The computation of Gauss quadrature rules for arbitrary weight functions using the Stieltjes algorithm is a purely sequential process, and the computational cost significantly increases when high accuracy is required. ParaStieltjes is a new algorithm to compute the recurrence coefficients of the associated orthogonal polynomials in parallel, from which the nodes and weights of the quadrature rule can then be obtained. ParaStieltjes is based on the time‐parallel Parareal algorithm for solving time‐dependent problems, and thus enlarges the applicability of this time parallel technique to a further, new area of scientific computing. We study ParaStieltjes numerically for different weight functions, and show that substantial theoretical speedup can be obtained when high accuracy is needed. We also present an asymptotic approximation for the node and weight distribution of Gauss quadrature rules, which can be used effectively in ParaStieltjes.

中文翻译：

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ParaStieltjes：对Stieltjes过程使用类似Parareal的方法并行计算高斯正交规则

使用Stieltjes算法计算任意权重函数的高斯正交规则是一个纯粹的顺序过程，当需要高精度时，计算成本会显着增加。ParaStieltjes是一种新算法，用于并行计算关联的正交多项式的递归系数，然后可以从中获得正交规则的节点和权重。ParaStieltjes基于时间并行Parareal算法，用于解决与时间有关的问题，因此将这种时间并行技术的适用性扩展到了科学计算的另一个新领域。我们研究ParaStieltjes对不同的权函数进行数值计算，结果表明当需要高精度时，可以获得相当大的理论加速。我们还提出了高斯正交规则的节点和权重分布的渐近逼近，可以在ParaStieltjes中有效地使用它。