No. 掲載種別 単著・共著区分 タイトル 著者 誌名 出版者 巻号頁 出版日 ISSN DOI URL 概要 1 研究論文（学術雑誌） 共著 Modified SE-Sinc approximation with boundary treatment over the semi-infinite interval and its error bound Tomoaki OKAYAMA and Ryota HAMADA JSIAM Letters The Japan Society for Industrial and Applied Mathematics 11, 5-7 2019/02/27 10.14495/jsiaml.11.5 The Sinc approximation is known as an efficient function approximation formula for functions that decay exponentially and are defined over the entire infinite interval. Even for functions that do not satisfy such conditions, Stenger constructed an approximation formula based on the Sinc approximation combining with the Single-Exponential (SE) transformation and introducing auxiliary basis functions. In this study, we improve the approximation formula by replacing the SE transformation and the auxiliary basis functions. Two kinds of error bounds for the modified formula are also given. 2 研究論文（学術雑誌） 共著 An optimal approximation formula for functions with singularities Ken'ichiro TANAKA, Tomoaki OKAYAMA and Masaaki SUGIHARA Journal of Approximation Theory Elsevier B.V. 234, 82-107 2018/10 10.1016/j.jat.2018.06.004 We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval (-1, 1) and possibly have algebraic singularities at the endpoints of the interval. As a space of such functions, we consider a Hardy space with the weight given by w_μ(z)=(1-z^2)^{μ/2} for μ>0, and formulate the optimality of an approximation formula for the functions in the space. Then, we propose an optimal approximation formula for the space for any μ>0, whereas μ is restricted as 0<μ<μ_* for a certain constant μ_* in the existing result. We also provide the results of numerical experiments to show the performance of the proposed formula. 3 研究論文（国際会議プロシーディングス） 共著 Improvement of Sinc-Nyström methods for initial value problems Ryota HARA and Tomoaki OKAYAMA Proceedings of the 2018 International Symposium on Nonlinear Theory and its Applications 651-654 2018/09 Nurmuhammad et al. proposed two Sinc-Nyström methods for initial value problems. The difference between the two methods lies in the variable transformations employed in them. One method uses a single-exponential (SE) transformation and the other uses a double-exponential (DE) transformation. This paper proposes new methods by replacing these transformations to achieve a high convergence rate for the method with the SE transformation, and to enable the inverse function of the DE transformation to be written with elementary functions. 4 研究論文（学術雑誌） 単著 Theoretical analysis of a Sinc-Nyström method for Volterra integro-differential equations and its improvement Tomoaki OKAYAMA Applied Mathematics and Computation Elsevier B.V. 324, 1-15 2018/05/01 10.1016/j.amc.2017.11.062 A Sinc-Nyström method for Volterra integro-differential equations was developed by Zarebnia (2010). The method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity. However, its exponential convergence has not been proved theoretically. In addition, to implement the method, the regularity of the solution is required, although the solution is an unknown function in practice. This paper reinforces the method by presenting two theoretical results: (1) the regularity of the solution is analyzed, and (2) its convergence rate is rigorously analyzed. Moreover, this paper improves the method so that a much higher convergence rate can be attained, and theoretical results similar to those listed above are provided. Numerical comparisons are also provided. 5 研究論文（学術雑誌） 単著 Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems Tomoaki OKAYAMA BIT Numerical Mathematics Springer Netherlands 58/ 1, 199-220 2018/03 10.1007/s10543-017-0663-z A Sinc-collocation method was proposed by Stenger, who also gave a theoretical analysis of the method in the case of a ``scalar'' equation. This paper extends the theoretical results to the case of a ``system'' of equations. Furthermore, this paper proposes a more efficient method by replacing the variable transformation employed in Stenger's method. The efficiency was confirmed by both a theoretical analysis and some numerical experiments. In addition to the existing and newly proposed Sinc-collocation methods, this paper also gives similar theoretical results for the Sinc-Nyström methods proposed by Nurmuhammad et al. In terms of computational cost, the newly proposed Sinc-collocation method is the most efficient among these methods. 