Calculation of Bessel Functions via the Summation of Series

Pleiades Publishing Ltd - Tập 12 - Trang 372-387 - 2020
E. A. Karatsuba1
1Federal Research Center “Informatics and Management” of the Russian Academy of Sciences, Dorodnitsyn Computing Center, Moscow, Russia

Tóm tắt

Two algorithms for efficient calculation of Bessel functions are presented: a fast algorithm with an increasing accuracy of computation and a computational algorithm for Bessel functions with large arguments.

Tài liệu tham khảo

Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge: Cambridge University Press, 1922.

Olver, F.W.J., The Asymptotic Expansion of Bessel Functions of Large Order, Philos. Trans. Royal Soc. London. Ser. A. Math. Phys. Sci., 1954, vol. 247, no. 930, pp. 328–368.

Thacher, H.C., New Backward Recurrences for Bessel Functions, Math. Comput., 1979, vol. 33, no. 146, pp. 744–764.

Baratella, P., Garetto, M., and Vinardi, G., Approximation of the Bessel Function Jν(x) by Numerical Integration, J. Comput. Appl. Math., 1981, vol. 7, no. 2, pp. 87–91.

Balla, K., Guk, O.S., and Vicsek, M., On the Computation of Bessel Functions of First Kind, Computing, 1993, vol. 50, no. 1, pp. 77–85.

Matviyenko, G., On the Evaluation of Bessel Functions, Appl. Comput. Harm. An., 1993, vol. 1, no. 1, pp. 116–135.

Temme, N.M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics, New York: Wiley, 1996.

Gautschi, W., Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae, BIT, 2002, vol. 42, no. 1, pp. 110–118.

Borwein, D., Borwein, J.M., and Chan, Y., The Evaluation of Bessel Functions via Exp-Arc Integrals, J. Math. An. Appl., 2008, vol. 341, no. 1, pp. 478–500.

Paris, R.B., High-Precision Evaluation of the Bessel Functions via Hadamard Series, J. Comput. Appl. Math., 2009, vol. 224, no. 1, pp. 84–100.

Harrison, J., Fast and Accurate Bessel Function Computation, Proc. of ARITH19, the 19th IEEE Conf. on Computer Arithmetic, 2009, pp. 104–113.

Jentschura, U.D. and Lötstedt, E., Numerical Calculation of Bessel, Hankel and Airy Functions, Comp. Phys. Comm., 2012, vol. 183, no. 3, pp. 506–519.

Krasikov, I., Approximations for the Bessel and Airy Functions with an Explicit Error Term, LMS J. Comput. Math., 2014, vol. 17, no. 1, pp. 209–225.

Heitman, Z., Bremer, J., Rokhlin, V., and Vioreanu, B., On the Asymptotics of Bessel Functions in the Fresnel Regime, Appl. Comput. Harm. An., 2015, vol. 39, no. 2, pp. 347–356.

Bremer, J., An Algorithm for the Rapid Numerical Evaluation of Bessel Functions of Real Orders and Arguments, Adv. Comput. Math., 2019, vol. 45, iss. 1, pp. 173–211.

Karatsuba, C.A., Fast Evaluation of Bessel Functions, Integral Transforms Spec. Func., 1993, vol. 1, no. 4, pp. 269–276.

Karatsuba, A.A., The Complexity of Computations, Trudy Steklov Mat. Inst., 1995, vol. 211, pp. 169–183.

Karatsuba, A.A., Comments to My Works, Written by Myself, Trudy Steklov Mat. Inst., 2013, vol. 282, suppl. 1, pp. S1–S23.

Kolmogorov, A.N., Information Theory and the Theory of Algorithms, Kluwer, 1993.

Karatsuba, A.A. and Ofman, Yu.P., Multiplication of Multidigit Numbers on Automata, Dokl. AN SSSR, 1963, vol. 145, no. 2, pp. 293/294.

Cooley, J.W. and Tukey, J.W., An Algorithm for the Machine Calculation of Complex Fourier Series, Math. Comput., 1965, vol. 19, pp. 297–301.

