Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities

Abstract:

Let $I[f]=\int _{-1}^1f(x) dx,$ where $f\in C^\infty (-1,1)$, and let $G_n[f]=\sum ^n_{i=1}w_{ni}f(x_{ni})$ be the $n$-point Gauss–Legendre quadrature approximation to $I[f]$. In this paper, we derive an asymptotic expansion as $n\to \infty$ for the error $E_n[f]=I[f]-G_n[f]$ when $f(x)$ has general algebraic-logarithmic singularities at one or both endpoints. We assume that $f(x)$ has asymptotic expansions of the forms \begin{align*} f(x)&\sim \sum ^\infty _{s=0}U_s(\log (1-x))(1-x)^{\alpha _s} \quad \text {as $x\to 1-$,} f(x)&\sim \sum ^\infty _{s=0}V_s(\log (1+x))(1+x)^{\beta _s} \quad \text {as $x\to -1+$,}\end{align*} where $U_s(y)$ and $V_s(y)$ are some polynomials in $y$. Here, $\alpha _s$ and $\beta _s$ are, in general, complex and $\Re \alpha _s,\Re \beta _s>-1$. An important special case is that in which $U_s(y)$ and $V_s(y)$ are constant polynomials; for this case, the asymptotic expansion of $E_n[f]$ assumes the form \[ E_n[f] \sim \sum ^\infty _{\substack {s=0\\ \alpha _s \not \in \mathbb {Z}^+}} \sum ^\infty _{i=1} a_{si} h^{\alpha _s+i} + \sum ^\infty _{\substack {s=0\\ \beta _s\not \in \mathbb {Z}^+}} \sum ^\infty _{i=1} b_{si} h^{\beta _s+i} \quad \text {as $n\to \infty $}, \] where $h=(n+1/2)^{-2}$, $\mathbb {Z}^+=\{0,1,2,\ldots \},$ and $a_{si}$ and $b_{si}$ are constants independent of $n$.