  Methods for the computation of classical Gaussian quadrature rules are
described which are effective both for small and large degree. These methods
are reliable because the iterative computation of the nodes has guaranteed
convergence, and they are fast due to their fourth-order convergence and its
asymptotic exactness for an appropriate selection of the variables. For
Gauss--Hermite and Gauss--Laguerre quadratures, local Taylor series can be used
for computing efficiently the orthogonal polynomials involved, with exact
initial values for the Hermite case and first values computed with a continued
fraction for the Laguerre case. The resulting algorithms have almost
unrestricted validity with respect to the parameters. Full relative precision
is 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 as
for the Laguerre nodes. These fast methods are exclusively based on convergent
processes, which, together with the high order of convergence of the underlying
iterative method, makes them particularly useful for high accuracy
computations. We show examples of very high accuracy computations (of up to
$1000$ digits of accuracy).
