  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.
