The method of steepest descent for estimating quadrature errors

This work presents an application of the method of steepest descent to estimate quadrature errors. The method is used to provide a unified approach to estimating the truncation errors which occur when Gauss–Legendre quadrature is used to evaluate the nearly singular integrals that arise as part of the two dimensional boundary element method. The integrals considered here are of the form ∫−11h(x)dx((x−a)2+b2)α, where h(x)h(x) is a “well-behaved” function, α>0α>0 and −1