Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights

Abstract

An efficient algorithm for the accurate computation of Gauss–Legendre and Gauss–Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton’s root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The -point quadrature rule is computed in operations to an accuracy of essentially double precision for any .