{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T03:55:51Z","timestamp":1769572551208,"version":"3.49.0"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,1,22]],"date-time":"2020-01-22T00:00:00Z","timestamp":1579651200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,1,22]],"date-time":"2020-01-22T00:00:00Z","timestamp":1579651200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,8]]},"DOI":"10.1007\/s11075-019-00852-1","type":"journal-article","created":{"date-parts":[[2020,1,22]],"date-time":"2020-01-22T06:02:52Z","timestamp":1579672972000},"page":"1381-1395","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Simple stopping criteria for the LSQR method applied to discrete ill-posed problems"],"prefix":"10.1007","volume":"84","author":[{"given":"Lothar","family":"Reichel","sequence":"first","affiliation":[]},{"given":"Hassane","family":"Sadok","sequence":"additional","affiliation":[]},{"given":"Wei-Hong","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,1,22]]},"reference":[{"key":"852_CR1","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1093\/imanum\/2.2.241","volume":"2","author":"ML Baart","year":"1982","unstructured":"Baart, M.L.: The use of auto-correlation for pseudorank determination in noisy ill-conditioned linear least-squares problems. IMA J. Numer. Anal. 2, 241\u2013247 (1982)","journal-title":"IMA J. Numer. Anal."},{"key":"852_CR2","doi-asserted-by":"publisher","first-page":"035002","DOI":"10.1088\/0266-5611\/24\/3\/035002","volume":"24","author":"F Bauer","year":"2008","unstructured":"Bauer, F., Kindermann, S.: The quasi-optimality criterion for classical inverse problems. Inverse Problems 24, 035002 (2008). (20pp)","journal-title":"Inverse Problems"},{"key":"852_CR3","doi-asserted-by":"publisher","first-page":"1795","DOI":"10.1016\/j.matcom.2011.01.016","volume":"81","author":"F Bauer","year":"2011","unstructured":"Bauer, F., Lukas, M.A.: Comparing parameter choice methods for regularization of ill-posed problems. Math. Comput. Simulation 81, 1795\u20131841 (2011)","journal-title":"Math. Comput. Simulation"},{"key":"852_CR4","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s10665-013-9677-y","volume":"93","author":"A Bouhamidi","year":"2015","unstructured":"Bouhamidi, A., Jbilou, K., Reichel, L., Sadok, H., Wang, Z.: Vector extrapolation applied to truncated singular value decomposition and truncated iteration. J. Eng. Math. 93, 99\u2013112 (2015)","journal-title":"J. Eng. Math."},{"key":"852_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s002110050382","volume":"81","author":"C Brezinski","year":"1998","unstructured":"Brezinski, C., Redivo Zaglia, M., Rodriguez, G., Seatzu, S.: Extrapolation techniques for ill-conditioned linear systems. Numer. Math. 81, 1\u201329 (1998)","journal-title":"Numer. Math."},{"key":"852_CR6","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/s11075-008-9163-1","volume":"49","author":"C Brezinski","year":"2008","unstructured":"Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with applications to regularization. Numer. Algorithms 49, 85\u2013104 (2008)","journal-title":"Numer. Algorithms"},{"key":"852_CR7","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s11075-008-9243-2","volume":"51","author":"C Brezinski","year":"2009","unstructured":"Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms 51, 61\u201376 (2009)","journal-title":"Numer. Algorithms"},{"key":"852_CR8","doi-asserted-by":"publisher","first-page":"1019","DOI":"10.1007\/s10543-017-0662-0","volume":"57","author":"C Fenu","year":"2017","unstructured":"Fenu, C., Reichel, L., Rodriguez, G., Sadok, H.: GCV for Tikhonov regularization by partial SVD. BIT Numer. Math. 57, 1019\u20131039 (2017)","journal-title":"BIT Numer. Math."},{"key":"852_CR9","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1098\/rsta.1953.0005","volume":"245","author":"L Fox","year":"1953","unstructured":"Fox, L., Goodwin, E.T.: The numerical solution of non-singular linear integral equations. Philos. Trans. Roy. Soc. London. Ser. A. 245, 501\u2013534 (1953)","journal-title":"Philos. Trans. Roy. Soc. London. Ser. A."},{"key":"852_CR10","doi-asserted-by":"publisher","first-page":"1008","DOI":"10.1023\/A:1021941328858","volume":"41","author":"M Hanke","year":"2001","unstructured":"Hanke, M.: On Lanczos based methods for the regularization of discrete ill-posed problems. BIT Numer. Math. 41, 1008\u20131018 (2001)","journal-title":"BIT Numer. Math."},{"key":"852_CR11","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1137\/1034115","volume":"34","author":"PC Hansen","year":"1992","unstructured":"Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561\u2013580 (1992)","journal-title":"SIAM Rev."