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[Author] Takatomi MIYATA(2hit)

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  • Long Time Integration for Initial Value Problems of Ordinary Differential Equations Using Power Series Arithmetic

    Takatomi MIYATA  Yasutaka NAGATOMO  Masahide KASHIWAGI  

     
    PAPER-Numerical Method & Optimization

      Vol:
    E84-A No:9
      Page(s):
    2230-2237

    In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

  • A New Dividing Method in Affine Arithmetic

    Shinya MIYAJIMA  Takatomi MIYATA  Masahide KASHIWAGI  

     
    LETTER

      Vol:
    E86-A No:9
      Page(s):
    2192-2196

    Affine arithmetic is a kind of interval arithmetic defined by Stolfi et al. In affine arithmetic, it is difficult to realize the efficient nonlinear binomial operations. The purpose of this letter is to propose a new dividing method which is able to supply more suitable evaluation than the old dividing method. And this letter also shows the efficiency of the new dividing method by numerical examples.