Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval

Naoya Yamanaka, Tomoaki Okayama, Shinichi Oishi

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    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.

    Original languageEnglish
    Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    PublisherSpringer Verlag
    Pages224-228
    Number of pages5
    Volume9582
    ISBN (Print)9783319328584
    DOIs
    Publication statusPublished - 2016
    Event6th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2015 - Berlin, Germany
    Duration: 2015 Nov 112015 Nov 13

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume9582
    ISSN (Print)03029743
    ISSN (Electronic)16113349

    Other

    Other6th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2015
    CountryGermany
    CityBerlin
    Period15/11/1115/11/13

    Fingerprint

    Infinite Interval
    Gamma function
    Error Bounds
    Overflow
    Evaluation
    Theorem
    Numerical Results
    Computing
    Form

    Keywords

    • Double exponential formula
    • Gamma function
    • Verified bound

    ASJC Scopus subject areas

    • Computer Science(all)
    • Theoretical Computer Science

    Cite this

    Yamanaka, N., Okayama, T., & Oishi, S. (2016). Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9582, pp. 224-228). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9582). Springer Verlag. https://doi.org/10.1007/978-3-319-32859-1_19

    Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval. / Yamanaka, Naoya; Okayama, Tomoaki; Oishi, Shinichi.

    Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9582 Springer Verlag, 2016. p. 224-228 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9582).

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Yamanaka, N, Okayama, T & Oishi, S 2016, Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 9582, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9582, Springer Verlag, pp. 224-228, 6th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2015, Berlin, Germany, 15/11/11. https://doi.org/10.1007/978-3-319-32859-1_19
    Yamanaka N, Okayama T, Oishi S. Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9582. Springer Verlag. 2016. p. 224-228. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-32859-1_19
    Yamanaka, Naoya ; Okayama, Tomoaki ; Oishi, Shinichi. / Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9582 Springer Verlag, 2016. pp. 224-228 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
    @inproceedings{6aa71aeadac9498ca6bb4d74183e1c3b,
    title = "Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval",
    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.",
    keywords = "Double exponential formula, Gamma function, Verified bound",
    author = "Naoya Yamanaka and Tomoaki Okayama and Shinichi Oishi",
    year = "2016",
    doi = "10.1007/978-3-319-32859-1_19",
    language = "English",
    isbn = "9783319328584",
    volume = "9582",
    series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
    publisher = "Springer Verlag",
    pages = "224--228",
    booktitle = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

    }

    TY - GEN

    T1 - Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval

    AU - Yamanaka, Naoya

    AU - Okayama, Tomoaki

    AU - Oishi, Shinichi

    PY - 2016

    Y1 - 2016

    N2 - 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.

    AB - 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.

    KW - Double exponential formula

    KW - Gamma function

    KW - Verified bound

    UR - http://www.scopus.com/inward/record.url?scp=84964059135&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84964059135&partnerID=8YFLogxK

    U2 - 10.1007/978-3-319-32859-1_19

    DO - 10.1007/978-3-319-32859-1_19

    M3 - Conference contribution

    AN - SCOPUS:84964059135

    SN - 9783319328584

    VL - 9582

    T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

    SP - 224

    EP - 228

    BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

    PB - Springer Verlag

    ER -