Quadrature formulas obtained by variable transformation and the DE-rule

摘要

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.