Enclosing Chebyshev Expansions in Linear Time

Abstract

We consider the problem of computing rigorous enclosures for polynomials represented in the Chebyshev basis. Our aim is to compare and develop algorithms with a linear complexity in terms of the polynomial degree. A first category of methods relies on a direct interval evaluation of the given Chebyshev expansion in which Chebyshev polynomials are bounded, e.g., with a divide-and-conquer strategy. Our main category of methods that are based on the Clenshaw recurrence includes interval Clenshaw with defect correction (ICDC), and the spectral transformation of Clenshaw recurrence rewritten as a discrete dynamical system. An extension of the barycentric representation to interval arithmetic is also considered that has a log-linear complexity as it takes advantage of a verified discrete cosine transform. We compare different methods and provide illustrative numerical experiments. In particular, our eigenvalue-based methods are interesting for bounding the range of high-degree interval polynomials. Some of the methods rigorously compute narrow enclosures for high-degree Chebyshev expansions at thousands of points in a few seconds on an average computer. We also illustrate how to employ our methods as an automatic a posteriori forward error analysis tool to monitor the accuracy of the Chebfun feval command.