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 394 articles , 1 standard article )

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  1. Bentbib, Abdeslem H.; Khouia, Asmaa; Sadok, Hassane: The LSQR method for solving tensor least-squares problems (2022)
  2. Brust, Johannes J.; Marcia, Roummel F.; Petra, Cosmin G.; Saunders, Michael A.: Large-scale optimization with linear equality constraints using reduced compact representation (2022)
  3. Jiao, Xiangmin; Chen, Qiao: Approximate generalized inverses with iterative refinement for (\epsilon)-accurate preconditioning of singular systems (2022)
  4. Lampe, Jörg; Voss, Heinrich: A survey on variational characterizations for nonlinear eigenvalue problems (2022)
  5. Potts, D.; Schmischke, M.: Learning multivariate functions with low-dimensional structures using polynomial bases (2022)
  6. Bauch, Jonathan; Nadler, Boaz; Zilber, Pini: Rank (2r) iterative least squares: efficient recovery of ill-conditioned low rank matrices from few entries (2021)
  7. Candiani, V.; Hyvönen, N.; Kaipio, J. P.; Kolehmainen, V.: Approximation error method for imaging the human head by electrical impedance tomography (2021)
  8. Chang, Xiaokai; Yang, Junfeng: A golden ratio primal-dual algorithm for structured convex optimization (2021)
  9. Chang, Xiao-Wen; Paige, Christopher C.; Titley-Peloquin, David: Structure in loss of orthogonality (2021)
  10. 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
  11. Jia, Zhongxiao; Li, Haibo: The joint bidiagonalization process with partial reorthogonalization (2021)
  12. Kämmerer, Lutz; Ullrich, Tino; Volkmer, Toni: Worst-case recovery guarantees for least squares approximation using random samples (2021)
  13. Ma, Yaonan; Liao, Li-Zhi: The Glowinski-Le Tallec splitting method revisited: a general convergence and convergence rate analysis (2021)
  14. McQuinn, Matthew: On the primordial information available to galaxy redshift surveys (2021)
  15. Montoison, Alexis; Orban, Dominique: TriCG and TriMR: two iterative methods for symmetric quasi-definite systems (2021)
  16. Morikuni, Keiichi: Projection method for eigenvalue problems of linear nonsquare matrix pencils (2021)
  17. Neubauer, Andreas: On Tikhonov-type regularization with approximated penalty terms (2021)
  18. Potts, Daniel; Schmischke, Michael: Approximation of high-dimensional periodic functions with Fourier-based methods (2021)
  19. 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)
  20. Samar, Mahvish; Li, Hanyu; Wei, Yimin: Condition numbers for the (K)-weighted pseudoinverse (L^\dagger_K) and their statistical estimation (2021)

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