A note on a verified automatic integration algorithm

Naoya Yamanaka, Masahide Kashiwagi, Shinichi Oishi, Takeshi Ogita

    Research output: Contribution to journalArticle

    Abstract

    A verified integration algorithm is proposed for calculating s-dimensional integrals over a finite domain using numerical computations. To construct an efficient verified numerical integrator, the truncation error and the rounding error need to be considered. It has been known that interval arithmetic is one of the most efficient methods of evaluating the rounding error. However, it is much slower than pure floating-point arithmetic, so that in an inclusion algorithm for integrals, the computational effort by the interval arithmetic tends to become a large part. To overcome this problem, an algorithm for evaluating the rounding error using floating-point computations is proposed. The proposed algorithm is based on calculating a priori error bounds for function evaluations and an accurate sum algorithm. With the use of the proposed algorithm and a inclusion algorithm for evaluating the truncation error, we propose an automatic inclusion algorithm. Numerical examples are presented for illustrating the effectiveness of the proposed algorithm.

    Original languageEnglish
    Pages (from-to)156-167
    Number of pages12
    JournalReliable Computing
    Volume15
    Issue number2
    Publication statusPublished - 2011 Jun

    Fingerprint

    Rounding error
    Interval Arithmetic
    Inclusion
    Truncation Error
    A Priori Error Bounds
    Numerical Integrators
    Digital arithmetic
    Floating-point Arithmetic
    Function evaluation
    Floating point
    Evaluation Function
    Numerical Computation
    Tend
    Numerical Examples

    Keywords

    • Numerical integration
    • Rounding error
    • Verification

    ASJC Scopus subject areas

    • Software
    • Applied Mathematics
    • Computational Mathematics

    Cite this

    A note on a verified automatic integration algorithm. / Yamanaka, Naoya; Kashiwagi, Masahide; Oishi, Shinichi; Ogita, Takeshi.

    In: Reliable Computing, Vol. 15, No. 2, 06.2011, p. 156-167.

    Research output: Contribution to journalArticle

    Yamanaka, N, Kashiwagi, M, Oishi, S & Ogita, T 2011, 'A note on a verified automatic integration algorithm', Reliable Computing, vol. 15, no. 2, pp. 156-167.
    Yamanaka, Naoya ; Kashiwagi, Masahide ; Oishi, Shinichi ; Ogita, Takeshi. / A note on a verified automatic integration algorithm. In: Reliable Computing. 2011 ; Vol. 15, No. 2. pp. 156-167.
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