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 286 articles , 1 standard article )

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  1. Arreckx, Sylvain; Orban, Dominique: A regularized factorization-free method for equality-constrained optimization (2018)
  2. Calvetti, D.; Pitolli, F.; Somersalo, E.; Vantaggi, B.: Bayes meets Krylov: statistically inspired preconditioners for CGLS (2018)
  3. Clempner, Julio B.; Poznyak, Alexander S.: A Tikhonov regularized penalty function approach for solving polylinear programming problems (2018)
  4. Estrin, Ron; Greif, Chen: SPMR: A family of saddle-point minimum residual solvers (2018)
  5. Jia, Zhongxiao; Yang, Yanfei: Modified truncated randomized singular value decomposition (MTRSVD) algorithms for large scale discrete ill-posed problems with general-form regularization^* (2018)
  6. McDonald, Eleanor; Pestana, Jennifer; Wathen, Andy: Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations (2018)
  7. Novati, P.: A convergence result for some Krylov-Tikhonov methods in Hilbert spaces (2018)
  8. Rao, Kaustubh; Malan, Paul; Perot, J. Blair: A stopping criterion for the iterative solution of partial differential equations (2018)
  9. Wettenhovi, Ville-Veikko; Kolehmainen, Ville; Huttunen, Joanna; Kettunen, Mikko; Gröhn, Olli; Vauhkonen, Marko: State estimation with structural priors in fMRI (2018)
  10. Ahmadi-Asl, Salman; Beik, Fatemeh Panjeh Ali: Iterative algorithms for least-squares solutions of a quaternion matrix equation (2017)
  11. Bakhos, Tania; Kitanidis, Peter K.; Ladenheim, Scott; Saibaba, Arvind K.; Szyld, Daniel B.: Multipreconditioned gmres for shifted systems (2017)
  12. Bellis, Cédric; Trabelsi, Manel; Frémy, Flavien: Reconstructing material properties by deconvolution of full-field measurement images: the conductivity case (2017)
  13. Bentbib, A. H.; El Guide, M.; Jbilou, K.: The block Lanczos algorithm for linear ill-posed problems (2017)
  14. Borges, Carlos; Gillman, Adrianna; Greengard, Leslie: High resolution inverse scattering in two dimensions using recursive linearization (2017)
  15. Breedveld, Sebastiaan; van den Berg, Bas; Heijmen, Ben: An interior-point implementation developed and tuned for radiation therapy treatment planning (2017)
  16. Calvetti, D.; Pitolli, F.; Prezioso, J.; Somersalo, E.; Vantaggi, B.: Priorconditioned CGLS-based quasi-MAP estimate, statistical stopping rule, and ranking of priors (2017)
  17. Chung, Julianne; Chung, Matthias: Optimal regularized inverse matrices for inverse problems (2017)
  18. Chung, Julianne; Saibaba, Arvind K.: Generalized hybrid iterative methods for large-scale Bayesian inverse problems (2017)
  19. Diamond, Steven; Boyd, Stephen: Stochastic matrix-free equilibration (2017)
  20. Gould, Nicholas; Scott, Jennifer: The state-of-the-art of preconditioners for sparse linear least-squares problems (2017)

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