The kink phenomenon in Fejér and Clenshaw-Curtis quadrature

The Fejér and Clenshaw–Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like O( −2N ), where is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw–Curtis rules is almost indistinguishable from that of the more celebrated Gauss–Legendre quadrature rule. For larger N , however, the error decreases at the rate O( −N ), i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon. Mathematics Subject Classification (2000) 65D32 · 41A55