{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:M5V5ASP7SFWU3OTYXHLD4DG42O","short_pith_number":"pith:M5V5ASP7","schema_version":"1.0","canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","source":{"kind":"arxiv","id":"2605.24692","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.24692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-23T18:10:12Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"e12f7ee780222643b5534259a9be10f767d09388dbddc9cb3093043f6380056e","abstract_canon_sha256":"c11f982ca3d519e54dcc6a8b4bac7d5497578d2fc5e0a62a0550d72e29ccc2e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:53.570368Z","signature_b64":"R7qcwvjYtSe8ak9ds/whmXSGqvg1aPQTY5h3JvN2zJ9ZTRQhpRnihTkcvjN0Zo+6uwq3KNgppE676soy7cykDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","last_reissued_at":"2026-05-26T01:03:53.569757Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:53.569757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.24692","created_at":"2026-05-26T01:03:53.569863+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.24692v1","created_at":"2026-05-26T01:03:53.569863+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24692","created_at":"2026-05-26T01:03:53.569863+00:00"},{"alias_kind":"pith_short_12","alias_value":"M5V5ASP7SFWU","created_at":"2026-05-26T01:03:53.569863+00:00"},{"alias_kind":"pith_short_16","alias_value":"M5V5ASP7SFWU3OTY","created_at":"2026-05-26T01:03:53.569863+00:00"},{"alias_kind":"pith_short_8","alias_value":"M5V5ASP7","created_at":"2026-05-26T01:03:53.569863+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O","json":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O.json","graph_json":"https://pith.science/api/pith-number/M5V5ASP7SFWU3OTYXHLD4DG42O/graph.json","events_json":"https://pith.science/api/pith-number/M5V5ASP7SFWU3OTYXHLD4DG42O/events.json","paper":"https://pith.science/paper/M5V5ASP7"},"agent_actions":{"view_html":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O","download_json":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O.json","view_paper":"https://pith.science/paper/M5V5ASP7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.24692&json=true","fetch_graph":"https://pith.science/api/pith-number/M5V5ASP7SFWU3OTYXHLD4DG42O/graph.json","fetch_events":"https://pith.science/api/pith-number/M5V5ASP7SFWU3OTYXHLD4DG42O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/action/storage_attestation","attest_author":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/action/author_attestation","sign_citation":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/action/citation_signature","submit_replication":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/action/replication_record"}},"created_at":"2026-05-26T01:03:53.569863+00:00","updated_at":"2026-05-26T01:03:53.569863+00:00"}