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Research on the double exponential transformation subroutine package

双指数变换子程序包的研究

基本信息

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

项目摘要

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.

假设给定对（a, b）的积分。由此可知，利用（a, b）映射到（-∽,∽）上的函数对积分进行变换，并对变换后的积分应用网格大小的梯形定则，可以得到精度较高的结果。1974年，首席研究员高桥（H. Takahashi）和森（M. Mori）发现，如果选择一种变换，变换后的函数以指数方式衰减两倍，结果将是最好的。他们提出了几种对各种积分的数值计算有用的具体变换。用这种方法得到的公式称为双指数公式。本研究计划的目的是为使用者提供一个二重指数公式的子程式包。我们开发的软件包包括有限区间积分子程序、端点奇点积分子程序、（0,∽）上缓慢衰减函数的积分子程序、缓慢衰减函数的傅里叶型积分子程序以及自动积分器。自动积分器是一个子例程，当用户给出定义积分器和容错值的函数子程序时，它返回一个结果，其错误预计在容错值之内。我们准备了一份说明如何使用软件包的手册，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。