a f (x)w(x) dx = n − k=0 wkf (xk) + Rn+1[f ], where w(x) is a weight function, {xk}k=0 are integration nodes, {wk} n k=0 are the corresponding weight coefficients, and Rn+1[f ] denotes the error term. During the past decades, various kinds of formulae of the above type have been developed. In this paper, we introduce a type of interpolatory quadrature, whose nodes are geometrically distributed ...

Inequalities are obtained for weighted integrals in terms of bounds involving the first derivative of the function. Quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case when the weight function w (x) ≡ 1. Applications to numerical integration are demonstrated.

In this paper Gauss-Radau and Gauss-Lobatto quadrature rules are presented to evaluate the rational integrals of the element matrix for a general quadrilateral. These integrals arise in finite element formulation for second order Partial Differential Equation via Galerkin weighted residual method in closed form. Convergence to the analytical solutions and efficiency are depicted by numerical ex...

Numerical approximations of definite integrals and related error estimations can be made using Simpson’s rules (inequalities). There are two well-known rules: 13 rule or quadrature formula 38 second formula. The aim the present paper is to extend several inequalities that hold for rule. More precisely, we prove a weighted version type inequality some Lipschitzian, bounded variations, convex fun...

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...

We introduce a finite class of weighted quadrature rules with the weight function |x|−2a exp −1/x2 on −∞,∞ as ∫−∞|x|−2a exp −1/x2 f x dx ∑n i 1 wif xi Rn f , where xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients, andRn f is the error value. We show that the above formula is valid only for the finite values of n. In...