Sigmoidal Transformations and the Trapezoidal Rule

A sigmoidal transformation is a one-to-one mapping of the compact interval [0,1] onto itself whose graph is S-shaped. After giving a formal definition, various mappings already given in the literature are reviewed in the light of the definition. At least one new transformation is introduced and criteria given for generating transformations having special properties. The use of these transformations in using the trapezoidal rule to evaluate the integral of f(x) over the limits [0,1] is then considered and asymptotic estimates of the truncation errors are obtained under different conditions. The paper concludes with some numerical examples.