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Multidomain spectral method for the Gauss hypergeometric function

Crespo, S., Fasondini, M., Klein, C., Stoilov, N., Vallée, C. (2019) Multidomain spectral method for the Gauss hypergeometric function. Numerical Algorithms, . pp. 1-35. ISSN 1017-1398. (doi:10.1007/s11075-019-00741-7) (KAR id:75517)


We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line R∪∞ , except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible.

Item Type: Article
DOI/Identification number: 10.1007/s11075-019-00741-7
Uncontrolled keywords: Hypergeometric function, Singular differential equations, Spectral methods
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Marco Fasondini
Date Deposited: 23 Jul 2019 15:16 UTC
Last Modified: 17 Jun 2020 23:00 UTC
Resource URI: (The current URI for this page, for reference purposes)
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