{"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/stability-of-the-lanczos-method-for-matrix","title":"Stability of the Lanczos Method for Matrix Function Approximation","arxiv_id":"1708.07788","date":"2017-08-25","proceeding":null,"authors":["Cameron Musco","Christopher Musco","Aaron Sidford"],"abstract":"The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \\times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[\\lambda_{min}(A), \\lambda_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision. We resolve this problem, proving that when $\\max_{x \\in [\\lambda_{min}, \\lambda_{max}]}|f(x)| \\le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\\log(nC\\|A\\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$. We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89]. For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $\\kappa$ where exact arithmetic Lanczos converges in $polylog(\\kappa)$ iterations, but Greenbaum's bound predicts $\\Omega(\\kappa^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(\\kappa^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(\\kappa)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions.","url_abs":"https://arxiv.org/abs/1708.07788v2","url_pdf":"https://arxiv.org/pdf/1708.07788v2.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":"stability-of-the-lanczos-method-for-matrix","repo_url":"https://github.com/cpmusco/fast-pcr","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1708.07788","atlas_url":"https://app.syntology.ai/?focus=1708.07788","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}