Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gauss-type quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a sign-variable measure, which arises in connection with Gauss-Kronrod quadrature, and power (or implicit) orthogonality enc...

Abstract: For finite positive Borel measures supported on the real line we consider a new type of quadrature rule with maximal algebraic degree of exactness, which involves function derivatives. We prove the existence of such quadrature rules and describe their basic properties. Also, we give an application of these quadrature rules to the solution of a Cauchy problem without solving it directl...

We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain. Mathe...

In this paper, the Gaussian type quadrature rules for fuzzy functions are discussed. The errors representation and convergence theorems are given. Moreover, four kinds of Gaussian type quadrature rules with error terms for approximate of fuzzy integrals are presented. The present paper complements the theoretical results of the paper by T. Allahviranloo and M. Otadi [T. Allahviranloo, M. Otadi,...

This paper is concerned with the approximation of matrix functionals defined by a large, sparse or structured, symmetric definite matrix. These functionals are Stieltjes integrals with a measure supported on a compact real interval. Rational Gauss quadrature rules that are designed to exactly integrate Laurent polynomials with a fixed pole in the vicinity of the support of the measure may yield...