].上海: 华东师范大学, 2018. http://cdmd.cnki.com.cn/Article/CDMD-10269-1018821497.htm [4] MILLANE R P, EADS J L. Polynomial approximations to Bessel functions[J]. IEEE Transactions on Antennas & Propagation, 2003, 51(1):1398-1400. http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_1101.2335 [5] MATVIYENKO G. On the evaluation of Bessel functions[J]. Applied and Computational Harmonic Analysis, 1993, 1(1):116-135. doi: 10.1006/acha.1993.1009 [6] ANDRUSYK A. Infinite series representations for Bessel functions of the first kind of integer order[EB/OL].[2018-10-20]. https://arxiv.org/abs/1210.2109 [7] HILDEBRAND F B. Introduction to Numerical Analysis[M]. New Delhi:Tata McGraw-Hill Publishing Company Limited, 1956. [8] BOßMANN F, PLONKA G, PETER T, et al. Sparse deconvolution methods for ultrasonic NDT[J]. Journal of Nondestructive Evaluation, 2012, 31(3):225-244. doi: 10.1007/s10921-012-0138-8 [9] ROY R, PAULRAJ A, KAILATH T. Estimation of signal parameters via rotational invariance techniquesESPRIT[C]//Proceedings of the SPIE. International Society for Optics and Photonics, 1986:94-101. https://academic.oup.com/imaiai [10] HUA Y, SARKAR T K. Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990, 38(5):814-824. doi: 10.1109/29.56027 [11] POTTS D, TASCHE M. Nonlinear approximation by sums of nonincreasing exponentials[J]. Applicable Analysis, 2011, 90(3/4):18. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=10.1080/00036810903569499 [12] YANAI T, FANN G I, GAN Z, et al. Multiresolution quantum chemistry in multiwavelet bases:Analytic derivatives for Hartree-Fock and density functional theory[J]. Journal of Chemical Physics, 2004, 121(7):2866-2876. doi: 10.1063/1.1768161 [13] BEYLKIN G, CRAMER R, FANN G, et al. Multiresolution separated representations of singular and weakly singular operators[J]. Applied & Computational Harmonic Analysis, 2007, 23(2):235-253. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=3740f9033b7e959718b00591a6455db9 [14] HANKE M. One shot inverse scattering via rational approximation[J]. Siam Journal on Imaging Sciences, 2012, 5(1):465-482. doi: 10.1137/110823985 [15] GOLUB G H, MILANFAR P, VARAH J. A Stable Numerical Method for Inverting Shape from Moments[M]. Society for Industrial and Applied Mathematics, 1999. [16] GIESBRECHT M, LABAHN G, LEE W S. Symbolic-numeric sparse polynomial interpolation in Chebyshev basis and trigonometric interpolation[C]//Proceedings of Computer Algebra in Scientific Computing (CASC 2004), 2004:195206. [17] CUYT A, LEE W S. Generalized eigenvalue relations:Spread polynomials and sine[R]. 2018.