6 研究論文（学術雑誌） 単著 Error estimates with explicit constants for the Sinc approximation over infinite intervals Tomoaki OKAYAMA Applied Mathematics and Computation Elsevier B.V. 319, 125-137 2018/02/15 10.1016/j.amc.2017.02.022 The Sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis. Even for other functions, the Sinc approximation works accurately when combined with a proper variable transformation. The convergence rate has been analyzed for typical cases including finite, semi-infinite, and infinite intervals. Recently, for verified numerical computations, a more explicit, “computable” error bound has been given in the case of a finite interval. In this paper, such explicit error bounds are derived for other cases. 7 研究論文（国際会議プロシーディングス） 共著 Explicit error bound for Muhammad-Mori's SE-Sinc indefinite integration formula over the semi-infinite interval Ryota HARA and Tomoaki OKAYAMA Proceedings of the 2017 International Symposium on Nonlinear Theory and its Applications 677-680 2017/12 URL In this paper, we consider two numerical integration formulas over the semi-infinite interval. First, Stenger proposed a formula by means of the Sinc indefinite integration and a single-exponential (SE) transformation. Second, Muhammad--Mori proposed another formula by replacing the SE transformation in Stenger's formula. An error bound of Stenger's formula has been already given. However, any error bound of Muhammad--Mori's formula has not yet been given. In this study, we give an error bound for Muhammad--Mori's formula, and compare the two formulas. 8 研究論文（国際会議プロシーディングス） 共著 Potential theoretic approach to design of accurate numerical integration formulas in weighted Hardy spaces Ken'ichiro TANAKA, Tomoaki OKAYAMA and Masaaki SUGIHARA Approximation Theory XV: San Antonio 2016 Springer, Cham 335-360 2017/07 10.1007/978-3-319-59912-0_17 We propose a method for designing accurate numerical integration formulas on weighted Hardy spaces, which are regarded as spaces of transformed integrands by some useful variable transformations such as the double-exponential transformation. We begin with formulating an optimality of numerical integration formulas in the space by using the norms of the error operators corresponding to those formulas. Then, we derive an expression of the minimum value of the norms, which gives a criterion for an optimal sequence of sampling points for numerical integration. Based on the expression, we propose an algorithm designing accurate numerical integration formulas on the space by a potential theoretic approach. The effectiveness of the designed formulas is supported by some numerical examples. 9 研究論文（学術雑誌） 共著 Error estimate with explicit constants for the trapezoidal formula combined with Muhammad-Mori's SE transformation for the semi-infinite interval Tomoaki OKAYAMA and Koichi MACHIDA JSIAM Letters The Japan Society for Industrial and Applied Mathematics 9, 45-47 2017/05/26 10.14495/jsiaml.9.45 An efficient quadrature formula, known as the Single-Exponential (SE) formula, was proposed by Stenger for the definite integral of an exponentially decaying function over the semi-infinite interval. The formula was derived by combining the trapezoidal formula with a SE transformation. An error bound of the formula was already given. In this study, we investigate another SE formula obtained by replacing the transformation with Muhammad-Mori's SE transformation. Its error bound was determined by theoretical analysis. Numerical comparisons of Stenger's SE formula with that of Muhammad-Mori's are given as well. 10 研究論文（学術雑誌） 共著 Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces Ken'ichiro TANAKA, Tomoaki OKAYAMA and Masaaki SUGIHARA IMA Journal of Numerical Analysis Oxford University Press 37/ 2, 861-904 2017/04 10.1093/imanum/drw022 We propose a method for designing accurate interpolation formulas on the real axis for function approximation in weighted Hardy spaces. Examples of interpolation formulas for functions in such spaces include the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single-exponential (SE) and double-exponential (DE) weights w, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions w. