The Chandrasekhar function revisited

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The need for an accurate (better than 10 significant digits) and fast algorithm for calculating the Chandrasekhar function, H(μ,ω), has stimulated the present analysis of different solutions of the relevant integral equation. It has been found that a very accurate analytical solution can be derived that is conveniently used in the range of small arguments, μ and ω. In a limited range of arguments, the H function can be expressed in terms of a rapidly converging series of Bernoulli constants. For example, the H function for μ=1 and ω=1 was readily calculated with an accuracy of 31 digits. A new algorithm, derived from an integral representation, is proposed for efficient calculations. Together with an algorithm published by Stibbs and Weir (1959), this algorithm was used in calculations of extensive tables of the H function with an accuracy of 21 significant digits. Based on the above analysis, a mixed algorithm optimized with respect to the execution time was designed. Program summaryProgram title: CHANDRAS_MIXCatalogue identifier: AEWW_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEWW_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1505No. of bytes in distributed program, including test data, etc.: 14655Distribution format: tar.gzProgramming language: FORTRAN 90.Computer: Any.Operating system: Windows XP, Windows 7, Windows 8, Linux.RAM: 1.0 MbClassification: 2.4, 7.2.Nature of problem: Algorithms derived from an integral representation of the H function generally exhibit a slow convergence which may considerably delay calculations involving integration of functions containing the H function. Furthermore, a set of reference values of the H function of a very high accuracy is useful in analysis of performance of different algorithms, and thus a relevant computational procedure is needed for that purpose.Solution method: Problem of slow convergence is circumvented by the derivation and use of an analytical solution sufficiently accurate in the range of small arguments. In the range of arguments exceeding 0.05, a new integral representation is derived that is rapidly converging and can be easily adjusted to calculations with accuracy of 21 significant digits. A mixed algorithm is constructed that is optimized with respect to the execution time.Restrictions: The arguments for the Chandrasekhar function, x and omega (notation used in the code), are restricted to the range: 0<=x<=1 and 0<=omega<=1.Unusual features: The Gauss–Legendre quadrature is used in calculations of the H function from different integral representations. The optimum number of abscissas, N, was found to be equal to 20. To control accuracy, the bipartition approach is used, i.e., calculations are repeated after halving the integration interval until the desired accuracy is reached.Running time: On average, about 11μs for both arguments of the Chandrasekhar function exceeding 0.05.