{"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/the-akhiezer-iteration","title":"The Akhiezer iteration","arxiv_id":"2312.02384","date":"2023-12-04","proceeding":null,"authors":["Cade Ballew","Thomas Trogdon"],"abstract":"We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint intervals. The iteration applies to shifted linear solves and can then be used for efficient matrix function approximation. Using the asymptotics of orthogonal polynomials, error bounds are provided. A key component in the efficiency of the method is the ability to compute the first $k$ orthogonal polynomial recurrence coefficients and the first $k$ weighted Stieltjes transforms of these orthogonal polynomials in $\\mathrm{O}(k)$ complexity using a numerical Riemann--Hilbert approach. For a special class of orthogonal polynomials, the Akhiezer polynomials, the method can be sped up significantly, with the greatest speedup occurring in the two interval case where important formulae of Akhiezer are employed and the Riemann--Hilbert approach is bypassed.","url_abs":"https://arxiv.org/abs/2312.02384v2","url_pdf":"https://arxiv.org/pdf/2312.02384v2.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":"the-akhiezer-iteration","repo_url":"https://github.com/cade-b/akhiezeriteration","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}