Barycentric-Remez algorithms for best polynomial approximation in the chebfun system - Pachon, R; Trefethen, L (BIT Numerical Mathematics (Springer), 2008)
Variants of the Remez algorithm for best polynomial approximation are presented based on two key features: the use of the barycentric interpolation formula to represent the trial polynomials, and the setting of the whole computation in the chebfun system, where the determination of local and global extrema at each iterative step becomes trivial. The new algorithms make it a routine matter to compute approximations of degrees in the hundreds, and as an example, we report approximation of |x| up to degree 10,000. Since barycentric formulas can also represent rational functions, the algorithms we introduce may also point the way to new methods for computing best rational approximations.