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 385 articles , 1 standard article )

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  1. Bauch, Jonathan; Nadler, Boaz; Zilber, Pini: Rank (2r) iterative least squares: efficient recovery of ill-conditioned low rank matrices from few entries (2021)
  2. Chang, Xiaokai; Yang, Junfeng: A golden ratio primal-dual algorithm for structured convex optimization (2021)
  3. Chang, Xiao-Wen; Paige, Christopher C.; Titley-Peloquin, David: Structure in loss of orthogonality (2021)
  4. 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
  5. Jia, Zhongxiao; Li, Haibo: The joint bidiagonalization process with partial reorthogonalization (2021)
  6. Kämmerer, Lutz; Ullrich, Tino; Volkmer, Toni: Worst-case recovery guarantees for least squares approximation using random samples (2021)
  7. Ma, Yaonan; Liao, Li-Zhi: The Glowinski-Le Tallec splitting method revisited: a general convergence and convergence rate analysis (2021)
  8. Montoison, Alexis; Orban, Dominique: TriCG and TriMR: two iterative methods for symmetric quasi-definite systems (2021)
  9. Morikuni, Keiichi: Projection method for eigenvalue problems of linear nonsquare matrix pencils (2021)
  10. Neubauer, Andreas: On Tikhonov-type regularization with approximated penalty terms (2021)
  11. Potts, Daniel; Schmischke, Michael: Approximation of high-dimensional periodic functions with Fourier-based methods (2021)
  12. Rezaiee-Pajand, Mohammad; Sarmadi, Hassan; Entezami, Alireza: A hybrid sensitivity function and Lanczos bidiagonalization-Tikhonov method for structural model updating: application to a full-scale bridge structure (2021)
  13. Samar, Mahvish; Li, Hanyu; Wei, Yimin: Condition numbers for the (K)-weighted pseudoinverse (L^\dagger_K) and their statistical estimation (2021)
  14. Sumbatyan, M. A.; Martynova, T. S.; Musatova, N. K.: Boundary element methods in diffraction of a point-source acoustic wave by a rigid infinite wedge (2021)
  15. 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)
  16. 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)
  17. Baechler, Gilles; Dümbgen, Frederike; Elhami, Golnoosh; Kreković, Miranda; Vetterli, Martin: Coordinate difference matrices (2020)
  18. Beik, Fatemeh P. A.; Jbilou, Khalide; Najafi-Kalyani, Mehdi; Reichel, Lothar: Golub-Kahan bidiagonalization for ill-conditioned tensor equations with applications (2020)
  19. Benvenuto, Federico; Jin, Bangti: A parameter choice rule for Tikhonov regularization based on predictive risk (2020)
  20. Bock, Hans Georg; Gutekunst, Jürgen; Potschka, Andreas; Garcés, María Elena Suaréz: A flow perspective on nonlinear least-squares problems (2020)

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