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 on a typical laptop.

描述了经证明具有收敛性的迭代方法，用于计算高斯-雅各比正交。该方法不需要对节点进行先验估计即可保证其四阶收敛。它们显示出通常比以前的方法快，并且对参数范围没有实际限制。求积分的节点和权重的唯一方法是基于收敛过程，再加上用于计算节点的定点方法的四阶收敛，这使得它成为高精度计算的理想方法，以至于如此在典型的笔记本电脑上，甚至可以计算具有数百万个节点和数千个数字的正交规则。