Variable transformations and Gauss-Legendre quadrature for integrals with endpoint singularities

Gauss-Legendre quadrature formulas have excellent convergence properties when applied to integrals $\int^1_0f(x) dx$ with $f\in C^\infty[0,1]$. However, their performance deteriorates when the integrands $f(x)$ are in $C^\infty(0,1)$ 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 singularities than those of $f(x)$. Thus, if $x=\psi(t)$ is a variable transformation that maps $[0,1]$ onto itself, we apply Gauss-Legendre quadrature to the transformed integral $\int^1_{0}f(\psi(t))\psi'(t) dt$, 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 define a new class of variable transformations we denote $\widetilde{\mathcal{S}}_{p,q}$, 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 quadrature 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 $\widetilde{\mathcal{S}}_{p,q}$. 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.