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

Let I[f ] = ∫ 1 −1 f(x) dx, where f ∈ C ∞(−1, 1), and let Gn[f ] = ∑n i=1 wnif(xni) be the n-point Gauss–Legendre quadrature approximation to I[f ]. In this paper, we derive an asymptotic expansion as n → ∞ for the error En[f ] = I[f ]−Gn[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 f(x) ∼ ∞ ∑ s=0 Us(log(1− x))(1− x)s as x → 1−, f(x) ∼ ∞ ∑ s=0 Vs(log(1 + x))(1 + x) βs as x → −1+, where Us(y) and Vs(y) are some polynomials in y. Here, αs and βs are, in general, complex and αs, βs > −1. An important special case is that in which Us(y) and Vs(y) are constant polynomials; for this case, the asymptotic expansion of En[f ] assumes the form En[f ] ∼ ∞ ∑