On the Computation of Recurrence Coefficients for Univariate Orthogonal Polynomials
On the Computation of Recurrence Coefficients for Univariate Orthogonal Polynomials
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DOI：
10.1007/s10915-021-01586-w
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发表时间：
2021-09-01
影响因子：
2.5
通讯作者：
Narayan, Akil
中科院分区：
文献类型：
--
作者：
Liu, Zexin;Narayan, Akil
与具有有限矩的真实的直线上的有限测度相关联的是关于该测度的正交多项式的三项公式中的递归系数。这些递归系数经常被输入到现代计算工具中，这些工具有助于评估和操作多项式的测量，并且这些任务是数值近似和求积的基础。虽然经典测度的递归系数是明确已知的，但非经典测度的递归系数通常必须进行数值计算。我们调查和审查现有的方法来计算这些单变量正交多项式族的递归系数，并提出了一种新的“预测-校正”算法的一般类的连续措施。我们结合联合收割机的预测-校正计划与一个新的混合算法，计算递归系数的一个相当广泛的一类措施，可以有连续和离散部分的稳定Lanczos程序。我们评估新算法对现有方法的准确性和效率。
Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel "predictor-corrector" algorithm for a general class of continuous measures. We combine the predictor-corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency.