Convergence acceleration of Gauss-Chebyshev quadrature formulae (Q1418856)

Let \(f\) be a function that is holomorphic in \((-1,1)\) and that has singularities of algebraic or combined algebraic/logarithmic form at the points \(\pm1\). To approximate the integral \(\int_{-1}^1 f(x) \,dx\), the authors suggest to rewrite the integral as \(\int_{-1}^1 g(x) (1-x^2)^{-1/2} \,dx\) with \(g(x) = f(x) (1-x^2)^{1/2}\), and then to use the Gauss-Chebyshev quadrature formula for \(g\). An asymptotic expansion for the error of this procedure is derived. Based on this expansion, extrapolation methods are applied to accelerate the convergence.