ParaStieltjes: Parallel computation of Gauss quadrature rules using a Parareal‐like approach for the Stieltjes procedure

Abstract

SummaryThe computation of Gauss quadrature rules for arbitrary weight functions using the Stieltjes algorithm is a purely sequential process, and the computational cost significantly increases when high accuracy is required. ParaStieltjes is a new algorithm to compute the recurrence coefficients of the associated orthogonal polynomials in parallel, from which the nodes and weights of the quadrature rule can then be obtained. ParaStieltjes is based on the time‐parallel Parareal algorithm for solving time‐dependent problems, and thus enlarges the applicability of this time parallel technique to a further, new area of scientific computing. We study ParaStieltjes numerically for different weight functions, and show that substantial theoretical speedup can be obtained when high accuracy is needed. We also present an asymptotic approximation for the node and weight distribution of Gauss quadrature rules, which can be used effectively in ParaStieltjes.