The idea of extending univariate quadrature formulae to cubature formulae that hold for spaces of polyharmonic functions is employed to obtain in a new way bivariate trapezoidal cubature rules. The notion of univariate monospline is extended to functions of two variables in terms of a solution of Poisson’s equation. This approach allows us to characterize the error of the trapezoidal cubature f...

Numerical integration is the study of how the numerical value of an integral can be found. Also called quadrature, which refers to nding a square whose area is the same as the area under a curve, it is one of the classical topics of numerical analysis. Of central interest is the process of approximating a deenite integral from values of the in-tegrand when exact mathematical integration is not ...

Transient analysis of nonlinear problems in structural and solid mechanics is mainly carried out using direct time integration of the equations of motion. For reliable solutions, a stable and efficient integration algorithm is desirable. Methods that are unconditionally stable in linear analyses appear to be a natural choice for use in nonlinear analyses, but unfortunately may not remain stable...

A new IIR (infinite impulse response) type digital fractional order differentiator (DFOD) is proposed by using a new family of first order digital differentiators expressed in the second order IIR filter form. The integer first order digital differentiators are obtained by the stable inversion of the weighted sum of Simpson integration rule and the trapezoidal integration rule. The distinguishi...

An interesting property of the midpoint rule and trapezoidal rule, which is expressed by the so-called Hermite–Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite–Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. I...

The numerical computation of P{X > x} can be accomplished in a variety of ways. An appealing class of methods may be derived from the contour integral connecting P{X > x} and its Fourier representation. Statisticians have largely focused on deriving saddlepoint approximations for this contour integral. The accuracy of such approximations is generally understood in vague terms only and, perhaps ...

A new class of quadrature rules for the integration of both regular and singular functions is constructed and analyzed. For each rule the quadrature weights are positive and the class includes rules of arbitrarily high-order convergence. The quadratures result from alterations to the trapezoidal rule, in which a small number of nodes and weights at the ends of the integration interval are repla...

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral I [f ]= ∫ 1 0 f (x) dx via a variable transformation x = φ(t) that maps [0,1] to itself, and then approximates the resulting transformed integral I [f ] = ∫ 1 0 f ( φ(t) ) φ′(t) dt by the trapezoidal rule. In this work, we propose ...

In this paper, we utilize a variant of the Grüss inequality to obtain some new perturbed trapezoid inequalities. We improve the error bound of the trapezoid rule in numerical integration in some recent known results. Also we give a new Iyengar’s type inequality involving a second order bounded derivative for the perturbed trapezoid inequality.