Yet another DE-Sinc indefinite integration formula

TL;DR

This paper introduces a new indefinite integration formula by replacing the tanh transformation with the double-exponential transformation in Stenger's existing Sinc-based formula, supported by theoretical analysis and numerical tests.

Contribution

It proposes a novel variation of Stenger's formula using the double-exponential transformation, enhancing convergence and computational efficiency.

Findings

The new formula shows improved convergence rates.

Numerical experiments confirm theoretical advantages.

The method maintains simplicity with matrix-vector form.

Abstract

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.