# 研究者詳細

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2019  On the global convergence of Schröder's iteration formula for real zeros of entire functions  共著
Journal of Computational and Applied Mathematics  ， Elsevier  ， 358  ， 136-145  ， 2019/10/01

The authors showed that the S2-$m$ formula of every odd order $m\ge3$ converges globally and monotonically to real zeros of polynomials on the real line. For Halley's formula, such the convergence property for real zeros of entire functions had been shown by Davies and Dawson.
In this paper, we extend both results by showing that the S2-$m$ formula of every odd order $m\ge5$ has the same convergence property for real zeros of entire functions.
By numerical examples, we illustrate the monotonic convergence of the formula of odd order $m=3,5$ and $7$ and the non-monotonic convergence of even order $m=2, 4$ and $6$.
Further, we compare several formulae of both first and second kinds in performance.

2019  Uniform approximation to finite Hilbert transform of oscillatory functions and its algorithm  共著
Journal of Computational and Applied Mathematics  ， Elsevier  ， 358  ， 327-342  ， 2019/10/01

Interpolating $f$ by a polynomial $p_n$ of degree $n$ and expanding in terms of the Chebyshev polynomials with $O(n\log n)$ operations by the FFT, we obtain an approximation $Q(p_n;c,\omega)\cong Q(f;c,\omega)$. We write $Q(p_n;c,\omega)$ as a sum of the sine and cosine integrals and an oscillatory integral of a polynomial of degree $n-1$. We efficiently evaluate the oscillatory integral of degree $n-1$, with a combination of authors' previous method and Keller's method.
For $f(z)$ analytic on the interval $[-1,1]$ in the complex plane $z$, the error of $Q(p_n;c,\omega)$ is bounded uniformly with respect to $c$ and $\omega$.
Numerical examples illustrate the performance of our method.

2019  Estimating convergence regions of Schröder’s iteration formula: how the Julia set shrinks to the Voronoi boundary   共著
Numerical Algorithms  ， Springer  ， 82/01  ， pp.183-199  ， 2019/09

For iterative formulae, convergence regions of initial values to zeros in the complex plane $z$ are essential. From numerical experiments, it is suggested that as order $m$ of the S2 formula grows, the complicated fractal structure of the boundary of convergence regions gradually diminishes.
We propose a method of estimating the convergence regions with the circles of Apollonius to verify this result for polynomials $f(z)$ with simple zeros. We indeed show that as $m$ grows, each region surrounded by the circles of Apollonius monotonically enlarges to the Voronoi cell of a zero of $f(z)$.
Numerical examples illustrate convergence regions for several values of $m$ and some polynomials.

2018  On the global convergence of Schröder’s iterative formulae for real roots of algebraic equations  共著
Journal of Computational and Applied Mathematics  ， Elsevier  ， 344  ， pp. 313-322  ， 2018/12/15

2017  Uniform approximation to Cauchy principal value integrals with logarithmic singularity  共著
Journal of Computational and Applied Mathematics  ， Elsevier  ， 327  ， pp.1-11  ， 2018/02/01

2017  A user-friendly method for computing indefinite integrals of oscillatory functions  共著
Journal of Computational and Applied Mathematics  ， Elsevier  ， 315  ， pp. 126-141  ， 2017/05/01

A comparison with a method of C–C type for definite integrals Q (f;1,ω) due to Domínguez et al. (2011) suggests that a smaller number of computations is required in our method. This is achieved by exploiting recurrence and normalization relations and their associated linear system. We show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension. For f(z) analytic on and inside an ellipse in the complex plane z the error of the approximation to Q(f;x,ω) of the improved method is shown to be bounded uniformly.
Numerical examples illustrate the stability and performance of the method.

2015  Error estimate for a corrected Clenshaw–Curtis quadrature rule  共著
Numerische Mathematik  ， Springer  ， 130/1  ， pp. 135-149  ， 2015/05

ecC-C則について，積分則の重みの高速計算法を示した．また重みの漸近正値性を示し，公式の数値的安定性を理論的に保証した．さらに，精密な誤差解析を行い，Fejér第2則，Fejér第2則，C-C則，Basu則，ecC-C則の順で精度が良いことを示した．
これらの結果を，いくつかの数値実験により確認した．

2014  平均母数に傾向性がある正規多群モデルにおける多重比較法に使用される分布の上側100αパーセント点  共著

2011  A POLYNOMIAL INTERPOLATION PROCESS AT QUASI-CHEBYSHEV NODES WITH THE FFT  共著
Mathematics of Computation  ， American Mathematical Society  ， 80/276  ， 2169-2184  ， 2011/10

2011  An approximation method for high-order fractional derivatives of algebraically singular functions  共著
Computers and Mathematics with Applications  ， Elsevier  ， 62  ， 930-937  ， 2011/08

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