Computation of Gauss-Kronrod quadrature rules

Recently Laurie presented a new algorithm for the computation of (2n + 1)-p oint Gauss-Kronrod quadrature rules with real nodes and positive weights. T his algorithm first determines a symmetric tridiagonal matrix of order 2n 1 from certain mixed moments, and then computes a partial spectral factori zation. We describe a new algorithm that does not require the entries of th e tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation ph ase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenpr oblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout th e paper we emphasize how the structure of the algorithm makes efficient imp lementation on parallel computers possible. Numerical examples illustrate t he performance of the algorithm.