Abstract
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 ${\mathcal{O}}(1)$ 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.
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