zbMATH — the first resource for mathematics

Convergence acceleration of Gauss-Chebyshev quadrature formulae. (English) Zbl 1036.65027
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.

65D32 Numerical quadrature and cubature formulas
65B05 Extrapolation to the limit, deferred corrections
41A55 Approximate quadratures
41A80 Remainders in approximation formulas
Full Text: DOI