Rational minimax approximation via adaptive barycentric representations
This work provides a practical and robust method for computing rational minimax approximations, which is important for numerical analysis and scientific computing applications where singularities are present.
The authors developed robust algorithms for rational minimax approximation on intervals with singularities, achieving type (80,80) approximations of |x| on [-1,1] in standard 16-digit floating-point arithmetic, a problem previously requiring 200-digit precision.
Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.