Sigmoidal--trapezoidal quadrature for ordinary and Cauchy principal value integrals

Consider the evaluation of If:=^^f201f(x) dx . Among all the quadrature rules for the approximate evaluation of this integral, the trapezoidal rule is known for its simplicity of construction and, in general, its slow rate of convergence to If. However, it is well known, from the Euler-Maclaurin formula, that if f is periodic of period 1, then the trapezoidal rule can converge very quickly to If. A sigmoidal transformation is a mapping of [0,1] onto itself and is such that when applied to If gives an integrand having some degree of periodicity. Applying the trapezoidal rule to the transformed integral gives an increased rate of convergence. In this paper, we explore the use of such transformations for both ordinary and Cauchy principal value integrals. By considering the problem in a suitably weighted Sobolev space, a very satisfactory analysis of the rate of convergence of the truncation error is obtained. This combination of a sigmoidal transformation followed by the trapezoidal rule gives rise to the so-called sigmoidal-trapezoidal quadrature rule of the title