On the computation of Patterson-type quadrature rules

Abstract

We describe a stable and efficient algorithm for computing positive suboptimal extensions of the Gaussian quadrature rule with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. These rules constitute a particular case of those first considered by Begumisa and Robinson (1991) and then by Patterson (1993) and have been proven to verify asymptotically good properties for a large class of weight functions. In particular, they may exist when the Gauss–Kronrod rule does not. The proposed algorithm is a nontrivial modification of the one introduced by Laurie (1997) for the Gauss–Kronrod quadrature, and it is based on the determination of an associated Jacobi matrix. The nodes and weights of the rule are then given as the eigenvalues and eigenvectors of the matrix, as in the classical Golub–Welsch algorithm (1969).