2021/01/28 by Zexin Liu, Liu, Zexin, Akil Narayan +1
Computer Science · Mathematics · #33D45 #42C10 #65D15 #FOS: Mathematics #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation #cs.NA #math.NA #msc:33D45 #msc:42C10 #msc:65D15
paper · pdf · doi:10.48550/arxiv.2101.11963
20 pages, 2 figures
openalex publication_date 2021/01/28 · arxiv created 2021/01/29 · arxiv updated 2021/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.