On Some Gauss and Lobatto Based Integration Formulae

On Some Gauss and Lobatto Based Integration Formulae

1. Introduction. The economy of the Gaussian quadrature formulae for carrying out numerical integration is to some extent reduced by the fact that an increase in the order of the formulae makes no use of previous integrand evaluations. Kronrod [1] has shown how the Gauss formula of degree 2n — 1 can be extended to one of degree 3rc + 2 by making use of the original n Gauss points and an additio...

Generalized Gauss – Radau and Gauss – Lobatto Formulae ∗

Computational methods are developed for generating Gauss-type quadrature formulae having nodes of arbitrary multiplicity at one or both end points of the interval of integration. Positivity properties of the boundary weights are investigated numerically, and related conjectures are formulated. Applications are made to moment-preserving spline approximation. AMS subject classification: 65D30.

Gauss-Lobatto Formulae and Extremal Problems with Polynomials

On Gautschi's conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several ...

Vandermonde systems on Gauss-Lobatto Chebyshev nodes

This paper deals with Vandermonde matrices Vn whose nodes are the Gauss–Lobatto Chebyshev nodes, also called extrema Chebyshev nodes. We give an analytic factorization and explicit formula for the entries of their inverse, and explore its computational issues. We also give asymptotic estimates of the Frobenius norm of both Vn and its inverse and present an explicit formula for the determinant o...