Iteration-Free Computation of Gauss-Legendre Quadrature Nodes and Weights

Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical and fast iteration-free method to compute individual Gauss–Legendre node-weight pairs in O(1) complexity and with double precision accuracy. An expansion for the barycentric interpolation weights for the Gauss–Legendre nodes is also derived. A C++ implementation is available online.