Further extension of a class of periodizing variable transformations for numerical integration

Abstract

Class S m variable transformations with integer m , for accurate numerical computation of finite-range integrals via the trapezoidal rule, were introduced and studied by the author. A representative of this class is the sin m -transformation. In a recent work of the author, this class was extended to arbitrary noninteger values of m , and it was shown that exceptionally high accuracies are achieved by the trapezoidal rule in different circumstances with suitable values of m . In another recent work by Monegato and Scuderi, the sin m -transformation was generalized by introducing two integers p and q , instead of the single integer m ; we denote this generalization as the sin p , q -transformation here. When p = q = m , the sin p , q -transformation becomes the sin m -transformation. Unlike the sin m -transformation which is symmetric, the sin p , q -transformation is not symmetric when p ≠ q , and this offers an advantage when the behavior of the integrand at one endpoint is quite different from that at the other endpoint. In view of the developments above, in the present work, we generalize the class S m by introducing a new class of nonsymmetric variable transformations, which we denote as S p , q , where p and q can assume arbitrary noninteger values, such that the sin p , q -transformation is a representative of this class and S m ⊂ S m , m . We provide a detailed analysis of the trapezoidal rule approximation following a variable transformation from the class S p , q , and show that, with suitable and not necessarily integer p and q , it achieves an unusually high accuracy when the integrand has algebraic endpoint singularities. We also illustrate our results with numerical examples via the sin p , q -transformation. Finally, we discuss the computation of surface integrals in R 3 containing point singularities with the help of class S p , q transformations.