Error estimates for interpolatory quadrature formulae

Abstract

Summary. In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pdlya type formulae.Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainder R[f] of a formula of order m can be represented as c .fc,"!(~). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases. Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed order m the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.

Error bounds for Gauss-Kronrod quadrature formulae

“…also Rabinowitz [13] for a proof of the nondefiniteness of R2/f+x . Brass and Förster [2], Brass and Schmeisser [3], and Monegato [7] proved upper bounds for c3n+2+K(R2*+x). In the following theorem, we give lower bounds as well as new upper bounds for c3n+2+K(R2/f+x) in the case w = 1 that improve the hitherto best-known bounds.…”

“…Therefore, if f is analytic in a region containing the interval [−1, 1], there exist positive constants 0 < r < 1 and M > 0 satisfying (2). The following theorem is proved in Sect.…”

“…The C-C rule, the Basu rule and the endpoint-corrected rule are of closed type having the endpoints ±1, while the Fejér rules of the first and second kind are the open type versions of the Basu rule and the C-C rule, respectively. The asymptotic convergence properties of the four C-C rule family are discussed in [2,5]. Table 1 The node polynomials and the nodes for the five rules of the C-C family…”

“…With the added hypothesis k = 1, the exact values of pm+l ;m are obtained in [6], [3]; with the same hypothesis, the asymptotics of pm-i+\{Q?tm+i)m) is given in [5].…”

Error estimate for a corrected Clenshaw–Curtis quadrature rule

“…Therefore, if f is analytic in a region containing the interval [−1, 1], there exist positive constants 0 < r < 1 and M > 0 satisfying (2). The following theorem is proved in Sect.…”

“…The C-C rule, the Basu rule and the endpoint-corrected rule are of closed type having the endpoints ±1, while the Fejér rules of the first and second kind are the open type versions of the Basu rule and the C-C rule, respectively. The asymptotic convergence properties of the four C-C rule family are discussed in [2,5]. Table 1 The node polynomials and the nodes for the five rules of the C-C family…”

Asymptotically Optimal Error Bounds for Quadrature Rules of Given Degree

“…With the added hypothesis k = 1, the exact values of pm+l ;m are obtained in [6], [3]; with the same hypothesis, the asymptotics of pm-i+\{Q?tm+i)m) is given in [5].…”