On computation of the Bessel function by summing up the series

E. A. Karatsuba

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 4, pp. 453-472.

Résumé

Two algorithms for an effective calculation of the Bessel function are presented: a fast algorithm with an increasing accuracy of computation and a computational algorithm for the case of a large argument of the Bessel function.

Bibliographie

[1] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922 | MR | Zbl

[2] F. W.J. Olver, “The asymptotic expansion of Bessel functions of large order”, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 247:930 (1954), 328–368 | DOI | MR

[3] H. C. Thacher, “New backward recurrences for Bessel functions”, Mathematics of Computation, 33:146 (1979), 744–764 | DOI | MR | Zbl

[4] P. Baratella, M. Garetto, G. Vinardi, “Approximation of the Bessel function $J_{\nu}(x)$ by numerical integration”, J. of Computational and Applied Mathematics, 7:2 (1981), 87–91 | DOI | MR | Zbl

[5] K. Balla, O. S. Guk, M. Vicsek, “On the computation of Bessel functions of first kind”, Computing, 50:1 (1993), 77–85 | DOI | MR | Zbl

[6] G. Matviyenko, “On the evaluation of Bessel functions”, Applied and Computational Harmonic Analysis, 1:1 (1993), 116–135 | DOI | MR | Zbl

[7] N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 1996 | MR | Zbl

[8] W. Gautschi, “Computation of Bessel and Airy functions and of related Gaussian quadrature formulae”, BIT, 42:1 (2002), 110–118 | DOI | MR | Zbl

[9] D. Borwein, J. M. Borwein, Y. Chan, “The evaluation of Bessel functions via exparc integrals”, J. of Mathematical Analysis and Applications, 341:1 (2008), 478–500 | DOI | MR | Zbl

[10] R. B. Paris, “High-precision evaluation of the Bessel functions via Hadamard series”, J. of Computational and Applied Mathematics, 224:1 (2009), 84–100 | DOI | MR | Zbl

[11] J. Harrison, “Fast and Accurate Bessel Function Computation”, Proc. of ARITH19, the 19th IEEE Conference on Computer Arithmetic, IEEE Computer Society Press, 2009, 104–113

[12] U. D. Jentschura, E. Lötstedt, “Numerical calculation of Bessel, Hankel and Airy functions”, Computer Physics Communications, 183:3 (2012), 506–519 | DOI | MR | Zbl

[13] I. Krasikov, “Approximations for the Bessel and Airy functions with an explicit error term”, LMS J. of Computation and Mathematics, 17:1 (2014), 209–225 | DOI | MR | Zbl

[14] Z. Heitman, J. Bremer, V. Rokhlin, B. Vioreanu, “On the asymptotics of Bessel functions in the Fresnel regime”, Applied and Computational Harmonic Analysis, 39:2 (2015), 347–356 | DOI | MR | Zbl

[15] J. Bremer, “An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments”, Advances in Computational Mathematics, 45:1 (2019), 173–211 | DOI | MR | Zbl

[16] C. A. Karatsuba, “Fast evaluation of Bessel functions”, Integral Transforms and Special Functions, 1:4 (1993), 269–276 | DOI | MR | Zbl

[17] Karatsuba A. A., “The complexity of computations”, Proc. Steklov Inst. Math., 211 (1995), 169–183 | MR | Zbl

[18] Karatsuba A. A., “Comments to my works”, Proc. Steklov Inst. Math., 282, suppl. 1 (2013), S1–S23 | DOI | DOI | MR | Zbl

[19] Kolmogorov A. N., Information theory and the theory of algorithms, Kluwer Academic Publishers, 1993 | MR | MR | Zbl

[20] Karatsuba A. A., Ofman Yu. P., “Multiplication of multidigit numbers on automata”, Soviet Physics-Doklady, 7 (1963), 595–596 | MR

[21] J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series”, Math. Comput., 19 (1965), 297–301 | DOI | MR | Zbl

[22] Toom A., “The complexity of a scheme of functional elements realizing the multiplication of integers”, Soviet Mathematics-Doklady, 3 (1963), 714–716 | MR | Zbl

[23] A. Schönhage, V. Strassen, “Schnelle Multiplikation grosser Zahlen”, Computing, 7 (1971), 281–292 | DOI | MR | Zbl

[24] A. Schönhage, A. F. W. Grotefeld, E. Vetter, Fast Algorithms A Multitape Turing Machine Implementation, BI-Wiss. Verl., Zürich, 1994 | MR | Zbl

[25] M. Fürer, “Faster integer multiplication”, SIAM J. on Computing, 39:3 (2009), 979–1005 | DOI | MR | Zbl

[26] V. Strassen, “Gaussian elimination is not optimal”, J. Numer. Math., 13 (1969), 354–356 | DOI | MR | Zbl

[27] D. Coppersmith, S. Winograd, “On the asymptotic complexity of matrix multiplication”, SIAM J. on Computing, 11:3 (1982), 472–492 | DOI | MR | Zbl

[28] B. C. Carlson, “Algorithms involving arithmetic and geometric means”, Amer. Math. Monthly, 78 (1971), 496–505 | DOI | MR | Zbl

[29] B. C. Carlson, “An algorithm for computing logarithms and arctangents”, Math. Comp., 26:118 (1972), 543–549 | DOI | MR | Zbl

[30] E. Salamin, “Computation $\pi$ of using arithmetic-geometric mean”, Math. Comp., 30:135 (1976), 565–570 | MR | Zbl

[31] J. M. Borwein, P. B. Borwein, Pi and the AGM — A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987 | MR | Zbl

[32] Karatsuba E. A., “On a new method for fast evaluation of transcendental functions”, Russian Math. Surveys, 46:2 (1991), 246–247 | DOI | MR | Zbl

[33] Karatsuba E. A., “Fast evaluation of transcendental functions”, Problems Inform. Transmission, 27:4 (1991), 339–360 | MR | Zbl

[34] Karatsuba E. A., “Fast calculation of the Riemann zeta function $\zeta(s)$ for integer values of the argument $s$”, Problems Inform. Transmission, 31:4 (1995), 353–362 | MR | Zbl

[35] Karatsuba E. A., “Fast evaluation of the Hurwitz zeta function and Dirichlet $L$-series”, Problems Inform. Transmission, 34:4 (1998), 342–353 | DOI | MR | Zbl

[36] E. A. Karatsuba, “Fast evaluation of hypergeometric function by FEE”, Computational Methods and Function Theory, eds. N. Papamichael, St. Ruscheweyh, E. B. Saff, World Sc. Pub., 1999, 303–314 | MR | Zbl

[37] E. A. Karatsuba, “On the computation of the Euler constant $\gamma$”, J. of Numerical Algorithms, 24 (2000), 83–97 | DOI | MR | Zbl

[38] E. A. Karatsuba, “Fast computation of $\zeta(3)$ and of some special integrals, using the polylogarithms, the Ramanujan formula and it's generalization”, BIT Numerical Mathematics, 41:4 (2001), 722–730 | DOI | MR | Zbl

[39] E. A. Karatsuba, “Fast computation of some special integrals of mathematical physics”, Scientific Computing, Validated Numerics, Interval Methods, eds. W. Krämer, J. W. von Gudenberg, 2001, 29–41 | DOI

[40] Karatsuba E. A., “Fast Catalan constant computation via the approximations obtained by the Kummer's type transformations”, Discrete Math. Appl., 23:5–6 (2013), 429–443 | DOI | MR | Zbl

[41] Karatsuba E. A., “On one method for fast approximation of zeta constants by rational fractions”, Problems Inform. Transmission, 50:2 (2014), 186–202 | DOI | MR | Zbl

[42] E. A. Karatsuba, “On the asymptotic representation of the Euler gamma function by Ramanujan”, J. Comput. Appl. Math., 135:2 (2001), 225–240 | DOI | MR | Zbl