  Iterative methods with certified convergence for the computation of
Gauss--Jacobi quadratures are described. The methods do not require a priori
estimations of the nodes to guarantee its fourth-order convergence. They are
shown to be generally faster than previous methods and without practical
restrictions on the range of the parameters. The evaluation of the nodes and
weights of the quadrature is exclusively based on convergent processes which,
together with the fourth order convergence of the fixed point method for
computing the nodes, makes this an ideal approach for high accuracy
computations, so much so that computations of quadrature rules with even
millions of nodes and thousands of digits are possible in a typical laptop.
