In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

An efficient algorithm for the accurate computation of Gauss–Legendre and Gauss– Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton’s root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The n-point quadrature rule is computed in O(n) operations to an accuracy of essentially double precision for any n ≥ 100.

Inequalities are obtained for quadrature rules in terms of upper and lower bounds of the first derivative of the integrand. Bounds of Ostrowski type quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case. Applications to numerical integration are demonstrated.

An inequality for a companion of Ostrowski's integral inequality is proved. Application to a composite quadrature rule is considered.

We develop parallel one-dimensional globally adaptive quadrature algorithms, building on NAG code D01AKF. Our most eeective strategy replaces D01AKF's error estimate ranking strategy by a tabulation approach. D01AKF uses 61-point Gauss-Kronrod (GK) quadrature. We also use the 21-point GK rule. A fuller discussion, with expanded results, is given in 7].

The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.