Generalized eigenvalue methods for Gaussian quadrature rules

Generalized Gaussian Quadrature Rules in Enriched Finite Element Methods

In this paper, we present new Gaussian integration schemes for the efficient and accurate evaluation of weak form integrals that arise in enriched finite element methods. For discontinuous functions we present an algorithm for the construction of Gauss-like quadrature rules over arbitrarily-shaped elements without partitioning. In case of singular integrands, we introduce a new polar transforma...

Gaussian Quadrature and the Eigenvalue Problem

where the nodes xk belong to the range of integration and the weights wk are computable. For example, this kind of formula always results when f̂ is a polynomial of degree less than n that interpolates to f at the nodes; i.e., f̂ (xk) = f (xk) for k = 1, . . . ,n. As we show below, once the nodes xk are fixed, it is easy to choose the weights wk so that if f is any polynomial of degree less than ...

Quadrature Rules for Fuzzy Integrals

Generalized anti-Gauss quadrature rules

Abstract. Gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis. Laurie proposed an (n+1)-point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n-point Gauss quadrature rule for all polynomials of degree up to 2n + 1. This rule is referred to as an anti-Gauss ru...

Generalized averaged Szegő quadrature rules

Szegő quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane. However, often it is difficult to determine the quadrature error. Recently, Spalević introduced generalized averaged Gauss quadrature rules for estimating the quadrature error obtained when applying Gauss quadrature over an interval on the real axis. We describe analogous quadrat...