{"paper":{"title":"LSMR: An iterative algorithm for sparse least-squares problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"cs.MS","authors_text":"David Fong, Michael Saunders","submitted_at":"2010-06-04T00:16:09Z","abstract_excerpt":"An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares problem $\\min \\norm{Ax-b}_2$, with $A$ being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation $A\\T Ax = A\\T b$, so that the quantities $\\norm{A\\T r_k}$ are monotonically decreasing (where $r_k = b - Ax_k$ is the residual for the current iterate $x_k$). In practice we observe that $\\norm{r_k}$ also decreases monotonically. Compared to LSQR, for which only $\\norm{r_k}$ is monotonic,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.0758","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"}