LSQR

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 340 articles , 1 standard article )

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  1. Baechler, Gilles; Dümbgen, Frederike; Elhami, Golnoosh; Kreković, Miranda; Vetterli, Martin: Coordinate difference matrices (2020)
  2. Cerdán, J.; Guerrero, D.; Marín, J.; Mas, J.: Preconditioners for rank deficient least squares problems (2020)
  3. Chang, Xiao-Wen; Kang, Peng; Titley-Peloquin, David: Error bounds for computed least squares estimators (2020)
  4. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)
  5. Jia, Zhongxiao: Approximation accuracy of the Krylov subspaces for linear discrete ill-posed problems (2020)
  6. Mohammady, Somaieh; Eslahchi, M. R.: Extension of Tikhonov regularization method using linear fractional programming (2020)
  7. Asgari, Z.; Toutounian, F.; Babolian, E.; Tohidi, E.: LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations (2019)
  8. Avron, Haim; Druinsky, Alex; Toledo, Sivan: Spectral condition-number estimation of large sparse matrices. (2019)
  9. Axelsson, Owe; Liang, Zhao-Zheng: Parameter modified versions of preconditioning and iterative inner product free refinement methods for two-by-two block matrices (2019)
  10. Berger, Peter; Gröchenig, Karlheinz; Matz, Gerald: Sampling and reconstruction in distinct subspaces using oblique projections (2019)
  11. Boiveau, Thomas; Ehrlacher, Virginie; Ern, Alexandre; Nouy, Anthony: Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods (2019)
  12. Busseti, Enzo; Moursi, Walaa M.; Boyd, Stephen: Solution refinement at regular points of conic problems (2019)
  13. Buttari, Alfredo; Orban, Dominique; Ruiz, Daniel; Titley-Peloquin, David: A tridiagonalization method for symmetric saddle-point systems (2019)
  14. Caruso, Noe Angelo; Novati, Paolo: Convergence analysis of LSQR for compact operator equations (2019)
  15. Chung, Julianne; Gazzola, Silvia: Flexible Krylov methods for (\ell_p) regularization (2019)
  16. Dahito, Marie-Ange; Orban, Dominique: The conjugate residual method in linesearch and trust-region methods (2019)
  17. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LNLQ: an iterative method for least-norm problems with an error minimization property (2019)
  18. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LSLQ: an iterative method for linear least-squares with an error minimization property (2019)
  19. Gazzola, Silvia; Noschese, Silvia; Novati, Paolo; Reichel, Lothar: Arnoldi decomposition, GMRES, and preconditioning for linear discrete ill-posed problems (2019)
  20. Gazzola, Silvia; Sabaté Landman, Malena: Flexible GMRES for total variation regularization (2019)

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