Fast algorithms for Jacobi expansions via nonoscillatory phase functions
This work provides more efficient numerical methods for problems involving Jacobi polynomials, benefiting computational mathematics and related fields.
The authors developed fast algorithms for Jacobi polynomial evaluation, Sturm-Liouville eigentransforms, and Gauss-Jacobi quadrature using nonoscillatory phase functions, achieving better performance than existing methods in most respects.
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.