MINRES swMATH ID: 13371 Software Authors: Christopher C. Paige, Michael A. Saunders, Sou-Cheng Choi, Dominique Orban, Umberto Emanuele Villa, Danielle Maddix Description: Implementation of a conjugate-gradient type method for solving sparse linear equations: Solve Ax=b or (A−sI)x=b. The matrix A−sI must be symmetric but it may be definite or indefinite or singular. The scalar s is a shifting parameter – it may be any number. The method is based on Lanczos tridiagonalization. You may provide a preconditioner, but it must be positive definite. MINRES is really solving one of the least-squares problems minimize ||Ax−b|| or ||(A−sI)x−b||. If A is singular (and s=0), MINRES returns a least-squares solution with small ||Ar|| (where r=b−Ax), but in general it is not the minimum-length solution. To get the min-length solution, use MINRES-QLP [2,3]. Similarly if A−sI is singular. Homepage: http://web.stanford.edu/group/SOL/software/minres/ Related Software: MINRES-QLP; LSQR; CRAIG; SparseMatrix; LSMR; Matlab; CUTEst; MUMPS; CGS; mctoolbox; SYM-ILDL; PETSc; SLEPc; Ipopt; ARPACK; Gurobi; LAPACK; Julia; Adam; AdaGrad Cited in: 44 Documents all top 5 Cited by 83 Authors 8 Orban, Dominique 7 Saunders, Michael A. 5 Paige, Christopher C. 3 Estrin, Ron 2 Choi, Sou-Cheng T. 2 Lim, Lek-Heng 2 Liu, Yang 2 Montoison, Alexis 2 Roosta, Fred 2 Titley-Péloquin, David 1 Al-Baali, Mehiddin 1 Athènes, Manuel 1 Bergou, El Houcine 1 Bouchard, Kristofer E. 1 Caliciotti, Andrea 1 Chang, Xiaowen 1 Chen, Qiao 1 Claeys, Xavier 1 Creuze, Jérôme 1 Cui, Tiangang 1 Curtis, Frank E. 1 Dahito, Marie-Ange 1 Dai, Hua 1 DeWeese, Michael R. 1 Diouane, Youssef 1 Dostál, Zdeněk 1 Duintjer Tebbens, Jurjen 1 Fasano, Giovanni 1 Frye, Charles G. 1 Ghannad, Alexandre 1 Giacomel, Lorenzo 1 Gondzio, Jacek 1 Goulart, Paul J. 1 Gould, Nick I. M. 1 Greif, Chen 1 Hallman, Eric 1 He, Qinglong 1 He, Ying 1 Hiptmair, Ralf 1 Il’in, Valeriĭ Pavlovich 1 Jiang, Xiaoye 1 Jiao, Xiangmin 1 Kalantzis, Vassilis 1 Kaur, Savneet 1 Knyazev, Andrew V. 1 Kungurtsev, Vyacheslav 1 Lauwens, Ben 1 Leveque, Santolo 1 Ligeralde, Andrew 1 Manguoglu, Murat 1 Marzouk, Youssef M. 1 Mehrmann, Volker 1 Meurant, Gérard A. 1 Moré, Jorge J. 1 Nakatsukasa, Yuji 1 Nocedal, Jorge 1 Olshanskii, Maxim A. 1 Pearson, John W. 1 Petit, Julien 1 Pospíšil, Lukáš 1 Potschka, Andreas 1 Rees, Tyrone 1 Roma, Massimo 1 Rontsis, Nikitas 1 Royer, Clément W. 1 Schork, Lukas 1 Simon, James 1 Simoncini, Valeria 1 Spantini, Alessio 1 Tenorio, Luis 1 Trinath, G. 1 Urzúa-Torres, Carolina 1 Varah, James M. 1 Vecharynski, Eugene 1 Wächter, Andreas 1 Wadia, Neha S. 1 Wang, Yanfei 1 Wild, Stefan M. 1 Willcox, Karen E. 1 Yao, Yuan 1 Ye, Yinyu 1 Zhang, Jianhua 1 Zuo, Qian all top 5 Cited in 20 Serials 11 SIAM Journal on Matrix Analysis and Applications 6 SIAM Journal on Scientific Computing 5 SIAM Journal on Optimization 3 Journal of Computational Physics 2 BIT 2 SIAM Journal on Numerical Analysis 2 Mathematical Programming. Series A. Series B 1 Inverse Problems 1 ACM Transactions on Mathematical Software 1 Journal of Computational and Applied Mathematics 1 Neural Computation 1 International Journal of Computer Mathematics 1 Linear Algebra and its Applications 1 SIAM Review 1 Numerical Linear Algebra with Applications 1 Computational and Applied Mathematics 1 Doklady Mathematics 1 Optimization Methods & Software 1 Mathematical Programming Computation 1 International Journal of Advances in Engineering Sciences and Applied Mathematics all top 5 Cited in 17 Fields 38 Numerical analysis (65-XX) 12 Operations research, mathematical programming (90-XX) 9 Linear and multilinear algebra; matrix theory (15-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Computer science (68-XX) 3 Systems theory; control (93-XX) 2 Statistics (62-XX) 2 Fluid mechanics (76-XX) 1 Combinatorics (05-XX) 1 Group theory and generalizations (20-XX) 1 Partial differential equations (35-XX) 1 Integral transforms, operational calculus (44-XX) 1 Integral equations (45-XX) 1 Optics, electromagnetic theory (78-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) Citations by Year