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. 11 研究論文（学術雑誌） 共著 Ganelius標本点を用いた関数近似公式 鵜島崇, 田中健一郎, 岡山友昭, 杉原 正顯 日本応用数理学会論文誌 日本応用数理学会 27/ 1, 1-20 2017/03/25 10.11540/jsiamt.27.1_1 本論文では，端点特異性をもつ解析関数の成す関数空間に対して，Ganelius 標本点を用いた関数近似公式を提案する．そして，この公式がSE-Sinc 関数近似公式よりも高精度となること，更には，最適な関数近似公式であることを示す． 12 研究論文（その他学術会議資料等） 共著 高等学校の統計分野における基本的な用語の定義の差異について 田中輝雄, 佐藤学, 齋藤夏雄, 関根光弘, 廣門正行, 岡山友昭 2017年度数学教育学会春季年会予稿集 数学教育学会 224-226 2017/03/25 高等学校における現行の教育課程から「データの分析」を含む「数学Ⅰ」が必履修となり，統計分野が重要視されている．特に，データと変量は重要な用語である．本稿では，「数学Ⅰ」と「数学Ｂ」において教科書によってこれらの用語の定義に差異があることを示す．そして，高等学校学習指導要領における記述も踏まえつつ，統計学の立場からこれらの定義の比較と考察を行う．さらに，統計分野を指導する際の留意点を述べる． 13 研究論文（その他学術会議資料等） 共著 高等学校の確率分野における基本的な用語の定義の差異について 田中輝雄, 佐藤学, 齋藤夏雄, 関根光弘, 廣門正行, 岡山友昭 2017年度数学教育学会春季年会予稿集 数学教育学会 146-148 2017/03/25 高等学校における現行の教育課程から「データの分析」を含む「数学Ⅰ」が必履修となり，統計分野が重要視されている．その基礎となるのが，確率分野である．本稿では，確率論の歴史と高等学校学習指導要領を踏まえ，「数学Ａ」と「数学Ｂ」の教科書において基本的な用語の定義に差異があることを示す．そして，測度論的確率論の立場からこれらの定義の比較と考察を行う．さらに，確率分野を指導する際の留意点を述べる． 14 研究論文（国際会議プロシーディングス） 共著 Verified error bounds for the real gamma function using double exponential formula over semi-infinite interval Naoya YAMANAKA, Tomoaki OKAYAMA and Shin'ichi OISHI Lecture Notes in Computer Science Springer International Publishing 9582, 224-228 2016/04/16 10.1007/978-3-319-32859-1_19 An algorithm is presented for computing verified error bounds for the value of the real gamma function. It has been shown that the double exponential formula is one of the most efficient methods for calculating integrals of the form. Recently, an useful evaluation based on the double exponential formula over the semi-infinite interval has been proposed. However, the evaluation would be overflow when applied to the real gamma function directly. In this paper, we present a theorem so as to overcome the problem in such a case. Numerical results are presented for illustrating effectiveness of the proposed theorem in terms of the accuracy of the calculation. 15 研究論文（国際会議プロシーディングス） 単著 Explicit error bound for modified numerical iterated integration by means of Sinc methods Tomoaki OKAYAMA Lecture Notes in Computer Science Springer International Publishing 9582, 202-217 2016/04/16 10.1007/978-3-319-32859-1_17 This paper reinforces numerical iterated integration developed by Muhammad–Mori in the following two points: (1) the approximation formula is modified so that it can achieve a better convergence rate in more general cases, and (2) an explicit error bound is given in a computable form for the modified formula. The formula works quite efficiently, especially if the integrand is of a product type. Numerical examples that confirm it are also presented. 16 研究論文（学術雑誌） 共著 Theoretical analysis of Sinc-Nyström methods for Volterra integral equations Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA Mathematics of Computation American Mathematical Society 84/ 293, 1189-1215 2015/05 10.1090/S0025-5718-2014-02929-3 In this paper, we present three theoretical results on Sinc-Nyström methods for Volterra integral equations of the first and second kind, which were proposed by Muhammad et al. Their methods involve the following issues: 1) it is difficult to determine the tuning parameter unless the solution is given, and 2) convergence has not been proved in a precise sense. In a mathematically rigorous manner, we present an implementable way to estimate the tuning parameter and a rigorous proof of the convergence with its rate explicitly revealed. Furthermore, we show that the resulting system is well conditioned. Numerical examples that support the theoretical results are also presented. 17 研究論文（学術雑誌） 共著 内積の誤用に対する指導について 関根光弘, 岡山友昭, 齋藤夏雄, 佐藤学, 廣門正行, 田中輝雄 2015年度数学教育学会春季年会発表論文集 数学教育学会 194-196 2015/03/23 ベクトルの内積は，スカラー同士の積とは異なる演算として高等学校の数学Bで導入されている．しかし大学入学後も，記号の濫用による誤答例が少なからず見受けられる．この点を踏まえ，教科教育法の講義において内積に関する問題への誤答例をいくつか提示し，それらに対する訂正と指導上の留意点を論述させた．提出された結果を受けて，内積に対する理解を深めるための教育実践を行ったことについて報告する． 18 研究論文（学術雑誌） 共著 非存在定理を大学初年次の数学教育において活用することのすすめ 廣門正行, 岡山友昭, 齋藤夏雄, 関根光弘, 佐藤学, 田中輝雄 2015年度数学教育学会春季年会発表論文集 数学教育学会 197-199 2015/03/23 数学の命題で存在に関するもの，特に「～が存在しない」という形の命題は一般に証明が難しい．一部の学生においては，これらの定理の論理的しくみを理解しようとせず，結論のみの丸暗記で置き換えようとする傾向が見られる．