Speaker
Description
Rational functions are fundamental to several non-linear approximation problems in, for example, model reduction, system identification, and PDE problems. Consequently, one is often interested in constructing an orthonormal basis of rational functions to ensure numerical stability and conditioning.
In this talk, we present a generalized framework for constructing such bases for rational function vectors of arbitrary length k. We show that the underlying recurrence relations can be represented by a specific structured pair of k-Hessenberg matrices. We propose two efficient algorithms to compute these bases:
- An updating algorithm using rotations,
- A rational Arnoldi-type algorithm.
To demonstrate the robustness of this framework, we apply it to the rational approximation of $\sqrt{z}$ on [0,1]. We show that our method successfully handles exponentially clustered poles and recovers the optimal lightning + polynomial convergence rates.