Fast algorithms for Jacobi expansions via nonoscillatory phase functions

2018/03/11 by James Bremer, Haizhao Yang, Bremer, James +1
Computer Science · Engineering · Physics and Astronomy · #33C45 #65D20 #65D32 #65R10 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optical Polarization and Ellipsometry

paper · pdf · doi:10.48550/arxiv.1803.03889

openalex publication_date 2018/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach is based on the well-known fact that Jacobi's differential equation admits a nonoscillatory phase function which can be loosely approximated via an affine function over much of its domain. Our algorithms perform better than currently available methods in most respects. We illustrate this with several numerical experiments, the source code for which is publicly available.

Related

• Fast Algorithms for the Multi-dimensional Jacobi Polynomial Transform
• Error bounds for the asymptotic expansions of the Jacobi polynomials
• On the numerical calculation of the roots of special functions\n satisfying second order ordinary differential equations
• Non-iterative computation of Gauss-Jacobi quadrature
• On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev--Jacobi transform