Based on the Sinc approximation combined with the tanh transformation, Haber derived an approximation formula for numerical indefinite integration over the finite interval (-1, 1). The formula uses a special function for the basis functions. In contrast, Stenger derived another formula, which does not use any special function but does include a double sum. Subsequently, Muhammad and Mori proposed a formula, which replaces the tanh transformation with the double-exponential transformation in Haber's formula. Almost simultaneously, Tanaka et al. proposed another formula, which was based on the same replacement in Stenger's formula. As they reported, the replacement drastically improves the convergence rate of Haber's and Stenger's formula. In addition to the formulas above, Stenger derived yet another indefinite integration formula based on the Sinc approximation combined with the tanh transformation, which has an elegant matrix-vector form. In this paper, we propose the replacement of the tanh transformation with the double-exponential transformation in Stenger's second formula. We provide a theoretical analysis as well as a numerical comparison.
翻译:根据Sinc近似值,加上Tanh变异,Haber为定时间隔(-1,1)中的数字无限期融合得出了一个近似公式,该公式对基函数使用一种特殊函数。相反,Stanger则得出另一种公式,该公式不使用任何特殊函数,但的确包含一个双倍数。随后,Muhammad和Mori提议了一个公式,用Haber公式的双重实验变异取代Tanh变异。几乎同时,Tana等人提议了另一种公式,该公式以Stanger公式中的相同替换为基础。他们报告说,该替换大大提高了Haber和Stenger公式的趋同率。除上述公式外,Stanger还根据Sinc近似值和Tangle变异的另一种不定期合并公式,后者具有优雅的矩阵-矢量表形式。在本文中,我们提议用Stanger第二个公式中的双加速变。我们提供了理论分析和数字比较。