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 358 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. Bringout, Gaël; Erb, Wolfgang; Frikel, Jürgen: A new 3D model for magnetic particle imaging using realistic magnetic field topologies for algebraic reconstruction (2020)
  5. Cerdán, J.; Guerrero, D.; Marín, J.; Mas, J.: Preconditioners for rank deficient least squares problems (2020)
  6. Chang, Xiao-Wen; Kang, Peng; Titley-Peloquin, David: Error bounds for computed least squares estimators (2020)
  7. Cornelis, Jeffrey; Vanroose, W.: Projected Newton method for noise constrained (\ell_p) regularization (2020)
  8. de Oliveira, F. R.; Ferreira, O. P.: Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations (2020)
  9. 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)
  10. Estrin, Ron; Friedlander, Michael P.; Orban, Dominique; Saunders, Michael A.: Implementing a smooth exact penalty function for equality-constrained nonlinear optimization (2020)
  11. 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)
  12. Hallman, Eric: Estimating the backward error for the least-squares problem with multiple right-hand sides (2020)
  13. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)
  14. Jia, Zhongxiao: Regularization properties of LSQR for linear discrete ill-posed problems in the multiple singular value case and best, near best and general low rank approximations (2020)
  15. Jia, Zhongxiao: Regularization properties of Krylov iterative solvers CGME and LSMR for linear discrete ill-posed problems with an application to truncated randomized SVDs (2020)
  16. Jia, Zhongxiao: Approximation accuracy of the Krylov subspaces for linear discrete ill-posed problems (2020)
  17. Jia, Zhongxiao; Yang, Yanfei: A joint bidiagonalization based iterative algorithm for large scale general-form Tikhonov regularization (2020)
  18. Mohammady, Somaieh; Eslahchi, M. R.: Extension of Tikhonov regularization method using linear fractional programming (2020)
  19. Reichel, Lothar; Sadok, Hassane; Zhang, Wei-Hong: Simple stopping criteria for the LSQR method applied to discrete ill-posed problems (2020)
  20. Schaub, Michael T.; Benson, Austin R.; Horn, Paul; Lippner, Gabor; Jadbabaie, Ali: Random walks on simplicial complexes and the normalized Hodge 1-Laplacian (2020)

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