# A novel algorithm for best rational approximation with applications to fractional diffusion

## Dr. Clemens Hofreither

Oct. 13, 2020, 3:30 p.m. S2 054

It has recently been understood that there is a deep connection between
rational approximation of functions of the type $x \mapsto x^\alpha$ and
numerical methods for elliptic fractional diffusion problems. However, in
particular the computation of best rational approximations in the maximum
norm of these functions has posed a significant obstacle in the practical
realization of numerical methods exploiting this link.

To overcome these difficulties, a novel algorithm for computing best
uniform rational approximations to real scalar functions in the setting of
zero defect is proposed in this talk. The method, dubbed BRASIL (best

rational approximation by successive interval length adjustment), is based
on the observation that the best rational approximation $r$ to a function
$f$ must interpolate $f$ at a certain number of interpolation nodes
$(x_j)$. Furthermore, the sequence of local maximum errors per interval
$(x_{j-1},x_j)$ must equioscillate. The proposed algorithm iteratively
rescales the lengths of the intervals with the goal of equilibrating the
local errors. The required rational interpolants are computed in a stable
way using the barycentric rational formula.

The new algorithm exhibits excellent numerical stability and computes best
rational approximations of high degree to many functions in a few seconds,
using only standard IEEE double-precision arithmetic. In some examples, we
demonstrate that it converges quickly in some situations where the current
state-of-the-art method, the \texttt{minimax} function from the
\texttt{Chebfun} package, fails.