A quadrature formula associated with a univariate spline quasi interpolant

Abstract

We study a new simple quadrature rule based on integrating a C 1 quadratic spline quasi-interpolant on a bounded interval. We give nodes and weights for uniform and non-uniform partitions. We also give error estimates for smooth functions and we show that our formula is a useful companion to Simpson's rule. (2000): 41A15, 65D07, 65D25, 65D32. AMS subject classification

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

“…We remark that similar results for S 2 in the univariate case can be found in [33]. Approximation order in case of both symmetric/uniform and nonuniform crisscross triangulations can be observed in the next section.…”

“…Finally a possible suitably weighted means of such integration rules can provide more accurate formulas [33] and this could be a further subject to be investigated.…”

On the solution of Fredholm integral equations based on spline quasi-interpolating projectors

“…In particular we consider quadratic (d = 2) and cubic (d = 3) splines, because such a choice lies on our experience of using such functions which has proved to be efficient in many integration problems (see e.g. [10][11][12][13]). …”

Error estimate and extrapolation of a quadrature formula derived from a quartic spline quasi-interpolant

“…However, the first author has shown [17,18] that integrating QIs of even degrees provides interesting quadrature formulae. These quadrature formulae are similar to the so called trapezoidal rules with endpoint corrections, see [7,8] for an example of this type of formulae.…”

“…By studying the sign structure of the Peano kernel, we derive an explicit formula of the quadrature error. It turns out, as for quadrature formulae in [18], that the signs of errors for composite Boole's formula and quartic spline formula are opposite for all integrand f such that f (6) maintains constant sign on the interval of integration. We show, using theoretical and practical arguments, why this phenomenon occurs.…”