Constructing Orthogonal Rational Function Vectors with an application in Rational Approximation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909995201724416 |
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| author | Vermeiren, Robbe |
| author_facet | Vermeiren, Robbe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_11317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constructing Orthogonal Rational Function Vectors with an application in Rational Approximation Vermeiren, Robbe Numerical Analysis 65F18 (Primary) 65D15 (Secondary) G.1.2; G.1.3; F.2.1 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. |
| title | Constructing Orthogonal Rational Function Vectors with an application in Rational Approximation |
| topic | Numerical Analysis 65F18 (Primary) 65D15 (Secondary) G.1.2; G.1.3; F.2.1 |
| url | https://arxiv.org/abs/2601.11317 |