{"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/fast-symmetric-factorization-of-hierarchical","title":"Fast symmetric factorization of hierarchical matrices with applications","arxiv_id":"1405.0223","date":"2014-05-01","proceeding":null,"authors":["Sivaram Ambikasaran","Michael O'Neil","Karan Raj Singh"],"abstract":"We present a fast direct algorithm for computing symmetric factorizations, i.e. $A = WW^T$, of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization scales as $\\mathcal{O}(n \\log^2 n)$ for hierarchically off-diagonal low-rank matrices. Once this factorization is obtained, the cost for inversion, application, and determinant computation scales as $\\mathcal{O}(n \\log n)$. In particular, this allows for the near optimal generation of correlated random variates in the case where $A$ is a covariance matrix. This symmetric factorization algorithm depends on two key ingredients. First, we present a novel symmetric factorization formula for low-rank updates to the identity of the form $I+UKU^T$. This factorization can be computed in $\\mathcal{O}(n)$ time if the rank of the perturbation is sufficiently small. Second, combining this formula with a recursive divide-and-conquer strategy, near linear complexity symmetric factorizations for hierarchically structured matrices can be obtained. We present numerical results for matrices relevant to problems in probability \\& statistics (Gaussian processes), interpolation (Radial basis functions), and Brownian dynamics calculations in fluid mechanics (the Rotne-Prager-Yamakawa tensor).","url_abs":"http://arxiv.org/abs/1405.0223v2","url_pdf":"http://arxiv.org/pdf/1405.0223v2.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":"fast-symmetric-factorization-of-hierarchical","repo_url":"https://github.com/sivaramambikasaran/HODLR","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1405.0223","atlas_url":"https://app.syntology.ai/?focus=1405.0223","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}