Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2021-07-20 , DOI: 10.1007/s10915-021-01586-w Zexin Liu 1 , Akil Narayan 1
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.
中文翻译:
关于单变量正交多项式递归系数的计算
与具有有限矩的实线上的有限度量相关联的是关于该度量的正交多项式的三项公式中的递推系数。这些递推系数通常是现代计算工具的输入,这些工具有助于评估和操作与度量相关的多项式,并且此类任务是数值逼近和求积的基础。尽管经典测度的递推系数是明确已知的,但非经典测度的递推系数通常必须进行数值计算。我们调查和审查了计算单变量正交多项式族的这些递推系数的现有方法,并提出了一种新的“预测器-校正器”算法,用于一般的连续测量类。我们将预测器-校正器方案与稳定的 Lanczos 程序相结合,用于一种新的混合算法,该算法可以计算具有连续和离散部分的相当广泛的度量类别的递归系数。我们在准确性和效率方面根据现有方法评估新算法。