{"paper":{"title":"Decentralized gradient algorithm for solution of a linear equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DC"],"primary_cat":"cs.SY","authors_text":"A. Stephen Morse, Brian D. O. Anderson, Shaoshuai Mou, Uwe Helmke","submitted_at":"2015-09-15T13:24:57Z","abstract_excerpt":"The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\\mathcal G$ with vertex set and edge set $\\mathcal V$ and $\\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solutio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04538","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"}