Rational approximations to the complex error function

D. Laurie, University of Stellenbosch, South Africa

Wednesday, May 29
at 12.15, CAB G 51

We start from Weideman's 1994 paper on evaluation of the complex error function (also known as the Faddeeva function) by a formula valid in all of the upper half-plane. That formula can be viewed as an (n-1,n) rational approximation in which the denominator has a single pole of appropriate multiplicity, and can be derived by transforming a certain auxiliary function to the upper half-plane to the unit disk by a Moebius transformation that maps the pole in question to infinity, followed by Taylor expansion around the origin. Instead of Taylor expansion, we use near-best rational approximation on the unit circle to obtain the same accuracy with $n$ reduced by a factor of more than two.

The technique used to obtain the near-best rational approximant is the Caratheodory-Fejér method of Trefethen and Gutknecht. A key step in this method is to find the n-th largest eigenvalue of a Hankel matrix formed from some of the Taylor coefficients of the auxiliary function. The free parameter in the Moebius transformation (i.e. the point is mapped to zero) is so chosen to minimize the magnitude of that eigenvalue.