Computing Gaussian quadrature rules with high relative accuracy

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...

Weighted quadrature rules with binomial nodes

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-...

Computing Discrepancies of Smolyak Quadrature Rules

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as a possible competitor to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadra-ture formulas consists in computing their L 2-discrepancy. Especially for larger dimensions, such computations are...

Quadrature Rules for Fuzzy Integrals

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...