Abstract
This presentation will summarize work done in the last years with several
coauthors on the use of the barycentric representation for
interpolating with rational functions. In a first part, the classical
nonlinear interpolation problem, with and without prescribed poles, will be
addressed. Moreover, a new interpolation/approximation problem which eliminates
the two pitfalls of classical rational interpolation, unattainable points and undesired
poles in the interval of interpolation, will be described (work in collaboration
with Hans Mittelmann).
The second part will deal with recent work on linear rational interpolation,
which consists in interpolating with a denominator depending on the nodes, but not on
the interpolated function. We could show that, for Cebysev points shifted with
analytic maps, a rational function presented in 1988 converges exponentially.
We will demonstrate numerically how such shifts lead to differentiation matrices
with drastically improved pseudospectrum, an important feature when solving time
evolution problems with ODE solvers (work in collaboration with Richard
Baltensperger and Benjamin Noel).
