Carathéodory-Fejér approximation

Compute the "best" rational approx. for the exponential on R_ (as Trefethen did).

In case you are looking for a fast and stable way to compute a rational approximation of the exponential on the negative real line you can stop here. This function will to the job!

Computing rational approximations of the exponential has a long and interesting history with deep links into pure maths. Some of this is summarised in

Talbot quadratures and rational approximations
L. N. Trefethen, J. A. C. Weideman and T. Schmelzer
BIT Numerical Mathematics (2006) 46, pp. 653-670.

However, one can apply this also to other interesting functions and two examples are discussed in

Computing the gamma function using contour integrals and rational approximations
T. Schmelzer and L. N. Trefethen
SIAM J. Numer. Anal., Vol. 45 (2007), No. 2, pp. 558-571.

and

Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
T. Schmelzer and L. N. Trefethen
ETNA, Volume 29, pp. 1-18, 2007.

Please start playing around with this method by looking into the file demo.m.