{"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/new-algebraic-fast-algorithms-for-n-body","title":"New Algebraic Fast Algorithms for $N$-body Problems in Two and Three Dimensions","arxiv_id":"2309.14085","date":"2023-09-25","proceeding":null,"authors":["Ritesh Khan","Sivaram Ambikasaran"],"abstract":"We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for $N$-body problems in $d$ dimensions, namely efficient $\\mathcal{H}^2_{*}$ (fully nested algorithm, i.e., $\\mathcal{H}^2$ matrix-like algorithm) and $(\\mathcal{H}^2 + \\mathcal{H})_{*}$ (semi-nested algorithm, i.e., cross of $\\mathcal{H}^2$ and $\\mathcal{H}$ matrix-like algorithms). The efficient $\\mathcal{H}^2_{*}$ and $(\\mathcal{H}^2 + \\mathcal{H})_{*}$ hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions, where the admissible clusters are the far-field and the vertex-sharing clusters. Due to the use of nested form of the bases, the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms ($\\mathcal{H}$ matrix algorithms). We rely on purely algebraic low-rank approximation techniques (e.g., ACA and NCA) and develop both algorithms in a black-box fashion. Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic (NCA or ACA-based compression) fast MVP algorithms in $2$D and $3$D. The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation. Notably, all the algorithms are developed in a similar fashion in $\\texttt{C++}$ and tested within the same environment, allowing for meaningful comparisons. The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard $\\mathcal{H}^2$ matrix algorithm with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/.","url_abs":"https://arxiv.org/abs/2309.14085v3","url_pdf":"https://arxiv.org/pdf/2309.14085v3.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":"new-algebraic-fast-algorithms-for-n-body","repo_url":"https://github.com/riteshkhan/h2weak","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"new-algebraic-fast-algorithms-for-n-body","repo_url":"https://github.com/riteshkhan/nhodlrdd","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}