{"paper":{"title":"Sharp Convergence Rates and Optimal Weights for Cimmino's Reflection Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Hemant Sharma","submitted_at":"2026-05-23T18:10:12Z","abstract_excerpt":"In this paper, Cimmino's classical reflection algorithm for solving the $n\\times n$ nonsingular linear system $A\\bx=\\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as $\\e^{(\\nu+1)}=M_w\\,\\e^{(\\nu)}$, where $M_w = I - A^\\top D_w A$, the error is shown to contract by the spectral radius $\\sprad(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, \\[\n  \\sprad(M_w) \\;=\\; |1-\\mu|\n  + \\tfrac{1}{2}\\sqrt{(w_1-w_2)^2 + 4w_1w_2\\cos^2\\!\\theta}, \\] where $\\mu=(w_1+w_2)/2$ and $\\th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24692","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.24692/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}