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 367 articles , 1 standard article )

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  1. Jakob S. Jørgensen, Evelina Ametova, Genoveva Burca, Gemma Fardell, Evangelos Papoutsellis, Edoardo Pasca, Kris Thielemans, Martin Turner, Ryan Warr, William R. B. Lionheart, Philip J. Withers: Core Imaging Library - Part I: a versatile Python framework for tomographic imaging (2021) arXiv
  2. Šušnjara, Anna; Verhnjak, Ožbej; Poljak, Dragan; Cvetković, Mario; Ravnik, Jure: Stochastic-deterministic boundary element modelling of transcranial electric stimulation using a three layer head model (2021)
  3. Zhang, Hui; Dai, Hua: The regularizing properties of global GMRES for solving large-scale linear discrete ill-posed problems with several right-hand sides (2021)
  4. Baechler, Gilles; Dümbgen, Frederike; Elhami, Golnoosh; Kreković, Miranda; Vetterli, Martin: Coordinate difference matrices (2020)
  5. Beik, Fatemeh P. A.; Jbilou, Khalide; Najafi-Kalyani, Mehdi; Reichel, Lothar: Golub-Kahan bidiagonalization for ill-conditioned tensor equations with applications (2020)
  6. Benvenuto, Federico; Jin, Bangti: A parameter choice rule for Tikhonov regularization based on predictive risk (2020)
  7. Bringout, Gaël; Erb, Wolfgang; Frikel, Jürgen: A new 3D model for magnetic particle imaging using realistic magnetic field topologies for algebraic reconstruction (2020)
  8. Calandra, Henri; Gratton, Serge; Riccietti, Elisa; Vasseur, Xavier: On iterative solution of the extended normal equations (2020)
  9. Cerdán, J.; Guerrero, D.; Marín, J.; Mas, J.: Preconditioners for rank deficient least squares problems (2020)
  10. Chang, Xiao-Wen; Kang, Peng; Titley-Peloquin, David: Error bounds for computed least squares estimators (2020)
  11. Coppé, Vincent; Huybrechs, Daan; Matthysen, Roel; Webb, Marcus: The AZ algorithm for least squares systems with a known incomplete generalized inverse (2020)
  12. Cornelis, Jeffrey; Vanroose, W.: Projected Newton method for noise constrained (\ell_p) regularization (2020)
  13. de Oliveira, F. R.; Ferreira, O. P.: Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations (2020)
  14. 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)
  15. Estrin, Ron; Friedlander, Michael P.; Orban, Dominique; Saunders, Michael A.: Implementing a smooth exact penalty function for equality-constrained nonlinear optimization (2020)
  16. Fu, Anqi; Zhang, Junzi; Boyd, Stephen: Anderson accelerated Douglas-Rachford splitting (2020)
  17. Gazzola, Silvia; Kilmer, Misha E.; Nagy, James G.; Semerci, Oguz; Miller, Eric L.: An inner-outer iterative method for edge preservation in image restoration and reconstruction (2020)
  18. Hallman, Eric: Estimating the backward error for the least-squares problem with multiple right-hand sides (2020)
  19. Hallman, Eric: Sharp 2-norm error bounds for LSQR and the conjugate gradient method (2020)
  20. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)

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