Dolomites Research Notes on Approximation

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.