{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:24:41Z","timestamp":1787336681480,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,4,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We propose iterative projection methods for solving square or rectangular consistent linear systems [Formula: see text]. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but even the smaller systems can be costly. We develop a process that appends one column to the sketching matrix each iteration and converges in a finite number of iterations whether the sketch is random or deterministic. In general, our process generates orthogonal updates to the approximate solution [Formula: see text]. By choosing the sketch to be the set of all previous residuals, we obtain a simple recursive update and convergence in at most [Formula: see text] iterations (in exact arithmetic). By choosing a sequence of identity columns for the sketch, we develop a generalization of the Kaczmarz method. In experiments on large sparse systems, our method (PLSS) with residual sketches is competitive with LSQR and LSMR and with residual and identity sketches compares favorably with state-of-the-art randomized methods.<\/jats:p>","DOI":"10.1137\/22m1509783","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T14:08:20Z","timestamp":1682690900000},"page":"A1012-A1037","source":"Crossref","is-referenced-by-count":1,"title":["PLSS: A Projected Linear Systems Solver"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3748-2440","authenticated-orcid":true,"given":"Johannes J.","family":"Brust","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California San Diego, La Jolla, CA 92093 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3800-4982","authenticated-orcid":true,"given":"Michael A.","family":"Saunders","sequence":"additional","affiliation":[{"name":"Department of Management Science and Engineering, Stanford University, Stanford, CA 94305 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,4,28]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/060673096"},{"key":"ref2","unstructured":"J. J. Brust , Code for Algorithm PLSS and Test Programs, https:\/\/github.com\/johannesbrust\/PLSS, 2022."},{"key":"ref3","unstructured":"C. Cartis , \nJ. Fiala , and \nZ. Shao , Hashing Embeddings of Optimal Dimension, with Applications to Linear Least Squares, https:\/\/doi.org\/10.48550\/ARXIV.2105.11815, https:\/\/arxiv.org\/abs\/2105.11815, 2021."},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1145\/1961189.1961199"},{"key":"ref5","unstructured":"T. A. Davis , \nY. Hu , and \nS. Kolodziej , SuiteSparse Matrix Collection, https:\/\/sparse.tamu.edu\/, 2015\u2013present."},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/10079687X"},{"key":"ref7","volume-title":"Matrix Computations","author":"Golub G. H.","year":"1996","edition":"3"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/15M1025487"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/026\/737400"},{"key":"ref12","first-page":"335","author":"Kaczmarz S.","year":"1937","journal-title":"Bull. Int. Acad. Polon. Sci. Lett. A"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1145\/2559902"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1100.0456"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/0711019"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"ref17","unstructured":"C. C. Paige  and \nA. M. Saunders , CRAIG: Sparse Equations, http:\/\/stanford.edu\/group\/SOL\/software\/craig\/, 2014\u20132022."},{"key":"ref18","unstructured":"Z. Qu , \nP. Richt\u00e1rik , \nM. Tak\u00e1c , and \nO. Fercoq , SDNA: Stochastic dual Newton ascent for empirical risk minimization, in International Conference on Machine Learning, 2016, pp. 1823\u20131832."},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1137\/18M1179249"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-008-9030-4"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1509783","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:26:24Z","timestamp":1787333184000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1509783"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,28]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4,30]]}},"alternative-id":["10.1137\/22M1509783"],"URL":"https:\/\/doi.org\/10.1137\/22m1509783","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,28]]}}}