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

Abstract:

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.