Abstract: Algorithms for computing the classical Gaussian quadrature rules (Gauss-Jacobi, Gauss-Laguerre, and Gauss-Hermite) are presented, based on globally convergent fourth order iterative methods combined with asymptotic approximations, which are applied in complementary regions of the parameter space. This approach yields methods that improve upon existing algorithms in speed, accuracy, and computational range. The MATLAB algorithm for Gauss-Jacobi is faster than previous métodos and lifts the upper restrictions on the parameters imposed by those methods; for example, for degrees up to 106 all nodes and weights can be computed within the underflow limit for -1

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