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The gamma function via interpolation

Abstract

A new computational framework for evaluation of the gamma function Γ(z) over the complex plane is developed. The algorithm is based on interpolation by rational functions, and generalizes the classical methods of Lanczos (SIAM J. Numer. Anal. B 1:86–96, (1964) and Spouge (SIAM J. Numer. Anal., 31(3):931–944, (1994) (which we show are also interpolatory). This framework utilizes the exact poles of the gamma function. By relaxing this condition and allowing the poles to vary, a near-optimal rational approximation is possible, which is demonstrated using the adaptive Antoulous Anderson (AAA) algorithm, developed in Nakatsukasa et al. (Appl. Math. Comp., 40:1494-1522, (2016)) and Nakatsukasa and Trefethen (SIAM J. Sci. Comp., 42(5):A3157–A3179, (2020)). The resulting approximations are competitive with Stirling’s formula in terms of overall efficiency.

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Notes

  1. 1.

    Despite the mismatch in arguments, Legendre’s notation Γ(z) has prevailed over Gauss’ notation π(z) = Γ(z + 1), which satisfies π(n) = n!. Although in certain cases [19, 38] the notation z! = Γ(z + 1) is instead adopted, to avoid confusion.

  2. 2.

    The integral of the first kind defines the beta function B(α,β) = Γ(α)Γ(β)/Γ(α + β).

  3. 3.

    in fact, this series is due to De Moivre. The series obtained by Stirling is

    $$ \ln z! \sim \ln(\sqrt{2\pi})+ Z\ln Z-Z+\sum\limits_{k\geq 1} \frac{B_{2k}\left( \frac{1}{2}\right)}{2k(2k-1)} \frac{1}{Z^{2k-1}}, \quad Z = z+\frac{1}{2}, $$

    which is slightly more accurate. See [11, 14].

  4. 4.

    Named after Jacob Bernoulli; Daniel was his nephew.

  5. 5.

    At least not explicitly. But this property is used indirectly in [30] and [22] to compute expansion coefficients.

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Acknowledgements

The author is very grateful to an anonymous peer reviewer for several key suggestions that greatly improved the quality of this manuscript. The author is especially indebted for the suggestion to examine the AAA algorithm.

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Correspondence to Matthew F. Causley.

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Causley, M.F. The gamma function via interpolation. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01204-8

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Keywords

  • Gamma function
  • Lanczos approximation
  • Spouge approximation
  • Stirling series
  • Interpolation
  • AAA approximation