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Methods for the computation of classical Gaussian quadrature rules aredescribed which are effective both for small and large degree. These methodsare reliable because the iterative computation of the nodes has guaranteedconvergence, and they are fast due to their fourth-order convergence and itsasymptotic exactness for an appropriate selection of the variables. ForGauss--Hermite and Gauss--Laguerre quadratures, local Taylor series can be usedfor computing efficiently the orthogonal polynomials involved, with exactinitial values for the Hermite case and first values computed with a continuedfraction for the Laguerre case. The resulting algorithms have almostunrestricted validity with respect to the parameters. Full relative precisionis reached for the Hermite nodes, without any accuracy loss and for any degree,and a mild accuracy loss occurs for the Hermite and Laguerre weights as well asfor the Laguerre nodes. These fast methods are exclusively based on convergentprocesses, which, together with the high order of convergence of the underlyingiterative method, makes them particularly useful for high accuracycomputations. We show examples of very high accuracy computations (of up to$1000$ digits of accuracy).

Fast, reliable and unrestricted iterative computation of Gauss--Hermite and Gauss--Laguerre quadratures