},{"key":"852_CR12","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719697","volume-title":"Rank-deficient and discrete ill-Posed problems: Numerical aspects of linear inversion","author":"PC Hansen","year":"1998","unstructured":"Hansen, P.C.: Rank-deficient and discrete ill-Posed problems: Numerical aspects of linear inversion. SIAM, Philadelphia (1998)"},{"key":"852_CR13","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s11075-007-9136-9","volume":"46","author":"PC Hansen","year":"2007","unstructured":"Hansen, P.C.: Regularization Tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189\u2013194 (2007)","journal-title":"Numer. Algorithms"},{"key":"852_CR14","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718836","volume-title":"Discrete inverse problems: insight and algorithms","author":"PC Hansen","year":"2010","unstructured":"Hansen, P.C.: Discrete inverse problems: insight and algorithms. SIAM, Philadelphia (2010)"},{"key":"852_CR15","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/j.cam.2014.06.004","volume":"273","author":"ME Hochstenbach","year":"2015","unstructured":"Hochstenbach, M.E., Reichel, L., Rodriguez, G.: Regularization parameter determination for discrete ill-posed problems. J. Comput. Appl. Math. 273, 132\u2013149 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"852_CR16","first-page":"233","volume":"38","author":"S Kindermann","year":"2011","unstructured":"Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233\u2013257 (2011)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"852_CR17","volume-title":"Computer solution of large linear systems","author":"G Meurant","year":"1999","unstructured":"Meurant, G.: Computer solution of large linear systems. Elsevier, Amsterdam (1999)"},{"key":"852_CR18","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718140","volume-title":"The Lanczos and conjugate gradient algorithms","author":"G Meurant","year":"2006","unstructured":"Meurant, G.: The Lanczos and conjugate gradient algorithms. SIAM, Philadelphia (2006)"},{"key":"852_CR19","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.cam.2005.05.028","volume":"193","author":"S Morigi","year":"2006","unstructured":"Morigi, S., Reichel, L., Sgallari, F., Zama, F.: Iterative methods for ill-posed problems and semiconvergent sequences. J. Comput. Appl. Math. 193, 157\u2013167 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"852_CR20","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1137\/0711019","volume":"11","author":"CC Paige","year":"1974","unstructured":"Paige, C.C.: Bidiagonalization of matrices and solutions of linear equations. SIAM J. Numer. Anal. 11, 197\u2013209 (1974)","journal-title":"SIAM J. Numer. Anal."},{"key":"852_CR21","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1145\/355984.355989","volume":"8","author":"CC Paige","year":"1982","unstructured":"Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Soft. 8, 43\u201371 (1982)","journal-title":"ACM Trans. Math. Soft."},{"key":"852_CR22","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/j.cam.2018.04.049","volume":"343","author":"Y Park","year":"2018","unstructured":"Park, Y., Reichel, L., Rodriguez, G., Yu, X.: Parameter determination for Tikhonov regularization problems in general form. J. Comput. Appl. Math. 343, 12\u201325 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"852_CR23","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1145\/321105.321114","volume":"9","author":"DL Phillips","year":"1962","unstructured":"Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9, 84\u201397 (1962)","journal-title":"J. Assoc. Comput. Mach."},{"key":"852_CR24","doi-asserted-by":"publisher","first-page":"740","DOI":"10.1137\/S1064827593252672","volume":"17","author":"T Regi\u0144ska","year":"1996","unstructured":"Regi\u0144ska, T.: A regularization parameter in discrete ill-posed problems. SIAM J. Sci. Comput. 17, 740\u2013749 (1996)","journal-title":"SIAM J. Sci. Comput."},{"key":"852_CR25","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s11075-012-9612-8","volume":"63","author":"L Reichel","year":"2013","unstructured":"Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63, 65\u201387 (2013)","journal-title":"Numer. Algorithms"},{"key":"852_CR26","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1016\/j.cam.2007.01.025","volume":"219","author":"L Reichel","year":"2008","unstructured":"Reichel, L., Sadok, H.: A new L-curve for ill-posed problems. J. Comput. Appl. Math. 219, 493\u2013508 (2008)","journal-title":"J. Comput. Appl. Math."},{"key":"852_CR27","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1007\/BF01739829","volume":"35","author":"MA Saunders","year":"1995","unstructured":"Saunders, M.A.: Solution of sparse rectangular systems using LSQR and Craig. BIT Numer. Math. 35, 588\u2013604 (1995)","journal-title":"BIT Numer. Math."},{"key":"852_CR28","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/0022-247X(72)90259-4","volume":"37","author":"CBJr Shaw","year":"1972","unstructured":"Shaw, C.B. Jr.: Improvement of the resolution of an instrument by numerical solution of an integral equation. J. Math. Anal. Appl. 37, 83\u2013112 (1972)","journal-title":"J. Math. Anal. Appl."},{"key":"852_CR29","volume-title":"The algebraic eigenvalue problem","author":"JH Wilkinson","year":"1965","unstructured":"Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford University Press, New York (1965)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00852-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00852-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00852-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:23:45Z","timestamp":1611188625000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00852-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,22]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["852"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00852-1","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,22]]},"assertion":[{"value":"25 August 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}