Abstract
An algorithm is presented for computing verified error bounds for the value of the real gamma function. It has been shown that the double exponential formula is one of the most efficient methods for calculating integrals of the form. Recently, an useful evaluation based on the double exponential formula over the semi-infinite interval has been proposed. However, the evaluation would be overflow when applied to the real gamma function directly. In this paper, we present a theorem so as to overcome the problem in such a case. Numerical results are presented for illustrating effectiveness of the proposed theorem in terms of the accuracy of the calculation.
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References
Rump, S.M.: Verified sharp bounds for the real gamma function over the entire floating-point range. Nonlinear Theor. Appl. IEICE 5, 339–348 (2014)
Kashiwagi, M.: Verified algorithm for special functions (in Japanese). http://verifiedby.me/kv/special/
Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Publ. RIMS Kyoto Univ. 9, 721–741 (1974)
Okayama, T.: Error estimates with explicit constants for Sinc quadrature and Sinc indefinite integration over infinite intervals. Reliable Comput. 19, 45–65 (2013)
kv Library. http://verifiedby.me/kv/
The GNU MPFR Library: http://www.mpfr.org/
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© 2016 Springer International Publishing Switzerland
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Yamanaka, N., Okayama, T., Oishi, S. (2016). Verified Error Bounds for the Real Gamma Function Using Double Exponential Formula over Semi-infinite Interval. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_19
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DOI: https://doi.org/10.1007/978-3-319-32859-1_19
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