Inequalities are obtained for quadrature rules in terms of upper and lower bounds of the first derivative of the integrand. Bounds of Ostrowski type quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case. Applications to numerical integration are demonstrated.

It is generally agreed that of all quadrature formulae, the trapezoidal rule, while being the simplest, is also the least accurate. There is, however, a rather general class of integrals for which the trapezoidal rule can be shown to be a highly accurate means of obtaining numerical values. Specifically, if the integrand is a periodic, even function with all derivatives continuous, then the int...

s A numerical iterative scheme is suggested to solve the Euler equations in two and three dimensions. The step of the iteration procedure consists of integration over the velocity which is here carried out by three di erent approximate integration methods, and in particular, by a special Monte Carlo technique. Regarding the Monte Carlo integration, we suggest a dependent sampling technique whic...

When doing error analysis for numerical quadrature, achieving good uniform bounds on higher order derivatives of the integrand is paramount. As undergraduates become increasingly adept with programmable calculators, numerical integration schemes such as Simpson’s Rule and the Trapezoidal Rule take on a new relevance. Although it may be a rare calculus class that dwells overmuch on error bounds ...

It is generally agreed that of all quadrature formulae, the trapezoidal rule, while being the simplest, is also the least accurate. There is, however, a rather general class of integrals for which the trapezoidal rule can be shown to be a highly accurate means of obtaining numerical values. Specifically, if the integrand is a periodic, even function with all derivatives continuous, then the int...

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...

In this paper, a novel algorithm for computing Physical Optics (PO) integrals is introduced. In this method, the integration problem is converted to an inverse problem by Levin’s integration algorithm. Furthermore, the singularities, that are possible to occur in the applications of Levin’s method, are handled by employing trapezoidal rule together with Levin’s method. Finally, the computationa...