On bicubic transformation for the numerical evaluation of Cauchy principal value integrals

Abstract

AbstractRecently a bicubic transformation was introduced to numerically compute the Cauchy principal value (CPV) integrals. Numerical results show that this new method converges faster than the conventional Gauss–Legendre quadrature rule when the integrand contains different types of singularity. Assume η is the singular point of a CPV integral. The point η divides the interval [−1, 1] into two parts: [−1, η] and [η, 1]. The bicubic transformation maps the intervals [−1, η] and [η, 1] to the interval [−1, 1] with the following constraints: it maps the point η − ϵ to μn, and η + ϵ to −μn, where μn is the largest Gaussian point of an n‐point Gauss–Legendre quadrature rule, and ϵ is a user‐supplied constant. The n‐point Gauss–Legendre quadruture rule is then applied. In contrast to ordinary expectation, further numerical experiment shows that smaller ϵ does not always produce better results. In this paper we are concerned with the selection of ϵ to yield rapid convergence of numerical integration when the bicubic transformation method is applied.