ここではこの傾向に焦点をあて，その定量的な分析を試みる．同時に，数学の命題をいかに暗記するかに重点を置く学生に発想の転換を促し，数学の面白さを味わってもらう教育法について考察する． 19 研究論文（国際会議プロシーディングス） 単著 Improvement of a Sinc-collocation method for Fredholm integro-differential equations Tomoaki OKAYAMA AIP Conference Proceedings AIP Publishing 1648/ 390008, 1-4 2015/03/10 10.1063/1.4912618 A Sinc-collocation method for Fredholm integro-differential equations was developed by Rashidinia–Zarebnia. Their numerical experiments show that the method converges exponentially. However, in reality, the method does not converge uniformly over the given interval [a, b]. In this study, the basis functions of the method are modified to enable the method to converge uniformly on [a, b]. Furthermore, this study further improves the method by modifying the variable transformation to attain a much higher convergence rate. 20 研究論文（国際会議プロシーディングス） 共著 High-precision eigenvalue bound for the Laplacian with singularities Xuefeng LIU, Tomoaki OKAYAMA and Shin'ichi OISHI Computer Mathematics (eds. Ruyong Feng, Wen-shin Lee and Yosuke Sato) Springer Berlin Heidelberg 311-323 2014/10/01 10.1007/978-3-662-43799-5_23 For the purpose of bounding eigenvalues of the Laplacian over a bounded polygonal domain, we propose an algorithm to give high-precision bound even in the case that the eigenfunction has singularities around reentrant corners. The algorithm is a combination of the finite element method and the Lehmann–Goerisch theorem. The interval arithmetic is adopted in floating point number computation. Since all the error in the computation, e.g., the function approximation error, the floating point number rounding error, are exactly estimated, the result can be mathematically correct. In the end of the chapter, there are computational examples over an L-shaped domain and a square-minus-square domain that demonstrate the efficiency of our proposed algorithm. 21 研究論文（学術雑誌） 共著 √(a^2)=|a|の大学数学教育における影響 田中輝雄, 齋藤夏雄, 佐藤学, 関根光弘, 廣門正行, 岡山友昭 2014年度数学教育学会秋季例会発表論文集 数学教育学会 89-91 2014/09/26 広島市立大学情報科学部では入学直後の1年生に対して，高校数学Ⅰ，Ⅱ，Ⅲ，Ａ，Ｂ，Ｃの全科目を出題範囲とする「数学基礎学力調査」を実施している．公式 √(a^2 )=|a| を問う設問の解答結果を踏まえ，数学Iで扱われるこの公式が，高校数学のどの単元で活用されているか，また，大学入試センターの試験問題でどのように取扱われているかを述べる．さらに，大学1年次で学習する解析学や線形代数学などへの影響について考察し，指導上の留意点について問題提起を行う． 22 （MISC）総説・解説（学術雑誌） 単著 第二種積分方程式に対する Sinc 選点法の改良とその理論解析 岡山友昭 応用数理 日本応用数理学会 24/ 3, 14-18 2014/09/25 10.11540/bjsiam.24.3_110 Sinc-collocation methods were developed for integral equations of the second kind by Rashidinia-Zarebnia, and their high convergence rates were confirmed by numerical experiments. However, runability and convergence of those methods have not been proved so far. Furthermore, those methods are in reality not implementable in application. Recent results by the present author, Matsuo and Sugihara have solved these issues, which are reviewed in this article. 23 研究論文（学術雑誌） 単著 Explicit error bound for the tanh rule and the DE formula for integrals with logarithmic singularity Tomoaki OKAYAMA JSIAM Letters The Japan Society for Industrial and Applied Mathematics 6, 9-11 2014/04/17 10.14495/jsiaml.6.9 The tanh rule and the double-exponential (DE) formula are known empirically and theoretically as quite efficient quadrature formulas, especially for integrals with endpoint singularity, including algebraic singularity and logarithmic singularity. Furthermore, in the case of integrals with algebraic singularity, explicit error bounds have been given for those formulas, which enables us to guarantee their approximation accuracy. In the case of integrals with logarithmic singularity, however, such explicit error bounds have not ever given thus far, although those formulas should work accurately in this case as well. This paper presents the desired theoretical explicit error bounds, with numerical experiments. 24 研究論文（学術雑誌） 単著 Error estimates with explicit constants for Sinc quadrature and Sinc indefinite integration over infinite intervals Tomoaki OKAYAMA Reliable Computing 19/ 1, 45-65 2013/12 URL The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation. Their convergence rates have been analyzed for typical cases including finite, semi-infinite, and infinite intervals. In addition, for verified automatic integration, more explicit error bounds that are computable have been recently given on a finite interval. In this paper, such explicit error bounds are given in the remaining cases on semi-infinite and infinite intervals. 