A Newton's method for best uniform rational approximation

Dr. Clemens Hofreither

May 31, 2022, 3:30 p.m. S2 054

We present a novel algorithm, inspired by the recent BRASIL algorithm,
for best uniform rational approximation 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. 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 with respect to the interpolation
nodes, and for the derivative of the nullspace of a full-rank matrix.