2023/06/01 by Zexin Liu, Akil Narayan
Computer Science · Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Matrix Theory and Algorithms #Scientific Research and Discoveries
paper · doi:10.1137/22m1477131
openalex publication_date 2023/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/26
.Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram–Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.Keywordsmultivariate orthogonal polynomialsrecurrence coefficient matricesStieltjes procedureMSC codes33D5042C0565D15