Gregory type quadrature based on quadratic nodal spline interpolation

Abstract

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

A quadrature formula associated with a univariate spline quasi interpolant

“…This QF has strong similarities with the class of trapezoidal rules with endpoint corrections of order 2 (see [4] and references therein). However, as is it shown below in Section 5.3, it gives slightly better approximations of integrals.…”

“…As I Q (f, I) is exact on Π 3 , the Peano kernel theorem (see e.g. [2, Chapter 3]) gives: 4 , we obtain:…”

“…For n large enough, one can writeE Q (f, I) = 23 5760 h 4 D 4 f (c) + O(h 5 ),therefore, the main part of the error is contained in the first term.5.3 Comparison with other integration rules and extrapolation.Let us compare the above error with errors in composite Simpson and Lacroix rules (see e.g [4]. for the latter): E S (f, I) := I(f, I) − I S (f, I) and E L (f, I) := I(f, I) − I L (f, I).…”

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

“…4, we show that K QB (t) ≤ 0, ∀t ∈ [0, n]. Then, from the mean value theorem, there exists c ∈ [0, n] such that n 0 K QB (t)f (7) (t)dt = f (7) (c) 2g (6) (a) + g (7) (ξ ) h 7 , and this completes the proof of the theorem.…”

“…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.…”