{"paper":{"title":"The Dual Kaczmarz Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anna Aboud, Emelie Curl, Eric S. Weber, M. Vaughan, Steven N. Harding","submitted_at":"2018-11-01T00:23:29Z","abstract_excerpt":"The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector $x$ in a (separable) Hilbert space from the inner-products $\\{\\langle x, \\phi_{n} \\rangle\\}$. The Kaczmarz algorithms defines a sequence of approximations from the sequence $\\{\\langle x, \\phi_{n} \\rangle\\}$; these approximations only converge to $x$ when $\\{\\phi_{n}\\}$ is ${effective}$. We dualize the Kaczmarz algorithm so that $x$ can be obtained from $\\{\\langle x, \\phi_{n} \\rangle\\}$ by using a second sequence $\\{\\psi_{n}\\}$ in the reconstruction. This allo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00169","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"}