Author
Filip, S
Nakatsukasa, Y
Trefethen, L
Beckermann, B
Journal title
SIAM Journal on Scientific Computing
Last updated
2024-03-27T17:19:15.243+00:00
Abstract
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.
Symplectic ID
698374
Download URL
http://arxiv.org/abs/1705.10132v2
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Publication type
Journal Article
Publication date
07 Aug 2018
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