Research on the double exponential transformation subroutine package

双指数变换子程序包的研究

• 批准号: 12640119
• 负责人: MORI Masatake
• 金额: $ 2.24万
• 依托单位: Tokyo Denki University (2001-2002)Kyoto University (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640119/
• 关键词: double exponential transformation; numerical integration; variable transformation; subroutine package; 二重指数関数型公式; DE変換; 数値解析; 最適公式

Suppose that an integral over (a, b) is given. Then it is known that, if you transform the integral using a function which maps (a, b) onto (-∽, ∽) and apply the trapezoidal rule with an mesh size to the integral after the transformation you will get a result with high precision. In 1974 H. Takahashi and M. Mori, the head investigator, found that if you choose a transformation by which the function after the transformation decays double exponentially the result will be the best, and they proposed several specific transformations useful for numerical evaluation of various kind of integrals. A formula obtained in this way is called the double exponential formula.The purpose of the present research project is to provide users with a subroutine package of the double exponential formulas. The package we developed includes subroutines for integrals over a finite interval, integrals with end point singularity, integrals of a slowly decaying function over (0, ∽), Fourier type integrals of a slowly decaying function, and it also includes automatic integrators. An automatic integrator is a subroutine that, when the user gives a function subprogram defining the integrand and an error tolerance, returns a result whose error is expected to lie in the tolerance. We prepared a manual which shows how to use the package and printed it in the report of the present research together with the entire source code written in FORTRAN.

假设给出了 (a, b) 上的积分，那么我们知道，如果使用将 (a, b) 映射到 (-∽, ∽) 的函数来变换积分，并应用具有网格大小的梯形规则。 1974 年，首席研究员 H. Takahashi 和 M. Mori 发现，如果选择一种变换，变换后的函数将呈双倍指数衰减。结果将是最好的，他们提出了几种对各种积分的数值评估有用的特定变换，以这种方式获得的公式称为双指数公式。本研究项目的目的是为用户提供一个子程序包。我们开发的包包括有限区间积分、具有端点奇点的积分、(0, ∽) 上缓慢衰减函数的积分、缓慢衰减函数的傅立叶型积分以及它还包括自动积分器，自动积分器是一个子程序，当用户给出定义被积数和误差容限的函数子程序时，它会返回一个结果，该结果的误差预计位于容差范围内。使用该包并将其与用 FORTRAN 编写的整个源代码一起打印在本研究报告中。

项目成果

Takuya Ooura: Journal of Computational and Applied Mathematics. (印刷中).

Takuy​​a Ooura：计算与应用数学杂志（正在出版）。

Kaname Amano: Contributions in Analytic Extension Formulas and their Applications. 15-25 (2001)

Kaname Amano：对解析扩展公式及其应用的贡献。

Saburou Saitoh: "Representations of analytic functions in terms of local values by means of the Riemann mapping function"Complex Variables. 45. 387-393 (2001)

Saburou Saitoh：“通过黎曼映射函数以局部值表示解析函数”复变量。

Takuya Ooura: "A continuous Euler transformation and its application to the Fourier transform of a slowly decaying function"Journal of Computational and Applied Mathematics. 130. 259-270 (2001)

Takuy​​a Ooura：“连续欧拉变换及其在缓慢衰减函数的傅里叶变换中的应用”计算与应用数学杂志。

Masatake Mori: "The double exponential transformation in numerical analysis"Journal of Computational and Applied Mathematics. 127. 287-296 (2001)

Masatake Mori：“数值分析中的双指数变换”计算与应用数学杂志。