{"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/randomized-gram-schmidt-process-with","title":"Randomized Gram-Schmidt process with application to GMRES","arxiv_id":"2011.05090","date":"2020-11-10","proceeding":null,"authors":["Oleg Balabanov","Laura Grigori"],"abstract":"A randomized Gram-Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical Gram-Schmidt process while being at least as numerically stable as the modified Gram-Schmidt process. Our approach is based on random sketching, which is a dimension reduction technique consisting in estimation of inner products of high-dimensional vectors by inner products of their small efficiently-computable random images, so-called sketches. In this way, an approximate orthogonality of the full vectors can be obtained by orthogonalization of their sketches. The proposed Gram-Schmidt algorithm can provide computational cost reduction in any architecture. The benefit of random sketching can be amplified by performing the non-dominant operations in higher precision. In this case the numerical stability can be guaranteed with a working unit roundoff independent of the dimension of the problem. The proposed Gram-Schmidt process can be applied to Arnoldi iteration and result in new Krylov subspace methods for solving high-dimensional systems of equations or eigenvalue problems. Among them we chose randomized GMRES method as a practical application of the methodology.","url_abs":"https://arxiv.org/abs/2011.05090v3","url_pdf":"https://arxiv.org/pdf/2011.05090v3.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":"randomized-gram-schmidt-process-with","repo_url":"https://github.com/obalabanov/randkrylov","is_official":0,"mentioned_in_paper":0,"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}