The use of rational functions in numerical quadrature

Abstract Quadrature problems involving functions that have poles outside the interval of integration can proÿtably be solved by methods that are exact not only for polynomials of appropriate degree, but also for rational functions having the same (or the most important) poles as the function to be integrated. Constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts. The superiority of such rational=polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

DedicatedtoProfessorNácere Hayek Calil on the occasion of his 75th birthday Abstract. We consider the convergence of Gauss-type quadrature formulas for the integral � ∞ f(x)ω(x)dx, whereωisaweight function on the half line 0 [0, ∞). The n-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials Λ−p,q−1 = { �q−1 k=−p akxk}, where p = p(n) is a sequence of integers satisfying 0 ≤ p(n) ≤ 2n and q = q(n) =2n − p(n). It is proved that under certain Carleman-type conditions for the weight and when p(n) orq(n) goesto∞, then convergence holds for all functions f for which fω is integrable on [0, ∞). Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line. 1