Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems
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| Format: | Preprint |
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2024
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| _version_ | 1866912247960305664 |
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| author | Gnazzo, Miryam Robol, Leonardo |
| author_facet | Gnazzo, Miryam Robol, Leonardo |
| contents | Given a nonlinear matrix-valued function $F(λ)$ and approximate eigenpairs $(λ_i, v_i)$, we discuss how to determine the smallest perturbation $δF$ such that $[F + δF](λ_i) v_i = 0$; we call the distance between the $F$ and $F + δF$ the backward error for this set of approximate eigenpairs. We focus on the case where $F(λ)$ is given as a linear combination of scalar functions multiplying matrix coefficients $F_i$, and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the $F_i$ have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the $δF_i$ are also given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_06327 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems Gnazzo, Miryam Robol, Leonardo Numerical Analysis 65H17, 65F15, 15A18 Given a nonlinear matrix-valued function $F(λ)$ and approximate eigenpairs $(λ_i, v_i)$, we discuss how to determine the smallest perturbation $δF$ such that $[F + δF](λ_i) v_i = 0$; we call the distance between the $F$ and $F + δF$ the backward error for this set of approximate eigenpairs. We focus on the case where $F(λ)$ is given as a linear combination of scalar functions multiplying matrix coefficients $F_i$, and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the $F_i$ have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the $δF_i$ are also given. |
| title | Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems |
| topic | Numerical Analysis 65H17, 65F15, 15A18 |
| url | https://arxiv.org/abs/2405.06327 |