{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/backward-errors-for-multiple-eigenpairs-in","title":"Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems","arxiv_id":"2405.06327","date":"2024-05-10","proceeding":null,"authors":["Miryam Gnazzo","Leonardo Robol"],"abstract":"Given a nonlinear matrix-valued function $F(\\lambda)$ and approximate eigenpairs $(\\lambda_i, v_i)$, we discuss how to determine the smallest perturbation $\\delta F$ such that $[F + \\delta F](\\lambda_i) v_i = 0$; we call the distance between the $F$ and $F + \\delta F$ the backward error for this set of approximate eigenpairs. We focus on the case where $F(\\lambda)$ 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 $\\delta F_i$ are also given.","url_abs":"https://arxiv.org/abs/2405.06327v2","url_pdf":"https://arxiv.org/pdf/2405.06327v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"backward-errors-for-multiple-eigenpairs-in","repo_url":"https://github.com/miryamgnazzo/backward-error-nonlinear","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}