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 343 articles , 1 standard article )

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  1. Gazzola, Silvia; Sabaté Landman, Malena: Flexible GMRES for total variation regularization (2019)
  2. Jozi, Meisam; Karimi, Saeed; Salkuyeh, Davod Khojasteh: An iterative method to compute minimum norm solutions of ill-posed problems in Hilbert spaces (2019)
  3. Karimi, Saeed; Jozi, Meisam: Weighted conjugate gradient-type methods for solving quadrature discretization of Fredholm integral equations of the first kind (2019)
  4. Keys, Kevin L.; Zhou, Hua; Lange, Kenneth: Proximal distance algorithms: theory and practice (2019)
  5. Matteo Ravasi, Ivan Vasconcelos: PyLops - A Linear-Operator Python Library for large scale optimization (2019) arXiv
  6. Meurant, Gérard; Tichý, Petr: Approximating the extreme Ritz values and upper bounds for the (A)-norm of the error in CG (2019)
  7. Paige, Christopher C.: Accuracy of the Lanczos process for the eigenproblem and solution of equations (2019)
  8. Pestana, J.: Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices (2019)
  9. Scott, Jennifer A.; Tůma, Miroslav: Sparse stretching for solving sparse-dense linear least-squares problems (2019)
  10. Slagel, J. Tanner; Chung, Julianne; Chung, Matthias; Kozak, David; Tenorio, Luis: Sampled Tikhonov regularization for large linear inverse problems (2019)
  11. Tibaut, Jan; Ravnik, Jure: Fast boundary-domain integral method for heat transfer simulations (2019)
  12. Akbari, Amir; Barton, Paul I.: An improved multi-parametric programming algorithm for flux balance analysis of metabolic networks (2018)
  13. Arreckx, Sylvain; Orban, Dominique: A regularized factorization-free method for equality-constrained optimization (2018)
  14. 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)
  15. Boelens, Arnout M. P.; Venturi, Daniele; Tartakovsky, Daniel M.: Parallel tensor methods for high-dimensional linear PDEs (2018)
  16. Calvetti, D.; Pitolli, F.; Somersalo, E.; Vantaggi, B.: Bayes meets Krylov: statistically inspired preconditioners for CGLS (2018)
  17. Clempner, Julio B.; Poznyak, Alexander S.: A Tikhonov regularized penalty function approach for solving polylinear programming problems (2018)
  18. Estrin, Ron; Greif, Chen: SPMR: A family of saddle-point minimum residual solvers (2018)
  19. Fougner, Christopher; Boyd, Stephen: Parameter selection and preconditioning for a graph form solver (2018)
  20. Hallman, Eric; Gu, Ming: LSMB: minimizing the backward error for least-squares problems (2018)

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