LSQR swMATH ID: 530 Software Authors: Paige, Christopher C.; Saunders, Michael A. Description: Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems. An iterative method is given for solving Ax = b and min|| Ax - b||2 , where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing LSQR with several other conjugate-gradient algorithms, indicating that LSQR is the most reliable algorithm when A is ill-conditioned. Homepage: http://dl.acm.org/citation.cfm?id=356000 Keywords: bidiagonalization; conjugate gradients; estimates of errors; numerical tests; sparse and very large systems; linear least squares Related Software: Regularization tools; LSMR; SparseMatrix; UTV; mctoolbox; Matlab; MINRES; IR Tools; AIR tools; RestoreTools; HSL; CGS; JDQZ; LAPACK; GKB-FP; CUTEst; JDQR; ErresTools; LSRN; PDCO Cited in: 415 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year LSQR: An algorithm for sparse linear equations and sparse least squares. Zbl 0478.65016Paige, Christopher C.; Saunders, Michael A. 1982 all top 5 Cited by 644 Authors 23 Reichel, Lothar 15 Saunders, Michael A. 11 Jia, Zhongxiao 8 Chung, Julianne M. 8 Orban, Dominique 8 Van der Vorst, Henk Albertus 7 Jbilou, Khalide 7 Paige, Christopher C. 7 Ravnik, Jure 6 Boyd, Stephen Poythress 6 Calvetti, Daniela 6 Golub, Gene Howard 6 Karimi, Saeed 6 Scott, Jennifer A. 6 Strakoš, Zdeněk 6 Titley-Péloquin, David 6 Viloche Bazán, Fermín S. 5 Gazzola, Silvia 5 Morigi, Serena 5 Morikuni, Keiichi 5 Nagy, James Gerard 5 Novati, Paolo 5 Sadok, Hassane 5 Sgallari, Fiorella 5 Wathen, Andrew John 4 Barlow, Jesse L. 4 Bentbib, Abdeslem Hafid 4 Borges, Leonardo Silveira 4 Bouhamidi, Abderrahman 4 Chung, Matthias 4 Estrin, Ron 4 Hajarian, Masoud 4 Hansen, Per Christian 4 Hayami, Ken 4 Hnětynková, Iveta 4 Hochstenbach, Michiel E. 4 Hyvönen, Nuutti 4 Jozi, Meisam 4 Kilmer, Misha Elena 4 Ling, Sitao 4 O’Leary, Dianne P. 4 Renaut, Rosemary Anne 4 Rodriguez, Giuseppe 4 Tenorio, Luis 4 Toutounian, Faezeh 4 Wang, Minghui 4 Žunič, Zoran 3 Brezinski, Claude 3 Chang, Xiaowen 3 Dongarra, Jack J. 3 Duff, Iain S. 3 Gratton, Serge 3 Hallman, Eric 3 Harhanen, Lauri 3 Huang, Ting-Zhu 3 Lukšan, Ladislav 3 Majander, H. 3 Potts, Daniel 3 Schmischke, Michael 3 Škerget, Leopold 3 Somersalo, Erkki 3 Tichý, Petr 3 Toint, Philippe L. 3 Tůma, Miroslav 3 Ullrich, Tino 3 Vuik, Cornelis 2 Al-Homidan, Suliman S. 2 Andreev, Roman 2 Austen, Gerrit 2 Avron, Haim 2 Baglama, James 2 Baur, Oliver 2 Beik, Fatemeh Panjeh Ali 2 Bierlaire, Michel 2 Björck, Åke 2 Boelens, Arnout M. P. 2 Chen, Scott Shaobing 2 Choi, Sou-Cheng T. 2 Cornelis, Jeffrey 2 Cui, Tiangang 2 Cunha, Maria Cristina C. 2 Dierckx, Paul 2 Donoho, David Leigh 2 El-Guide, Mohamed 2 Entezami, Alireza 2 Ferreira, Orizon Pereira 2 Gill, Philip E. 2 Goldfarb, Donald 2 Gondzio, Jacek 2 González-Lima, María D. 2 Greif, Chen 2 Gröchenig, Karlheinz 2 Hannukainen, Antti 2 Harker, Matthew 2 Herrmann, Felix J. 2 Hriberšek, Matjaž 2 Hyndman, Rob J. 2 Jia, Zhigang 2 Khouia, Asmaa 2 Knepper, Sarah ...and 544 more Authors all top 5 Cited in 107 Serials 35 SIAM Journal on Scientific Computing 27 Journal of Computational and Applied Mathematics 22 SIAM Journal on Matrix Analysis and Applications 22 Linear Algebra and its Applications 19 BIT 19 Numerical Algorithms 16 Inverse Problems 16 Journal of Computational Physics 12 Applied Numerical Mathematics 11 Applied Mathematics and Computation 10 Computational Optimization and Applications 10 Engineering Analysis with Boundary Elements 8 Numerical Linear Algebra with Applications 7 Computers & Mathematics with Applications 7 Journal of Scientific Computing 7 SIAM Review 6 Journal of Optimization Theory and Applications 6 Mathematical Programming. 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