Constructing Orthonormal Rational Function Vectors with an application in Rational Approximation

TL;DR

The paper reframes the construction of orthonormal bases for rational function vectors as a pencil-based inverse eigenvalue problem with $k$-Hessenberg recurrences, and proposes two algorithms: an updating method and a rational vector Krylov (Arnoldi) approach. Both methods extend to $k$-component vectors and are analyzed through their associated pencils $(H,K)$ and unitary transform operations, with an isometric connection to rational Krylov spaces. Numerical experiments show the updating algorithm yields superior orthogonality and stability, and when applied to approximating $\sqrt{z}$ on $[0,1]$ it reproduces the optimal lightning-plus-polynomial convergence rate under exponentially clustered poles. The framework provides a robust, structured approach to rational approximation problems with near-singular behavior, with potential extensions to downdating and more efficient pencil representations.

Abstract

We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation of the inverse generalized eigenvalue problem by Van Buggenhout et al.\ (2022), we extend it to rational vectors of arbitrary length $k$, where the recurrence relations are represented by a pair of $k$-Hessenberg matrices, i.e., matrices with possibly $k$ nonzero subdiagonals. An updating algorithm based on similarity transformations using rotations and a Krylov-type algorithm related to the rational Arnoldi method are derived. The performance is demonstrated on the rational approximation of $\sqrt{z}$ on $[0,1]$, where the optimal lightning + polynomial convergence rate of Herremans, Huybrechs, and Trefethen (2023) is successfully recovered. This illustrates the robustness of the proposed methods for handling exponentially clustered poles near singularities.

Figures (3)

• Figure 1: Left: errors $\operatorname{err}_Q$ and $\operatorname{err}_{\bm{\phi}}$. Right: errors $\operatorname{err}_{\bm{p}}$ and $\operatorname{err}_r$ for the updating (red) and Krylov-type (blue) algorithms, with equidistant nodes on the complex unit circle and equidistant poles on a circle of radius $3/2$. All errors are averaged over 5 runs.
• Figure 2: Left: errors $\operatorname{err}_Q$ and $\operatorname{err}_{\bm{\phi}}$. Right: errors $\operatorname{err}_{\bm{p}}$ and $\operatorname{err}_r$ for the updating (red) and Krylov-type (blue) algorithms, with equidistant nodes on the complex unit circle and two of them close to each other. All errors are averaged over 5 runs.
• Figure 3: Left: $\|f - \phi_{i}(z)\|_{\infty}$ for $2 \le i \le N_1 + 2$ (blue) and $N_1 + 3 \le i \le N$ (red). Grey dots indicate the optimal approximations selected for certain $N$ following the theoretical minimax convergence speed $\mathcal{O}\left(\exp(-\pi \sqrt{2N})\right)$. Right: Error $\|f - r\|_{\infty}$ for different values of N and a comparison of the evaluations $r(0)$ (see eq. \ref{['eq:rwithzero']}) and $\widehat{r}(0)$ (see eq. \ref{['eq:rwithoutzero']}).

Theorems & Definitions (15)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4