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 242 articles , 1 standard article )

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  1. Ahmadi-Asl, Salman; Beik, Fatemeh Panjeh Ali: Iterative algorithms for least-squares solutions of a quaternion matrix equation (2017)
  2. Bellis, Cédric; Trabelsi, Manel; Frémy, Flavien: Reconstructing material properties by deconvolution of full-field measurement images: the conductivity case (2017)
  3. Borges, Carlos; Gillman, Adrianna; Greengard, Leslie: High resolution inverse scattering in two dimensions using recursive linearization (2017)
  4. Chung, Julianne; Chung, Matthias: Optimal regularized inverse matrices for inverse problems (2017)
  5. Diamond, Steven; Boyd, Stephen: Stochastic matrix-free equilibration (2017)
  6. Morikuni, Keiichi: Multistep matrix splitting iteration preconditioning for singular linear systems (2017)
  7. Novati, Paolo: Some properties of the Arnoldi-based methods for linear ill-posed problems (2017)
  8. Renaut, Rosemary A.; Horst, Michael; Wang, Yang; Cochran, Douglas; Hansen, Jakob: Efficient estimation of regularization parameters via downsampling and the singular value expansion, downsampling regularization parameter estimation (2017)
  9. Renaut, Rosemary A.; Vatankhah, Saeed; Ardestani, Vahid E.: Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems (2017)
  10. Zwaan, Ian N.; Hochstenbach, Michiel E.: Multidirectional subspace expansion for one-parameter and multiparameter Tikhonov regularization (2017)
  11. Adcock, Ben; Platte, Rodrigo B.: A mapped polynomial method for high-accuracy approximations on arbitrary grids (2016)
  12. Andreev, Roman: Wavelet-in-time multigrid-in-space preconditioning of parabolic evolution equations (2016)
  13. Bazán, Fermín S.V.; Kleefeld, Andreas; Leem, Koung Hee; Pelekanos, George: Sampling method based projection approach for the reconstruction of 3D acoustically penetrable scatterers (2016)
  14. Bello Cruz, J.Y.; Ferreira, O.P.; Prudente, L.F.: On the global convergence of the inexact semi-smooth Newton method for absolute value equation (2016)
  15. Dong, