Optimal selection formulas of mesh size and truncation numbers for the double-exponential formula

Abstract

The double-exponential formula is a quadrature formula that applies the trapezoidal formula to the infinite integral of double-exponentially decaying functions. Its high efficiency has been recognized since the 1970s, and in 1997, it was demonstrated that the formula is optimal in some sense. To achieve its best performance, its mesh size and truncation numbers are optimally selected depending on a given positive integer. However, existing selection formulas are not optimal. In this paper, we propose optimal selection formulas for the mesh size and truncation numbers, giving a far better error bound than the existing one.