{"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/a-stable-one-synchronization-variant-of","title":"A stable one-synchronization variant of reorthogonalized block classical Gram--Schmidt","arxiv_id":"2411.07077","date":"2024-11-11","proceeding":null,"authors":["Erin Carson","Yuxin Ma"],"abstract":"The block classical Gram--Schmidt (BCGS) algorithm and its reorthogonalized variant are widely-used methods for computing the economic QR factorization of block columns $X$ due to their lower communication cost compared to other approaches such as modified Gram--Schmidt and Householder QR. To further reduce communication, i.e., synchronization, there has been a long ongoing search for a variant of reorthogonalized BCGS variant that achieves $O(u)$ loss of orthogonality while requiring only \\emph{one} synchronization point per block column, where $u$ represents the unit roundoff. Utilizing Pythagorean inner products and delayed normalization techniques, we propose the first provably stable one-synchronization reorthogonalized BCGS variant, demonstrating that it has $O(u)$ loss of orthogonality under the condition $O(u) \\kappa^2(X) \\leq 1/2$, where $\\kappa(\\cdot)$ represents the condition number. By incorporating one additional synchronization point, we develop a two-synchronization reorthogonalized BCGS variant which maintains $O(u)$ loss of orthogonality under the improved condition $O(u) \\kappa(X) \\leq 1/2$. An adaptive strategy is then proposed to combine these two variants, ensuring $O(u)$ loss of orthogonality while using as few synchronization points as possible under the less restrictive condition $O(u) \\kappa(X) \\leq 1/2$. As an example of where this adaptive approach is beneficial, we show that using the adaptive orthogonalization variant, $s$-step GMRES achieves a backward error comparable to $s$-step GMRES with BCGSI+, also known as BCGS2, both theoretically and numerically, but requires fewer synchronization points.","url_abs":"https://arxiv.org/abs/2411.07077v2","url_pdf":"https://arxiv.org/pdf/2411.07077v2.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":"a-stable-one-synchronization-variant-of","repo_url":"https://github.com/katlund/BlockStab","is_official":1,"mentioned_in_paper":0,"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}