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掲載誌名 Journal name,出版機関名 Publishing organization,巻/号 Vol./no.,頁数 Page nos.,発行年月(日) Date
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   

概要(Abstract) Schröder's formula of the second kind of order $m$ of convergence (S2-$m$ formula) is a generalization of Newton's ($m=2$) and Halley's ($m=3$) iterative formulae for finding zeros of functions.
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. 

備考(Remarks) 著者:Hiroshi Sugiura,Takemitsu Hasegawa .
共同研究につき本人担当分抽出不可能.  

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   

概要(Abstract) For the finite Hilbert transform of oscillatory functions $Q(f;c,\omega)$ $=\pv f(x)$ $e^{i\omega x}/(x-c)\,dt$ with a smooth function $f$ and real $\omega\neq0$, for $c\in(-1,1)$ in the sense of Cauchy principal value or for $c=\pm1$ of Hadamard finite-part, we present an approximation method of Clenshaw-Curtis type and its algorithm.
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. 

備考(Remarks) 著者:Takemitsu Hasegawa, Hiroshi Sugiura.
共同研究につき本人担当分抽出不可能.  

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   

概要(Abstract) Schröder’s iterative formula of the second kind (S2 formula) for finding zeros of a function $f(z)$ is a generalization of Newton’s formula to an arbitrary order $m$ of convergence.
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.  

備考(Remarks) 著者:Tomohiro Suzuki, Hiroshi Sugiura and Takemitsu Hasegawa
 

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   

概要(Abstract) Schröder's formulae of the first (S1) and second (S2) kind of order m of convergence are generalizations of Newton's (m=2) and Halley's (S2, m=3) iterative formulae for finding zeros of functions. Davies and Dawson show that for entire functions with only real zeros, Halley's formula converges globally and monotonically to their zeros, independently of the initial value on the real line. We show that the S2 formulae of odd order enjoy the same convergence feature for polynomials with only real zeros. Numerical examples illustrate this. We illustrate no monotonic convergence of the S1 formulae and of the S2 formulae of even order. 

備考(Remarks)  

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   

概要(Abstract) An approximation of Clenshaw–Curtis type is given for Cauchy principal value integrals of logarithmically singular functions with a given function f. Using a polynomial p of degree N interpolating f at the Chebyshev nodes we obtain an approximation of the integral. We expand p in terms of Chebyshev polynomials with O(N log N ) computations by using the fast Fourier transform. Our method is efficient for smooth functions f, for which p converges to f fast as N grows, and so simple to implement. This is achieved by exploiting three-term inhomogeneous recurrence relations in three stages to evaluate the approximation of integral. For f analytic on the interval [−1, 1] in the complex plane z , the error of the approximation is shown to be bounded uniformly. Using numerical examples we demonstrate the performance of the present method. 

備考(Remarks)  

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   

概要(Abstract) For indefinite integrals Q(f;x,ω) on the interval [-1,1] with the oscillatory weight exp(iωx), Torii and the first author (Hasegawa and Torii, 1987) developed a quadrature method of Clenshaw–Curtis (C–C) type. Its improvement was made and combined with Sidi's mW-transformation by Sidi and the first author (Hasegawa and Sidi, 1996) to compute infinite oscillatory integrals. The improved method per se, however, has not been elucidated in its attractive features, which here we reveal with new results and its detailed algorithm.
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. 

備考(Remarks)  

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

概要(Abstract) Takemitsu Hasegawa, Hiroshi Sugiura 

備考(Remarks)  効率的な数値積分公式として著名なFejér第1則,Fejér第2則,Clenshaw-Curts則(C-C則), Basu則に加えて,端補正Clenshaw-Curts則(ecC-C則)を提案した.ecC-C則は,標本点を追加しつつ積分精度を上げてゆく自動積分に適している.すなわち,n点ecC-C則の精度が不足するときn-3点の標本点を追加して,より高精度な2n-3点ecC-C則を計算することができる.
 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   
日本統計学会誌  , 日本統計学会  , 44/2  , pp.271-314  , 2015/03   

概要(Abstract) 分散が同一で平均に傾向性があるk 群の正規分布モデルを考える.白石(2014)は, Hayter(1990)とLee and Spurrier(1995)のシングルステップの多重比較検定を優越する閉検定手順の理論を構築した.2群間のt 検定統計量の最大値の分布の上側100α%点を使って,優越性の証明を行うことができる.また,Williams(1971)の逐次棄却型検定法で使われる分布の上側100α%点を求める具体的な計算式を示す.また, Williams (1971) の逐次棄却型検定法で使われる分布の上側100α%点を求める計算式を示す.つぎに,これらの計算に現れる密度関数の性質を明らかにし,それらがLund and Bowers(1992)とStenger(1993)のsinc近似で効率的に近似できることを示す.最後にsinc 近似法の例として, Hayter(1990)のシングルステップの多重比較検定を優越する閉検定手順を実行するための上側100α% 点を求める計算アルゴリズムを与える.また,その有効性を数値実験により示す. 

備考(Remarks)  

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   

概要(Abstract) 従来,標本点数を増やしつつ適切な精度の補間多項式を構成する方法として,FFTに基づくChebyshev補間のアルゴリズムが用いられてきた.Chebyshev補間は安定かつ高精度でFFTを用いて高速に構成できる.欠点は,標本点数の増加比率が2と大きいことである.この論文では,増加比率を任意に小さく出来る,quasi-Chebyshev補間を提案し,その高速アルゴリズムを構成した.quasi-Chebyshev補間はChebyshev補間のある種の一般化になっている.また,quasi-Chebyshev補間の補間誤差の理論を確立し,具体的に,最小誤差の補間系列を,増加比率に応じて3種設計して見せた.. 

備考(Remarks) 著者:Hiroshi Sugiura,Takemitsu Hasegawa .共同研究につき本人担当分抽出不可能. 

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   

概要(Abstract) 原点に代数的特異点のある関数の高階の分数階微分の高速で高精度な計算法を提案する. 我々の方法は,有限区間で滑らかな関数の,任意の階数の分数階導関数を一様な精度で近似することが出来る.手法として,関数のChebyshev級数展開と3項漸化式を用い,効率的な計算が可能である.この方法は,任意の階数の分数階微分を扱えるという点で,2009年に発表した論文No.58の一般化である. 

備考(Remarks) 著者:Takemitsu Hasegawa, Hiroshi Sugiura.共同研究につき本人担当分抽出不可能. 

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