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Computing Hypergeometric Functions Rigorously

ACM Transactions on Mathematical SoftwareVolume 45Issue 3Article No.: 30pp 1–26https://doi.org/10.1145/3328732
Published:08 August 2019Publication History
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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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