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.

When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable rational functions. The latter are to be chosen so as to match the most important poles of the integrand. We describe two methods for generating such quadrature ...

Gauss quadrature rules associated with a nonnegative measure support on (part of) the real axis find many applications in Scientific Computing. It is important to be able estimate error when replacing an integral by ℓ-node rule order choose suitable number of nodes. A classical approach this evaluate (2ℓ+1)-node Gauss–Kronrod rule. However, 2ℓ+1 nodes might not exist. The generalized averaged f...

In this paper Gauss-Radau and Gauss-Lobatto quadrature rules are presented to evaluate the rational integrals of the element matrix for a general quadrilateral. These integrals arise in finite element formulation for second order Partial Differential Equation via Galerkin weighted residual method in closed form. Convergence to the analytical solutions and efficiency are depicted by numerical ex...

We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. Our algorithm com...

The existence of (standard) Gauss quadrature rules with respect to a nonnegative measure dμ with support on the real axis easily can be shown with the aid of orthogonal polynomials with respect to this measure. Efficient algorithms for computing the nodes and weights of an n-point Gauss rule use the n × n symmetric tridiagonal matrix determined by the recursion coefficients for the first n orth...

Fully symmetric interpolatory integration rules are constructed for multidimensional inte-grals over innnite integration regions with a Gaussian weight function. The points for these rules are determined by successive extensions of the one dimensional three point Gauss-Hermite rule. The new rules are shown to be eecient and only moderately unstable.