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 351 articles , 1 standard article )

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  1. Baechler, Gilles; Dümbgen, Frederike; Elhami, Golnoosh; Kreković, Miranda; Vetterli, Martin: Coordinate difference matrices (2020)
  2. Beik, Fatemeh P. A.; Jbilou, Khalide; Najafi-Kalyani, Mehdi; Reichel, Lothar: Golub-Kahan bidiagonalization for ill-conditioned tensor equations with applications (2020)
  3. Benvenuto, Federico; Jin, Bangti: A parameter choice rule for Tikhonov regularization based on predictive risk (2020)
  4. Cerdán, J.; Guerrero, D.; Marín, J.; Mas, J.: Preconditioners for rank deficient least squares problems (2020)
  5. Chang, Xiao-Wen; Kang, Peng; Titley-Peloquin, David: Error bounds for computed least squares estimators (2020)
  6. de Oliveira, F. R.; Ferreira, O. P.: Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations (2020)
  7. Drake, Kathryn P.; Wright, Grady B.: A fast and accurate algorithm for spherical harmonic analysis on HEALPix grids with applications to the cosmic microwave background radiation (2020)
  8. Estrin, Ron; Friedlander, Michael P.; Orban, Dominique; Saunders, Michael A.: Implementing a smooth exact penalty function for equality-constrained nonlinear optimization (2020)
  9. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)
  10. Jia, Zhongxiao: Approximation accuracy of the Krylov subspaces for linear discrete ill-posed problems (2020)
  11. Jia, Zhongxiao; Yang, Yanfei: A joint bidiagonalization based iterative algorithm for large scale general-form Tikhonov regularization (2020)
  12. Mohammady, Somaieh; Eslahchi, M. R.: Extension of Tikhonov regularization method using linear fractional programming (2020)
  13. Reichel, Lothar; Sadok, Hassane; Zhang, Wei-Hong: Simple stopping criteria for the LSQR method applied to discrete ill-posed problems (2020)
  14. Schaub, Michael T.; Benson, Austin R.; Horn, Paul; Lippner, Gabor; Jadbabaie, Ali: Random walks on simplicial complexes and the normalized Hodge 1-Laplacian (2020)
  15. Selvitopi, Oguz; Acer, Seher; Manguoğlu, Murat; Aykanat, Cevdet: The effect of various sparsity structures on parallelism and algorithms to reveal those structures (2020)
  16. Asgari, Z.; Toutounian, F.; Babolian, E.; Tohidi, E.: LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations (2019)
  17. Avron, Haim; Druinsky, Alex; Toledo, Sivan: Spectral condition-number estimation of large sparse matrices. (2019)
  18. Axelsson, Owe; Liang, Zhao-Zheng: Parameter modified versions of preconditioning and iterative inner product free refinement methods for two-by-two block matrices (2019)
  19. Berger, Peter; Gröchenig, Karlheinz; Matz, Gerald: Sampling and reconstruction in distinct subspaces using oblique projections (2019)
  20. Boiveau, Thomas; Ehrlacher, Virginie; Ern, Alexandre; Nouy, Anthony: Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods (2019)

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