Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes, and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated, and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants for the Simpson rule.Mathematics Subject Classification (1991): 41A55, 41A15, 41A05, 65D32, 65D30, 65D07, 65D05
Using a Peano kernel technique, Jackson-type estimates with respect to the maximum norm are derived for the quadratic nodal spline interpolation error. The explicitly calculated error constants are shown to grow linearly with respect to the local mesh ratio parameter, and are, at least for the important special case of a uniform spline knot sequence, significantly smaller than those previously calculated by different methods.
Academic Press
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.