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.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 299 articles , 1 standard article )

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  1. Tibaut, Jan; Ravnik, Jure: Fast boundary-domain integral method for heat transfer simulations (2019)
  2. Arreckx, Sylvain; Orban, Dominique: A regularized factorization-free method for equality-constrained optimization (2018)
  3. Bentbib, A. H.; El Guide, M.; Jbilou, K.; Onunwor, E.; Reichel, L.: Solution methods for linear discrete ill-posed problems for color image restoration (2018)
  4. Calvetti, D.; Pitolli, F.; Somersalo, E.; Vantaggi, B.: Bayes meets Krylov: statistically inspired preconditioners for CGLS (2018)
  5. Clempner, Julio B.; Poznyak, Alexander S.: A Tikhonov regularized penalty function approach for solving polylinear programming problems (2018)
  6. Estrin, Ron; Greif, Chen: SPMR: A family of saddle-point minimum residual solvers (2018)
  7. Hallman, Eric; Gu, Ming: LSMB: minimizing the backward error for least-squares problems (2018)
  8. Jia, Zhongxiao; Yang, Yanfei: Modified truncated randomized singular value decomposition (MTRSVD) algorithms for large scale discrete ill-posed problems with general-form regularization (2018)
  9. Kim, Kyoum Sun; Yun, Jae Heon: Image deblurring using global PCG method with Kronecker product preconditioner (2018)
  10. Ling, Si-Tao; Wang, Ming-Hui; Cheng, Xue-Han: A new implementation of LSMR algorithm for the quaternionic least squares problem (2018)
  11. McDonald, Eleanor; Pestana, Jennifer; Wathen, Andy: Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations (2018)
  12. Novati, P.: A convergence result for some Krylov-Tikhonov methods in Hilbert spaces (2018)
  13. Rao, Kaustubh; Malan, Paul; Perot, J. Blair: A stopping criterion for the iterative solution of partial differential equations (2018)
  14. Scott, Jennifer; Tůma, Miroslav: A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows (2018)
  15. Vuik, C.: Krylov subspace solvers and preconditioners (2018)
  16. Wettenhovi, Ville-Veikko; Kolehmainen, Ville; Huttunen, Joanna; Kettunen, Mikko; Gröhn, Olli; Vauhkonen, Marko: State estimation with structural priors in fMRI (2018)
  17. Yue, Hangrui; Yang, Qingzhi; Wang, Xiangfeng; Yuan, Xiaoming: Implementing the alternating direction method of multipliers for big datasets: a case study of least absolute shrinkage and selection operator (2018)
  18. Yun, Jae Heon: Parallel performance of the $g\ell$-pcg method for image deblurring problems (2018)
  19. Ahmadi-Asl, Salman; Beik, Fatemeh Panjeh Ali: Iterative algorithms for least-squares solutions of a quaternion matrix equation (2017)
  20. Bakhos, Tania; Kitanidis, Peter K.; Ladenheim, Scott; Saibaba, Arvind K.; Szyld, Daniel B.: Multipreconditioned GMRES for shifted systems (2017)

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