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Computing hypergeometric functions rigorously

Fredrik Johansson 1
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function pFq and computation of high-order parameter derivatives.
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https://hal.inria.fr/hal-01336266
Contributor : Fredrik Johansson <>
Submitted on : Wednesday, June 22, 2016 - 5:39:44 PM
Last modification on : Thursday, January 11, 2018 - 6:22:36 AM

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  • HAL Id : hal-01336266, version 1

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Fredrik Johansson. Computing hypergeometric functions rigorously. 2016. ⟨hal-01336266v1⟩

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