二重指数関数型数値積分公式を用いた行列実数乗の計算  [in Japanese] A Note on Computing the Matrix Fractional Power Using the Double Exponential Formula  [in Japanese]

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<p><b>概要.</b> 本論文では数値積分による行列実数乗A<sup>α</sup>の計算を考える.A<sup>α</sup>の積分表示は減衰の遅い被積分関数の広義積分で表される.これまでは,変数変換により積分区間を有限にした後にGauss求積を用いて計算されてきたが,Aの条件数やαによっては収束性が悪化しうる.本論文では新たなアプローチとして二重指数関数型積分公式に着目し,積分区間の設定方法等をまとめたアルゴリズムを示したのち,その有用性を数値的に検証する.</p>

<p><i>Abstract.</i> The matrix fractional power A<sup>α</sup> can be represented as an improper integral for a slowly decaying function over the half infinite interval. Conventional approaches to approximating the integral are to apply Gaussian quadrature after some variable transformations; however, the convergence could be slow for an ill-conditioned matrix or a specific value of α. To avoid such situation, we consider the Double Exponential (DE) formula instead of conventional approaches, propose algorithms with settings of the integration interval, and then we show some numerical results.</p>

Journal

  • Transactions of the Japan Society for Industrial and Applied Mathematics

    Transactions of the Japan Society for Industrial and Applied Mathematics 28(3), 142-161, 2018

    The Japan Society for Industrial and Applied Mathematics

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