Extended lagrange interpolation and nonclassical Gauss quadrature formulae

Abstract

Extensions of quadrature formulae are of importance, for example, in the construction of automatic integrators, but many sequences fail to exist in usable form. Using the theory of quasiorthogonality and reinterpreting it in terms of the standard orthogonal polynomials, we find an approximation of the integral by a convex combination of the nth Gaussian quadrature sum and a quadrature of a fixed form of highest degree of precision. We therefore prove that a variety of extended quadratures with positive weights and interior nodes can be computed directly by standard software for ordinary Gauss quadrature formulae. We also discuss how the technique can be applied to compute Gauss extensions of Gauss-Radau and Gauss-Lobatto quadrature formulae.