{"paper":{"title":"Jacobi method for symmetric $4\\times4$ matrices converges for every cyclic pivot strategy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Erna Begovic, Vjeran Hari","submitted_at":"2017-01-09T23:21:55Z","abstract_excerpt":"The paper studies the global convergence of the Jacobi method for symmetric matrices of size $4$. We prove global convergence for all $720$ cyclic pivot strategies. Precisely, we show that inequality $S(A^{[t+3]})\\leq\\gamma S(A^{[t]})$, $t\\geq1$, holds with the constant $\\gamma<1$ that depends neither on the matrix $A$ nor on the pivot strategy. Here $A^{[t]}$ stands for the matrix obtained from $A$ after $t$ full cycles of the Jacobi method and $S(A)$ is the off-diagonal norm of $A$. We show why three consecutive cycles have to be considered. The result has a direct application on the $J$-Jac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02387","kind":"arxiv","version":2},"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"}