Liu Zexin, Narayan Akil
Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute, the University of Utah.
J Sci Comput. 2021 Sep;88(3). doi: 10.1007/s10915-021-01586-w. Epub 2021 Jul 20.
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.
与实直线上具有有限矩的有限测度相关联的是关于该测度的正交多项式三项公式中的递推系数。这些递推系数常常是现代计算工具的输入，这些工具便于对关于该测度的多项式进行求值和运算，而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通过数值计算得到。我们调研并回顾了用于计算单变量正交多项式族递推系数的现有方法，并针对一般类别的连续测度提出了一种新颖的“预测 - 校正”算法。我们将预测 - 校正方案与稳定的兰索斯过程相结合，得到一种新的混合算法，该算法可计算一类相当广泛的、可能同时具有连续部分和离散部分的测度的递推系数。我们从准确性和效率方面将新算法与现有方法进行了评估。
