NEP_MiniMax: An Approach for NEPs Based on Matrix-valued Minimax Approximations
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| Format: | Preprint |
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2026
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| author | Zhang, Chenkun Gu, Jiawei Zhang, Lei-Hong |
| author_facet | Zhang, Chenkun Gu, Jiawei Zhang, Lei-Hong |
| contents | We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(λ)\mathbf{u}= 0$ on compact continua $Ω\subset \mathbb{C}$. The method combines two key components: (1) a rational minimax approximation scheme where the {m-d-Lawson} algorithm constructs a minimax rational approximation for the vector-valued function from $T(x)$'s split form, yielding a matrix-valued rational approximation $R^*(x) = P^*(x)/q^*(x) \approx T(x)$, and (2) a structure-exploiting linearization technique. The minimax approximation guarantees uniform accuracy while generally keeping $R^*(x)$ pole-free in $Ω$. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^*(λ) \mathbf{u}= 0$ via a strong linearization that exactly preserves eigenvalue multiplicities. Numerical experiments on benchmarks from the NLEVP collection demonstrate competitiveness with state-of-the-art methods (e.g., Beyn, NLEIGS, SV-AAA) in efficiency and accuracy, with theoretical error bounds directly relating eigenpair approximations to the rational approximation quality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13794 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | NEP_MiniMax: An Approach for NEPs Based on Matrix-valued Minimax Approximations Zhang, Chenkun Gu, Jiawei Zhang, Lei-Hong Numerical Analysis 65F15, 65H17, 41A20, 65D15 We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(λ)\mathbf{u}= 0$ on compact continua $Ω\subset \mathbb{C}$. The method combines two key components: (1) a rational minimax approximation scheme where the {m-d-Lawson} algorithm constructs a minimax rational approximation for the vector-valued function from $T(x)$'s split form, yielding a matrix-valued rational approximation $R^*(x) = P^*(x)/q^*(x) \approx T(x)$, and (2) a structure-exploiting linearization technique. The minimax approximation guarantees uniform accuracy while generally keeping $R^*(x)$ pole-free in $Ω$. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^*(λ) \mathbf{u}= 0$ via a strong linearization that exactly preserves eigenvalue multiplicities. Numerical experiments on benchmarks from the NLEVP collection demonstrate competitiveness with state-of-the-art methods (e.g., Beyn, NLEIGS, SV-AAA) in efficiency and accuracy, with theoretical error bounds directly relating eigenpair approximations to the rational approximation quality. |
| title | NEP_MiniMax: An Approach for NEPs Based on Matrix-valued Minimax Approximations |
| topic | Numerical Analysis 65F15, 65H17, 41A20, 65D15 |
| url | https://arxiv.org/abs/2603.13794 |