Fast and accurate computation of classical Gaussian quadratures

Abstract: Algorithms for computing classical Gaussian quadrature rules (Gauss-Jacobi, Gauss-Laguerre, and Gauss-Hermite) are presented, based on globally convergent fourth-order iterative methods and asymptotic approximations, which are applied in complementary regions of the parameter space. The combination of these approaches results in methods that surpass previous algorithms in terms of speed, accuracy, and computational range (practically unrestricted). The Gauss-Radau and Gauss-Lobatto variants are also considered, along with the computation of the associated barycentric weights. Arbitrary accuracy algorithms are also provided for the symmetric cases (Gauss-Gegenbauer and Gauss-Hermite).