Fast Construction of the Fejér and Clenshaw-Curtis Quadrature Rules

We present an elegant algorithm for stably and quickly generating the weights of Fejér's quadrature rules and of the Clenshaw-Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional region