{"paper":{"title":"Residual-Weighted Randomized Jacobi: Sharpened Bounds via Residual Concentration and Asynchronous Extension","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DC","cs.NA"],"primary_cat":"math.NA","authors_text":"Evan Coleman","submitted_at":"2026-05-31T13:29:29Z","abstract_excerpt":"We study randomized stationary methods for symmetric positive definite linear systems in which component $j$ is selected with probability proportional to $|r_j|^\\ell$. This power-weighted family interpolates continuously between uniform randomized Jacobi as $\\ell \\to 0$ and Gauss--Southwell greedy relaxation as $\\ell \\to \\infty$. For the central case $\\ell = 2$, we sharpen the standard one-step convergence analysis using the inverse participation ratio (IPR) $\\nu^2(r) = n\\|r\\|_4^4/\\|r\\|_2^4$, which equals $1$ when the residual is uniform and grows toward $n$ as it concentrates. The resulting b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01232","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/2606.01232/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"}