25 研究論文（学術雑誌） 共著 DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods. Part II: Indefinite integration Ken'ichiro TANAKA, Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA Numerische Mathematik Springer Berlin Heidelberg 125/ 3, 545-568 2013/11/01 10.1007/s00211-013-0541-9 In this paper, the theoretical convergence rate of the Sinc indefinite integration combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. Although the DE transformation is considered as an enhanced version of the SE transformation for Sinc-related methods, the function space for which the DE transformation is suitable is smaller than that for SE, and therefore, there exist some examples such that the DE transformation is not better than the SE transformation. Even in such cases, however, some numerical observations in the literature suggest that there is almost no difference in the convergence rates of SE and DE. In fact, recently, the observations have been theoretically explained for two explicit approximation formulas: the Sinc quadrature and the Sinc approximation. The conclusion is that in such cases, the DE’s rate is slightly lower, but almost the same as that of the SE. The contribution of this study is the derivation of the same conclusion for the Sinc indefinite integration. Numerical examples that support the theoretical result are also provided. 26 研究論文（学術雑誌） 共著 DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods. Part I: Definite integration and function approximation Tomoaki OKAYAMA, Ken'ichiro TANAKA, Takayasu MATSUO and Masaaki SUGIHARA Numerische Mathematik Springer Berlin Heidelberg 125/ 3, 511-543 2013/11/01 10.1007/s00211-013-0540-x In this paper, the theoretical convergence rate of the trapezoidal rule combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. It is well known that the DE transformation enables the rule to achieve a much higher rate of convergence than the SE transformation, and the convergence rate has been analyzed and justified theoretically under a proper assumption. Here, it should be emphasized that the assumption is more severe than the one for the SE transformation, and there actually exist some examples such that the trapezoidal rule with the SE transformation achieves its usual rate, whereas the rule with DE does not. Such cases have been observed numerically, but no theoretical analysis has been given thus far. This paper reveals the theoretical rate of convergence in such cases, and it turns out that the DE’s rate is almost the same as, but slightly lower than that of the SE. 27 研究論文（学術雑誌） 共著 Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA Numerische Mathematik Springer Berlin Heidelberg 124/ 2, 361-394 2013/06/01 10.1007/s00211-013-0515-y The original form of the Sinc approximation is efficient for functions whose boundary values are zero, but not for other functions. The typical way to treat general boundary values is to introduce auxiliary basis functions, and in fact such an approach has been taken commonly in the literature. However, the approximation formula in each research is not exactly the same, and still other formulas can be derived as variants of existing formulas. The purpose of this paper is to sum up those existing formulas and new ones, and to give explicit proofs of those convergence theorems. 28 研究論文（学術雑誌） 単著 A note on the Sinc approximation with boundary treatment Tomoaki OKAYAMA JSIAM Letters 日本応用数理学会 5, 1-4 2013/02/07 10.14495/jsiaml.5.1 The original form of the Sinc approximation is efficient for functions whose boundary values are zero, but not for other functions. The typical way to treat general boundary values is to introduce auxiliary basis functions, and in fact such an approach has been taken commonly in the literature. However, the approximation formula in each research is not exactly the same, and still other formulas can be derived as variants of existing formulas. The purpose of this paper is to sum up those existing formulas and new ones, and to give explicit proofs of those convergence theorems. 