Gaussian quadrature with weight function $x\sp{n}$ on the interval $(-1,\,1)$

Gaussian interval quadrature rule for exponential weights

Interval quadrature formulae of Gaussian type on R and R+ for exponential weight functions of the form w(x) = exp(−Q(x)), where Q is a continuous function on its domain and such that all algebraic polynomials are integrable with respect to w, are considered. For a given set of nonoverlapping intervals and an arbitrary n, the uniqueness of the n-point interval Gaussian rule is proved. The result...

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 Universalityof the Gaussian Quadrature FormulaKnut

Several deenitions of universality of an n-point quadrature formula Q n are discussed. Universality means that Q n is able to compete with the respective optimal formula in many classes of functions. It is proved in a certain sense that the Gaus-sian quadrature formula satisses such a universality criterion. The underlying classes of functions are In each of these classes, we loose at most the ...

Quadrature rules on unbounded interval

After some remarks on the convergence order of the classical gaussian formula for the numerical evaluation of integrals on unbounded interval, the authors develop a new quadrature rule for the approximation of such integrals of interest in the practical applications. The convergence of the proposed algorithm is considered and some numerical examples are given.

The Gaussian Quadrature Method