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Multidomain spectral method for the Gauss hypergeometric function

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Abstract

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 \(\mathbb {R}\cup {\infty }\), 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.

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Notes

  1. The Kummer group of symmetries of the hypergeometric equation is essentially generated by Moebius transformations which leave the set of singularities 0, 1, ∞ invariant; note that such a symmetry does not exist for Fuchsian equations with four singularities, but our approach can still be applied as discussed in Section 6.

  2. In Fig. 5, the results for the Fourier spectral method are plotted against N and not n, as the axis label indicates.

  3. The next section gives an example where the condition numbers of the system (12) become large. For the example considered in Fig. 11, for 𝜖 = 10− 1, 10− 5, 10− 10, the condition numbers of the system (18)–(19) are on the order of 104, 108 and 1013, respectively, and those of (20)–(21) are on the order of 106, 1014 and 1024, respectively.

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Acknowledgements

We thank C. Lubich for helpful remarks.

Funding

This work was partially supported by the PARI and FEDER programs in 2016 and 2017, by the ANR-FWF project ANuI by the isite BFC via the project NAANoD and by the Marie-Curie RISE network IPaDEGAN. M. Fasondini acknowledges financial support from the EPSRC grant EP/P026532/1.

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  1. C. Vallée

    Present address: Univ. Rennes, Inria, CNRS, IRISA, Inserm, Empenn U1228, Campus de Beaulieu, F-35042, Rennes, France

Authors and Affiliations

  1. Institut de Mathématiques de Bourgogne, Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, BP 47870, 21078, Dijon Cedex, France

    S. Crespo, C. Klein, N. Stoilov & C. Vallée

  2. School of Mathematics, Statistics and Actuarial Science, University of Kent, Sibson Building, Parkwood Road, Canterbury, Kent, CT2 7FS, UK

    M. Fasondini

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  1. S. Crespo
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Correspondence to C. Klein.

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Crespo, S., Fasondini, M., Klein, C. et al. Multidomain spectral method for the Gauss hypergeometric function. Numer Algor 84, 1–35 (2020). https://doi.org/10.1007/s11075-019-00741-7

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