On the Convergence of the Vector-Fitting Algorithm

Vector fitting (VF), first published by Gustavesen and Semlyen in 1999, is widely used for constructing rational models from measured or full-wave electromagnetic simulated frequency-domain responses (
S
-,
Y
-, or
Z
-parameters). As pointed out by Grivet-Talocia and Bandinu in 2006, to date there is no convergence analysis of the pole relocation iteration in VF. The goal of this paper is to elucidate this issue. It will be shown that the iteration seeks the roots of a set of coupled multivariate rational equations. For noise-free measurements, it is shown that there is no iteration involved, assuming that the number of starting poles is chosen greater or equal to the order of the underlying system. For noisy data, the VF iteration may not find any solution due to the fact that all stationary points of the fixed point iteration are repelling. Therefore, in case the iteration does not converge, we propose to incorporate the Newton step in the VF iteration, thus guaranteeing local convergence. Lastly, we provide a short review of variable projection as an alternative to VF.