On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple and stable Chebyshev–Jacobi transform

Abstract

Received on ; revised on ]We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev-Lobatto points. This is achieved via a decomposition of Hahn's interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw-Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch's modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability.

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

“…It is also not clear a priori how numerically stable it is to convert down many degrees. But recent work by Slevinsky [44,45,46] shows great promise in this regard for fast and stable algorithms with general Jacobi polynomial bases. Slevinsky showed how to use a divide-and-conquer butterfly method to compute the fast conversion from arbitrary Jacobi parameters to parameter values −1 < a, b ≤ 0.…”

Fast algorithms for Jacobi expansions via nonoscillatory phase functions

“…form a basis in its space of solutions. By repeatedly differentiating (33) and (34), it can be shown that ψ is a phase function for (32) if and only if its derivative satisfies the nonlinear ordinary differential equation…”

“…transforms we apply, the Jacobi transform can be implemented easily by combining the method of [33] with the nonuniform fast Fourier transform (see, for instance, [23] and [22] for an approach of this type for applying the Legendre transform and its inverse). Other methods for applying the Jacobi transform, some of which have lower asymptotic complexity than the algorithm of [33] and the approach of this paper, are available.…”

“…Fortunately, the running time of this precomputation procedure is quite small. Indeed, for most values of n, it requires less time than a single application of the Jacobi-Chebyshev transform via [33]. Once the precomputation phase has been dispensed with, we our algorithm for the application of the Jacobi transform is roughly ten times faster than the algorithm of [33].…”

Fast algorithms for the multi-dimensional Jacobi polynomial transform

“…Almost all such algorithms can be placed into one of two categories. Algorithms in the first category, such as [25,26,27,28,2,29,30,31,39], make use of the structure of the connection matrices which take the coefficients in the expansion of a function in terms of one set of Jacobi polynomials to the coefficients in the expansion of the same function for a different set of Jacobi polynomials. They typically operate by computing the Chebyshev coefficients of expansion and then applying a connection matrix or a series of connection matrices to obtain the coefficients in the desired expansion.…”