29 研究論文（学術雑誌） 単著 第二種積分方程式に対するSinc法の実用的視点からの理論解析 岡山友昭 日本応用数理学会論文誌 日本応用数理学会 22/ 3, 181-212 2012/09/25 10.11540/jsiamt.22.3_181 第二種積分方程式に対するSinc法の開発が2000年初頭より盛んになり，その数値解法の性能の高さが報告されている．ただし，それらの多くには共通して次の二つの難点があった．1)解が与えられない限り，スキームの実装が困難なこと，及び2)数値解の収束が保証されていないことである．実用上，この状況は望ましくない．近年，理論解析によってこの二点が解決されてきており，本論文ではそれらの研究について概説する． 30 研究論文（学術雑誌） 共著 On boundedness of the condition number of the coefficient matrices appearing in Sinc-Nyström methods for Fredholm integral equations of the second kind Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA JSIAM Letters 日本応用数理学会 3, 81-84 2011/11/30 10.14495/jsiaml.3.81 Sinc-Nyström methods for Fredholm integral equations of the second kind have been independently proposed by Muhammad et al. and Rashidinia-Zarebnia. They also gave error analyses, but the results did not claim the convergence of their schemes in a precise sense. This is because in their error estimates there remained an unestimated term: the norm of the inverse of the coefficient matrix of the resulting linear system. In this paper, we estimate the term theoretically to complete the convergence estimate of their methods. Furthermore, we also prove the boundedness of the condition number of each coefficient matrix. 31 研究論文（学術雑誌） 共著 Improvement of a Sinc-collocation method for Fredholm integral equations of the second kind Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA BIT Numerical Mathematics Springer Netherlands 51/ 2, 339-366 2011/06/01 10.1007/s10543-010-0289-x A Sinc-collocation scheme for Fredholm integral equations of the second kind was proposed by Rashidinia–Zarebnia in 2005. In this paper, two improved versions of the Sinc-collocation scheme are presented. The first version is obtained by improving the scheme so that it becomes more practical, and natural from a theoretical view point. Then it is rigorously proved that the convergence rate of the modified scheme is exponential, as suggested in the literature. In the second version, the variable transformation employed in the original scheme, the “tanh transformation”, is replaced with the “double exponential transformation”. It is proved that the replacement improves the convergence rate drastically. Numerical examples which support the theoretical results are also given. 32 研究論文（学術雑誌） 共著 A fast verified automatic integration algorithm using double exponential formula Naoya YAMANAKA, Tomoaki OKAYAMA, Shin'ichi OISHI and Takeshi OGITA Nonlinear Theory and Its Applications, IEICE The Institute of Electronics, Information and Communication Engineers 1/ 1, 119-132 2010/10/01 10.1587/nolta.1.119 A fast verified automatic integration algorithm is proposed for calculating univariate integrals over finite intervals. This algorithm is based on the double exponential formula proposed by Takahasi and Mori. The double exponential formula uses a certain trapezoidal rule. This trapezoidal rule is determined by fixing two parameters, the width h of a subdivision of a finite interval and the number n of subdivision points of this subdivision. A theorem is presented for calculating h and n as a function of a given tolerance of the verified numerical integration of a definite integral. An efficient a priori method is also proposed for evaluating function calculation errors including rounding errors of floating point calculations. Combining these, a fast algorithm is proposed for verified automatic integration. Numerical examples are presented for illustrating effectiveness of the proposed algorithm. 33 研究論文（学術雑誌） 共著 Approximate formulae for fractional derivatives by means of Sinc methods Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA Journal of Concrete and Applicable Mathematics Eudoxus Press, LLC 8/ 3, 470-488 2010/07 In this paper, two new approximate formulae for fractional derivatives are developed by means of Sinc methods. The difference of the two formulae is the variable transformations incorporated; the single exponential transformation and the double exponential transformation. We give error analysis of the formulae, and show that these formulae archive exponential convergence. Numerical examples that confirm the analysis are also given. 