Institute of Mathematics

Talk

Modul:   MAT870  Zurich Colloquium in Applied and Computational Mathematics

Computation of Best Exponential Sums Approximating 1/x in the Maximum Norm

Prof. Dr. Wolfgang Hackbusch talk

Speaker invited by: Prof. Dr. Stefan Sauter

Date: 25.09.19  Time: 16.15 - 17.45  Room: ETH HG E 1.2

Exponential sums consist of the terms a_i*exp(-b_ix). Such approximations for the functions 1/x and 1/sqrt(x) are of particular interest. We give examples of their use in quantum chemistry and tensor calculus.

The best approximation of 1/x with respect to the maximum norm is theoretically well understood, and rather sharp error estimates are known. The error decays exponentially in the number of terms. The optimal exponential sum is characterised by the equioscillation property. The Remez algorithm is the standard method in the case of polynomial approximation.

The existing literature about the numerical computation of the best approximation by exponential sums shows that all authors faced severe numerical difficulties. This may lead to the wrong impression that the problem is illposed. In the lecture a stable method is described which is used to compute best approximations for various parameters up to 57 terms.

Literature: W. Hackbusch: Computation of best L^{\infty} exponential sums for 1/x by Remez' algorithm. Comput. Vis. in Sci. (2019) 20:1-11