Double Exponential Formulas for Numerical Integration

Abstract

A family of numerical quadrature formulas is introduced by application of the trapezoidal rule to infinite integrals which result from the given integrals \int^b_a f(x)dx by suitable variable transformations x = \phi(u) . These formulas are characterized by having double exponential asymptotic behavior of the integrands in the resulting infinite integrals as u →± ∞ , and it is shown both analytically and numerically that such formulas are generally optimal with respect to the ecomony of the number of sampling points.