34 研究論文（学術雑誌） 共著 第二種積分方程式に対するSinc選点法の改良とその理論解析 岡山友昭, 松尾宇泰, 杉原正顯 日本応用数理学会論文誌 日本応用数理学会 20/ 2, 71-113 2010/06/25 10.11540/jsiamt.20.2_71 本論文では，Rashidinia-Zarebnia (2005, 2007)によって提案されたSinc選点法を用いた第二種積分方程式の数値解法に対し,次の2つの成果を報告する．1) Rashidinia-Zarebniaによるスキームをより実装に即した形に修正し，そのスキームに対して厳密な収束証明を与える．2)元のスキームで用いられていた一重指数関数型変換に代えて二重指数関数型変換(DE変換)を採用したスキームを提案し，理論解析と数値実験により，収束が格段に改善されることを示す． 35 研究論文（学術雑誌） 共著 Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA Journal of Computational and Applied Mathematics Elsevier B.V. 234/ 4, 1211-1227 2010/06/15 10.1016/j.cam.2009.07.049 In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far. 36 研究論文（学術雑誌） 共著 Error estimates with explicit constants for the tanh rule and the DE formula for indefinite integrals Tomoaki OKAYAMA, Takayasu MATSUO and Masaaki SUGIHARA JSIAM Letters The Japan Society for Industrial and Applied Mathematics 2, 13-16 2010/03/12 10.14495/jsiaml.2.13 The tanh rule and the double-exponential (DE) formula are known as efficient quadrature rules for ``definite integrals'' over a finite interval (a, b). In this note we consider a numerical method for ``indefinite integrals'' obtained by applying the tanh rule or the DE formula to the integration over the interval (a, x) for each x. For these methods the conventional error analyses yield error estimates depending on x, which are impractical. We here present error estimates that do not depend on x, and furthermore, with explicit constants. 37 研究論文（国際会議プロシーディングス） 共著 A fast automatic integration algorithm using double exponential formula based on verification theory Naoya YAMANAKA, Tomoaki OKAYAMA, Shin'ichi OISHI and Takeshi OGITA Proceedings of the 24th International Technical Conference on Circuits/Systems, Computers and Communications 161-164 2009/07 URL A fast automatic integration algorithm of calculating univariate integrals over finite interval using numerical computations is proposed. The proposed algorithm is based on the double exponential formula. In order that the formula works accurately and efficiently, we have presented a theorem for verified computations. In this paper, we propose an approximate integration algorithm based on this theorem. Numerical results are presented showing the performance of the proposed algorithm. 38 研究論文（大学，研究機関紀要） 共著 A fast verified automatic integration algorithm using double exponential formula Naoya YAMANAKA, Tomoaki OKAYAMA, Shin'ichi OISHI and Takeshi OGITA 数理解析研究所講究録 京都大学数理解析研究所 1638, 146-158 2009/04 URL A fast verified automatic integration algorithm of calculating univariate integrals over finite interval using numerical computations is proposed. The proposed algorithm is applicable to the double exponential formula for numerical integration proposed by H. Takahashi and M. Mori. To get highly accurate integral value using the formula, how to decide the two parameters h and n is critical. In this paper, we present a theorem to get an adequate pair h and n for a given tolerance. Furthermore, we propose a fast verified automatic integration algorithm based on the theorem with a priori error algorithm for rounding error. Numerical results are presented showing the performance of the proposed algorithm. 39 研究論文（大学，研究機関紀要） 共著 第二種積分方程式に対するSinc法とその理論解析 岡山友昭, 松尾宇泰, 杉原正顯 数理解析研究所講究録 京都大学数理解析研究所 1638, 38-55 2009/04 URL Sinc法はSinc関数近似に基づく数値計算法の総称であり，その近似性能の高さが注目され，微分方程式の数値解法や，積分方程式の数値解法など，各種の問題に応用されてきている．ただし，積分方程式の数値解法には，大きく分けて以下の二つの難点があった．一つ目は，実装に解の情報が必要なことである．あるパラメータを定めるのに，解のなめらかさの情報が必要であるが，実用上は解は求めるべき未知関数であり，直接調べることはできないことが問題となる．二つ目は，収束証明が厳密に与えられていないことである．近似解を求めるための連立1次方程式の可解性や，真の解に収束することが厳密に証明されておらず，数値解法がきちんと動作するかどうかは実際に実行してみるまでわからないという状態であった．そこで本論文では理論解析を行い，これらの二つの難点を克服できることを示す． 40 研究論文（学術雑誌） 共著 自動微分とDE公式を用いた非整数階微分の数値計算 岡山友昭, 村重淳 日本応用数理学会論文誌 日本応用数理学会 16/ 1, 51-65 2006/03/25 10.11540/jsiamt.16.1_51 This paper proposes a new method of numerical calculation for fractional derivatives. Con-ventional methods using difference approxi-mation have some defects such as rounding errors and computation time. In this paper, fractional derivatives are expressed as the combination of ordinary derivatives and integrals which are calculated using automatic differentiation and double exponential formula, respectively. Numerical examples show that the proposed method gives high-precision results with practical computational cost.