Toom, A., The Complexity of a Scheme of Functional Elements Realizing the Multiplication of Integers, Dokl. ANSSSR, 1963, vol. 150, pp. 496–498.

Schönhage, A. and Strassen, V., Schnelle Multiplikation grosser Zahlen, Computing, 1971, vol. 7, pp. 281–292.

Schönhage, A., Grotefeld, A.F.W., and Vetter, E., Fast Algorithms—A Multitape Turing Machine Implementation, Zürich: BI-Wiss.-Verl., 1994.

Fürer, M., Faster Integer Multiplication, SIAM J. Comput., 2009, vol. 39, no. 3, pp. 979–1005.

Strassen, V., Gaussian Elimination Is not Optimal, J. Num. Math., 1969, vol. 13, pp. 354–356.

Coppersmith, D. and Winograd, S., On the Asymptotic Complexity of Matrix Multiplication, SIAM J. Comput., 1982, vol. 11, no. 3, pp. 472–492.

Carlson, B.C., Algorithms Involving Arithmetic and Geometric Means, Am. Math. Monthly, 1971, vol. 78, pp. 496–505.

Carlson, B.C., An Algorithm for Computing Logarithms and Arctangents, Math. Comp., 1972, vol. 26, no. 118, pp. 543–549.

Salamin, E., Computation of π Using Arithmetic-Geometric Mean, Math. Comp., 1976, vol. 30, no. 135, pp. 565–570.

Borwein, J.M. and Borwein, P.B., Pi and the AGM—A Study in Analytic Number Theory and Computational Complexity, New York: Wiley, 1987.

Karatsuba, E.A., On a New Method for Fast Evaluation of Transcendental Functions, Usp. Mat. Nauk, 1991, vol. 46, no. 2(278), pp. 219/220.

Karatsuba, E.A., Fast Evaluation of Transcendental Functions, Probl. Per. Inform., 1991, vol. 27, no. 4, pp. 87–100.

Karatsuba, E.A., Fast Calculation of the Riemann Zeta Function ζ(s) for Integer Values of the Arguments, Probl. Per. Inform., 1995, vol. 31, no. 4, pp. 69–80.

Karatsuba, E.A., Fast Evaluation of the Hurwitz Zeta Function and Dirichlet L-Series, Probl. Per. Inform., 1998, vol. 34, no. 4, pp. 342–353.

Karatsuba, E.A., Fast Evaluation of Hypergeometric Function by FEE, in Comput. Methods Funct. Theory, Papamichael, N., Ruscheweyh, St., and Saff, E.B., Eds., World Sci. Pub., 1999, pp. 303–314.

Karatsuba, E.A., On the Computation of the Euler Constant γ, J. Num. Algor., 2000, vol. 24, pp. 83–97.

Karatsuba, E.A., Fast Computation of ζ(3) and of Some Special Integrals, Using the Polylogarithms, the Ramanujan Formula and It’s Generalization, BIT Num. Math., 2001, vol. 41, no. 4, pp. 722–730.

Karatsuba, E.A., Fast Computation of Some Special Integrals of Mathematical Physics, in Scientific Computing, Validated Numerics, Interval Methods, Krämer, W. and von Gudenberg, J.W., Eds., 2001, pp. 29–41.

Karatsuba, E.A., Fast Catalan Constant Computation via the Approximations Obtained by the Kummer’s Type Transformations, Discr. Math. Appl., 2013, vol. 23, nos. 5/6, pp. 429–443.

Karatsuba, E.A. On One Method for Fast Approximation of Zeta Constants by Rational Fractions, Probl. Per. Inform., 2014, vol. 50, no. 2, pp. 77–95.

Karatsuba, E.A., On the Asymptotic Representation of the Euler Gamma Function by Ramanujan, J. Comput. Appl. Math., 2001, vol. 135, no. 2, pp. 225–240.

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Pleiades Publishing Ltd

Tập 12

372-387

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