A note on extended Gaussian quadrature rules

Some Remarks on the Construction of Extended Gaussian Quadrature Rules

We recall some results from a paper by Szego on a class of polynomials which are related to extended Gaussian quadrature rules. We show that a very efficient algorithm, for the computation of the abscissas of the rules in question, was already described in that paper. We also point out that this method extends to rules for integrals with an ultraspherical-type weight function. A bound for the e...

Gaussian quadrature rules using function derivatives

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...

On the Construction of Gaussian Quadrature Rules from Modified Moments

Given a weight function a>(x) on (a, ß), and a system of polynomials {p*(x)}"=oi with degree pk{x) = k, we consider the problem of constructing Gaussian quadrature rules fff(x)w(x)dx = Yf= i V/ift"') from "modified moments" vk = ^pk(x)w(x)dx. Classical procedures take pk{x) = xk, but suffer from progressive ill-conditioning as n increases. A more recent procedure, due to Sack and Donovan, takes...

Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method

New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadra...

Quadrature Rules for Fuzzy Integrals