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Fast and reliable high accuracy computation of Gauss--Jacobi quadrature

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.