Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair
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| Format: | Preprint |
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2025
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| _version_ | 1866911302026264576 |
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| author | Kanhya, Prince Raj, Udit |
| author_facet | Kanhya, Prince Raj, Udit |
| contents | We study the problem of determining whether a prescribed eigenpair $(λ,x)$
can be made an exact eigenpair of a nonnegative Hankel matrix through the smallest
possible structured perturbation. The task reduces to check the feasibility of a
set of linear constraints that encode both the Hankel structure and entrywise
nonnegativity. When the feasibility set is nonempty, we compute the minimum-norm
perturbation $ΔH$ such that $(H+ΔH)x=λx$. When no such perturbation
exists, we compute the nearest nonnegative Hankel matrix in a residual sense by
minimizing $\|(H+ΔH)x-λx\|_{2}$ subject to the imposed constraints.
Because closed-form formulas for the structured backward error are generally
unavailable, our method provides a fully numerical and optimization-based
framework for evaluating eigenpair sensitivity under nonnegativity-preserving
Hankel perturbations. Numerical examples illustrate both feasible and infeasible cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair Kanhya, Prince Raj, Udit Numerical Analysis We study the problem of determining whether a prescribed eigenpair $(λ,x)$ can be made an exact eigenpair of a nonnegative Hankel matrix through the smallest possible structured perturbation. The task reduces to check the feasibility of a set of linear constraints that encode both the Hankel structure and entrywise nonnegativity. When the feasibility set is nonempty, we compute the minimum-norm perturbation $ΔH$ such that $(H+ΔH)x=λx$. When no such perturbation exists, we compute the nearest nonnegative Hankel matrix in a residual sense by minimizing $\|(H+ΔH)x-λx\|_{2}$ subject to the imposed constraints. Because closed-form formulas for the structured backward error are generally unavailable, our method provides a fully numerical and optimization-based framework for evaluating eigenpair sensitivity under nonnegativity-preserving Hankel perturbations. Numerical examples illustrate both feasible and infeasible cases. |
| title | Construction of the Nearest Nonnegative Hankel Matrix for a Prescribed Eigenpair |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.04812 |