Convergence acceleration of some Gaussian quadrature formulas for analytic functions

Let (S_n) be the sequence given by the Jacobi-Gauss quadrature method when the integrand is an analytic function with a lopatilluric singularity or with a branch point on the real axis, and S its limi. We give an asymptotic representation of the errors S − S_n and of S_n+s − S_n, which leads to building other sequences which give a better approximation of the exact value of the integral than S_n. All the results are illustrated by numerical examples.     

Convergence Acceleration of Gauss–Chebyshev Quadrature Formulae

The aim of this paper is to accelerate, via extrapolation methods, the convergence of the sequences generated by the Gauss–Chebyshev quadrature formula applied to functions holomorphic in ]?1,1[ and possessing, in the neighborhood of 1 or ?1, an asymptotic expansion with log?(1±x)(1±x)^α, (1±x)^α, α>?1, as elementary elements.     

Gaussian quadrature and acceleration of convergence

The aim of this paper is to take up again the study done in previous papers, to the case where the integrand possesses an algebraic singularity within the interval of integration. The singularities or poles close to the interval of integration considered in this paper are only real or purely imaginary. This revised version was published online in August 2006 with corrections to the Cover Date.