A form of Gauss-Quadrature rule over [0,1] has been investigated that involves the derivative of the integrand at the pre-assigned left or right end node. This situation arises when the underlying polynomials are orthogonal with respect to the weight function (): 1 x x ω = − over [0,1]. Along the lines of Golub's work, the nodes and weights of the quadrature rule are computed from a Jacobi-type...

Last Time: We discussed quadrature rules, which are means of estimating integrals of functions using a weighted sum of function values. Among these, one of the simplest is the trapezoidal rule, which derives its name from the fact that it attempts to approximate the area under the curve using a trapezoid. Although it can be proved the accuracy of this rule increases quadratically with the numbe...

We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the “half-point rule”, indicating that optimal rules involve a number of points roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order o...

Abstract. It is well-known that the trapezoidal rule, while being only second-order accurate in general, improves to spectral accuracy if applied to the integration of a smooth periodic function over an entire period on a uniform grid. More precisely, for the function that has a square integrable derivative of order r the convergence rate is o ( N−(r−1/2) ) , where N is a number of grid nodes. ...

For the construction of an interpolatory integration rule on the unit circle T with n nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers pn and qn, pn + qn = n − 1, which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bou...

We study the order of convergence Galerkin variational integrators for ordinary differential equations. approximate a (Lagrangian) problem by restricting space curves to set polynomials degree at most $s$ and approximating action integral using quadrature rule. show that, if rule is sufficiently accurate, thus obtained $2s$.

Few Neurologists will be unfamiliar with referral letters stating that the patient has “Rhomberg’s sign” or some “mild rhombergism” (capitalisation variable), a particularly irritating misnomer for pedants among us. As Henry Higgins fictional Professor of Phonetics [1] observed: ‘Does same thing hold true in India, Pickering? Is there peculiar habit not only dropping letter like “h”, but using ...

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...

The detection of insu ciently resolved or ill-conditioned integrand structures is critical for the reliability assessment of the quadrature rule outputs. We discuss a method of analysis of the pro le of the integrand at the quadrature knots which allows inferences approaching the theoretical 100% rate of success, under error estimate sharpening. The proposed procedure is of the highest interest...

We propose a new quadrature rule for Cauchy principal value integrals based on quadratic spline quasi-interpolants which have an optimal approximation order and satisfy boundary interpolation conditions. In virtue of these spline properties, we can prove uniform convergence for sequences of such quadratures and provide uniform error bounds. A computational scheme for the quadrature weights is g...