A Newton method for best uniform rational approximation

We present a novel algorithm, inspired by the recent BRASIL algorithm, for best uniform rational approximation of real continuous functions on real intervals based on a formulation of the problem as a nonlinear system of equations and barycentric interpolation. We derive a closed form for the Jacobian of the system of equations and formulate a Newton's method for its solution. The resulting method for best uniform rational approximation can handle singularities and arbitrary degrees for numerator and denominator. We give some numerical experiments which indicate that it typically converges globally and exhibits superlinear convergence in a neighborhood of the solution. A software implementation of the algorithm is provided. Interesting auxiliary results include formulae for the derivatives of barycentric rational interpolants with respect to the interpolation nodes, and for the derivative of the nullspace of a full-rank matrix.

Automated summary

The algorithm is based on a formulation of the problem as a nonlinear system of equations and barycentric interpolation for best uniform rational approximation of real continuous functions on real intervals, and can handle singularities and arbitrary degrees for numerator and denominator. Numerical experiments show that it typically converges globally and exhibits superlinear convergence in a neighborhood of the solution. Interesting auxiliary results include formulae for the derivatives of barycentric rational interpolants and for the derivative of the nullspace of a full-rank matrix.