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We propose an IMT-type quadrature formula which achieves the same asymptotic error estimate as the DE formula. The point of the idea is to optimize the parameters of the IMT-type transformation depending on the number of sampling points. We also show the performance of our IMT-type quadrature formula by numerical examples.

The double exponential formula is known to be very powerful for evaluation of various kinds of integrals, in particular integrals with end point singularities or integrals over the half infinite interval. It is also known that a weak point of this formula is the inefficiency when applied to a slowly decaying oscillatory integral over the half infinite interval such as I = ∫0∞f1(x) sin x dx, f1(x) is an algebraic function. In this paper we propose a new type of the double exponential formula which is quite efficient for evaluation of the integral mentioned above. It is based on such a transformation that makes the points of the formula after the transformation approach to the zeros of sin x double exponentially for large x.

This paper gives a survey of the results known to date about quadrature formulas obtained by variable transformation followed by an application of the trapezoidal rule with an equal mesh size. It has been shown that a formula obtained by an appropriate transformation is in general efficient and also robust against the end point singularity. Various kinds of useful transformations together with the asymptotic error behaviors of the resulting quadrature formulas are summarized. In particular special emphasis is placed on an asymptotically optimal formula called the double exponential formula, abbreviated as the DE-rule, which is characterized by the double exponential decrease of its weights in the neighborhood of the end points of the transformed interval of integration.