Research on the double exponential transformation subroutine package

假设给定（a，B）上的一个积分。然后，我们知道，如果你使用一个将（a，B）映射到（-ε，ε）上的函数来变换积分，并在变换后将具有网格尺寸的梯形法则应用于积分，你将得到一个高精度的结果。1974年H. Takahashi和M.首席研究员Mori发现，如果你选择一个变换，变换后的函数以双指数方式衰减，结果将是最好的，他们提出了几个对各种积分的数值计算有用的具体变换。本文的目的是提供一个双指数公式的子程序包。我们开发的软件包包括在有限区间上的积分，端点奇异积分，在（0，λ）上的缓慢衰减函数的积分，缓慢衰减函数的傅立叶型积分的子程序，它还包括自动积分器。自动积分器是一个子例程，当用户给出定义被积函数和误差容差的函数子程序时，该子例程返回一个其误差预计位于容差内的结果。我们准备了一本手册，说明如何使用该软件包，并将其与用FORTRAN编写的整个源代码一起打印在本研究的报告中。

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.