An IMT-Type Double Exponential Formula for Numerical Integration

Abstract

A quadrature formula for evaluation of improper integrals over ( — 1 , 1 ) is presented, which is obtained in such a way that the interval of integration (—1, 1) is transformed into (—°°, °°) by x=tanh(A sinh 2Bu/(l — M)) and that the trapezoidal rule with an equal mesh size is subsequently applied to the transformed integral. Asymptotic error analysis is made by means of the method of contour integral and a comparison with the IMTrule and the double exponential formula is given along with some numerical examples. §1. The IMT-ruIe and the Double Exponential Formula In 1969 Iri, Moriguti and Takasawa [4] presented a formula for numerical integration which is useful in particular in evaluating integrals with end point singularities. Their idea is to apply a variable transform to the given integral such that the function values of the transformed integrand vanish at the both end points of the transformed integral together with all its derivatives. This formula is known as the IMT-rule [2, p. 114] and is thought to be one of the most efficient formulas for integrands with end point singularities [3]. On the other hand, Takahasi and Mori [9] proposed a family of formulas using a different kind of variable transform based on the asymptotic optimality of the trapezoidal rule for integrals over the infinite interval ( — 00, oo) [8, pp. 74-76]. Consider an integral