VARIABLE TRANSFORMATIONS AND GAUSS-LEGENDRE QUADRATURE FOR INTEGRALS WITH ENDPOINT SINGULARITIES

摘要

Gauss-Legendre quadrature formulas have excellent convergence properties when applied to integrals f~1_0 f(x) dx with f __ C~__ [0, 1]. However, their performance deteriorates when the integrands f(x) are in C~__(0, 1) but are singular at x = 0 and/or x = 1.One way of improving the performance of Gauss-Legendre quadrature in such cases is by combining it with a suitable variable transformation such that the transformed integrand has weaker sin_gularities than those of f(x). Thus, if x = _×(t) is a variable transformation that maps [0, 1] onto itself, we apply Gauss-Legendre quadrature to the trans_formed integral f~1_O f(_×(t))_×'(t) dt, whose singularities at t = 0 and/or t = 1 are weaker than those of f (x) at x = 0 and/or x = 1. In this work, we first de_fine a new class of variable transformations we denote S_(p,q), where p and q are two positive parameters that characterize it. We also give a simple and easily computable representative of this class. Next, by invoking some recent results by the author concerning asymptotic expansions of Gauss_Legendre quadra_ture approximations as the number of abscissas tends to infinity, we present a thorough study of convergence of~the combined approximation procedure, with variable transformations from S_(p,q). We show how optimal results can be obtained by adjusting the parameters p and q of the variable transformation in an appropriate fashion. We also give numerical examples that confirm the theoretical results.