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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.

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