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Algorithms and codes for the Macdonald function: recent progress and comparisons

JOURNAL OF computationAL AND APPLIED MATHEMATICS 2003年第1期161卷 179-192页

作者： Fabijonas, BR Lozier, DW Rappoport, JM So Methodist Univ Dept Math Dallas TX 75275 USA NIST Informat Technol Lab Gaithersburg MD 20899 USA Russian Acad Sci Moscow 117335 Russia

The modified Bessel function K-iv(x), also known as the Macdonald function, finds application in the Kontorovich-Lebedev integral transform when x and v are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared.

关键词： Macdonald functions Bessel functions computation of special functions

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Locating and computing in parallel all the simple roots of special functions using PVM

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JOURNAL OF computationAL AND APPLIED MATHEMATICS 2001年第1-2期133卷 545-554页

作者： Plagianakos, VP Nousis, NK Vrahatis, MN Univ Patras Dept Math UPAIRC GR-26500 Patras Greece

An algorithm is proposed for locating and computing in parallel and with certainty all the simple roots of any twice continuously differentiable function in any specific interval. To compute with certainty all the roots, the proposed method is heavily based on the knowledge of the total number of roots within the given interval. To obtain this information we use results from topological degree theory and, in particular, the Kronecker-Picard approach. This theory gives a formula for the computation of the total number of roots of a system of equations within a given region, which can be computed in parallel, With this tool in hand, we construct a parallel procedure for the localization and isolation of all the roots by dividing the given region successively and applying the above formula to these subregions until the final domains contain at the most one root. The subregions with no roots are discarded, while for the rest a modification of the well-known bisection method is employed for the computation of the contained root. The new aspect of the present contribution is that the computation of the total number of zeros using the Kronecker-Picard integral as well as the localization and computation of all the roots is performed in parallel using the parallel virtual machine (PVM). PVM is an integrated set of software tools and libraries that emulates a general-purpose, flexible, heterogeneous concurrent computing framework on interconnected computers of varied architectures. The proposed algorithm has large granularity and low synchronization, and is robust. It has been implemented and tested and our experience is that it can massively compute with certainty all the roots in a certain interval. Performance information from massive computations related to a recently proposed conjecture due to Elbert (this issue, J. Comput. Appl. Math. 133 (2001) 65-83) is reported. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： Elbert's conjecture special functions Bessel functions computation of special functions construction of tables of special functions parallel and distributed algorithms parallel virtual machine zero isolation Kronecker-Picard theory topological degree theory computing simple roots bisection method

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Convergence acceleration of Taylor sections by convolution

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CONSTRUCTIVE APPROXIMATION 1999年第4期15卷 523-536页

作者： Muller, J Univ Trier FB 4 D-54286 Trier Germany

Taylor sections S-n(f) of an entire function f often provide easy computable polynomial approximants of f. However, while the rate of convergence of (S-n(f))(n) is nearly optimal on circles around the origin, this is no longer true for other plane sets as, for example, real compact intervals. The aim of this paper is to construct for certain families of (entire) functions sequences of polynomial approximants which are computable with essentially the same effort as Taylor sections and which have a better rate of convergence on some parts of the plane. The resulting method may be applied, for example, to (modified) Bessel functions, to confluent hypergeometric functions, or to parabolic cylinder functions.

关键词： convolution Hadamard product convergence acceleration computation of special functions

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Numerical algorithms for uniform Airy-type asymptotic expansions

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NUMERICAL ALGORITHMS 1997年第2期15卷 207-225页

作者： Temme, NM CWI NL-1090 GB AMSTERDAMNETHERLANDS

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.

关键词： uniform asymptotic expansions turning points Airy-type expansions Bessel functions computation of special functions

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Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

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NUMERISCHE MATHEMATIK 2010年第2期116卷 269-289页

作者： Luis Lopez, Jose Temme, Nico M. Univ Publ Navarra Dept Ingn Matemat & Informat Pamplona 31006 Spain Ctr Wiskunde & Informat NL-1098 XG Amsterdam Netherlands

Expansions in terms of Bessel functions are considered of the Kummer function (1) F (1)(a;c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.

关键词： Kummer functions Confluent hypergeometric functions Bessel functions Asymptotic expansions computation of special functions

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Numerical methods for the computation of the confluent and Gauss hypergeometric functions

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NUMERICAL ALGORITHMS 2017年第3期74卷 821-866页

作者： Pearson, John W. Olver, Sheehan Porter, Mason A. Univ Kent Sch Math Stat & Actuarial Sci Cornwallis Bldg East Canterbury CT2 7NF Kent England Univ Sydney Sch Math & Stat Sydney NSW 2006 Australia Univ Oxford Math Inst Oxford Ctr Ind & Appl Math Andrew Wiles BldgWoodstock Rd Oxford OX2 6GG England

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss-Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide "roadmaps" with our recommendation for which methods should be used in each situation.

关键词： computation of special functions Confluent hypergeometric function Gauss hypergeometric function

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Parabolic cylinder functions of large order

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JOURNAL OF computationAL AND APPLIED MATHEMATICS 2006年第1-2期190卷 453-469页

作者： Jones, DS Univ Dundee Dept Math Dundee Scotland

The asymptotic behaviour of parabolic cylinder functions of large real order is considered. Various expansions in terms of elementary functions are derived. They hold uniformly for the variable in appropriate parts of the complex plane. Some of the expansions are doubly asymptotic with respect to the order and the complex variable which is an advantage for computational purposes. Error bounds are determined for the truncated versions of the asymptotic series. (c) 2005 Published by Elsevier B.V.

关键词： parabolic cylinder functions uniform asymptotic expansions computation of special functions

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Parabolic cylinder functions of large order

Parabolic cylinder functions of large order

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International Conference on Mathematics and Its Applications

作者： Jones, DS Univ Dundee Dept Math Dundee Scotland

关键词： parabolic cylinder functions uniform asymptotic expansions computation of special functions

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computation of the reverse generalized Bessel polynomials and their zeros

COMPUTATIONAL AND MATHEMATICAL METHODS

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computationAL AND MATHEMATICAL METHODS 2021年第6期3卷

作者： Dunster, T. Mark Gil, Amparo Ruiz-Antolin, Diego Segura, Javier San Diego State Univ Dept Math & Stat San Diego CA 92182 USA Univ Cantabria Dept Matemat Aplicada & CC Comp ETSI Caminos Santander 39005 Spain Univ Cantabria Dept Matemat Estadist & Comp Santander Spain

It is well known that one of the most relevant applications of the reverse Bessel polynomials theta n(z) is filter design. In particular, the poles of the transfer function of a Bessel filter are basically the zeros of theta n(z). In this article we discuss an algorithm to compute the zeros of reverse generalized Bessel polynomials theta n(z;a). A key ingredient in the algorithm will be a method to compute the polynomials. For this purpose, we analyze the use of recurrence relations and asymptotic expansions in terms of elementary functions to obtain accurate approximations to the polynomials. The performance of all the numerical approximations will be illustrated with examples.

关键词： computation of special functions numerical approximations reverse Bessel polynomials

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