On convergence rates of Fejér and Gauss–Chebyshev quadrature rules

Quadrature Rules for Fuzzy Integrals

Convergence guarantees for kernel-based quadrature rules in misspecified settings

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super√ n convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging settings where integrands are expensive to evaluate or where integrands are high dimensional. These rules are based on the assumption that the integrand has a certain ...

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 o...

Quadrature rules on unbounded interval

After some remarks on the convergence order of the classical gaussian formula for the numerical evaluation of integrals on unbounded interval, the authors develop a new quadrature rule for the approximation of such integrals of interest in the practical applications. The convergence of the proposed algorithm is considered and some numerical examples are given.

On the Convergence Rates of Gauss and Clenshaw-Curtis Quadrature for Functions of Limited Regularity

We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw–Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n−s−1) for some s > 0, Clenshaw–Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johns...