The estimation of the quadrature error a Gauss rule when applied to approximation an integral determined by real-valued integrand and nonnegative measure with support on real axis is important problem in scientific computing. Laurie developed anti-Gauss rules as aid estimate this error. Under suitable conditions associated give upper lower bounds for value desired integral. It then natural use ...

We consider a problem that arises in the evaluation of computer graphics illumination models. In particular, there is a need to nd a nite set of wavelengths at which the illumination model should be evaluated. The result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene. These values are th...

Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f}. These are quadrature formulas with n positive weights and with the n−m remaining nodes real a...

Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...

A new boundary integral formulation of the second kind for exterior Stokes flow is introduced. The formulation is stable, complete, singularity-free, and natural for bodies of complicated shape and topology. We prove an existence and uniqueness result for the formulation for arbitrary flows and illustrate its performance via several numerical examples using a Nyström method with Gauss–Legendre ...

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The m...

In recent years a number of authors have considered an error analysis for quadrature rules of Newton-Cotes type. In particular, the mid-point, trapezoid and Simpson rules have been investigated more recently ([2], [4], [5], [6], [11]) with the view of obtaining bounds on the quadrature rule in terms of a variety of norms involving, at most, the first derivative. In the mentioned papers explicit...