Eine Restdarstellung bei Gregoryschen Quadraturverfahren ungerader Ordnung

On the generalized Euler-Maclaurin formula

“…which mainly result from (12) and the strictly monotonous decreasing of the Laplace numbers (2). Hence (17) and (18) read (see also (12)) E r1 n = −(nB r r+3 + 2B r r+4 )h r+4 f (r+3) (ξ), |E r1 n | ≤ |B r r+3 |(b − a)h r+3 M r+3 , n ≥ r, r = 1, 3, 5, .…”

“…and (5) Example 2 For r = 1 (Durand's rule) and s = 3 spline methods (as have been used, e.g., in [2,8]) show the classical remainder (17); due to the numerical data from above, this can be optimally estimated:…”

“…Example 4 For odd r and s = 1 we have the Brass formula with classical remainder [2]. The condition (19) and therefore the optimal error estimate are ensured by the inequalities…”

“…are the Bernoulli numbers, (1) reduces for r = 0 to the Euler-Maclaurin summation formula (note that L 1 = 1 2 ) or for s = 0 to the Gregory quadrature formula, respectively. In (1), the remainder term E rs n is of order O(h r+s+2 ) and vanishes for polynomials up to degree r + s. "Classical" representations of the remainder are known for r = 0 [4], for r even, s = 0 [1,7], and for r odd, s = 1 [2].…”