{"paper":{"title":"Symmetric indefinite triangular factorization revealing the rank profile matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.NA","authors_text":"Clement Pernet (CASYS), Jean-Guillaume Dumas (CASYS)","submitted_at":"2018-02-26T15:37:00Z","abstract_excerpt":"We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization  $\\mathbf{P}^T\\mathbf{A}\\mathbf{P} = \\mathbf{L}\\mathbf{D}\\mathbf{L}^T$  where $\\mathbf{P}$ is a permutation matrix, $\\mathbf{L}$ is lower triangular with a unit diagonal and  $\\mathbf{D}$ is symmetric block diagonal with $1{\\times}1$ and $2{\\times}2$ antidiagonal blocks. The novel algorithm requires $O(n^2r^{\\omega-2})$ arithmetic operations. Furthermore, experimental results demonstrate that our algorithm ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.10453","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}