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

Kzaz, M.; Prévost, M.

Numer. Algorithms 34, No. 2-4, 379-391 (2003).

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.

Reviewer: Kai Diethelm (Braunschweig)

MSC:

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

Keywords:

Gauss-Chebyshev quadrature; asymptotic expansion; convergence acceleration; analytic function; singular integrand; extrapolation methods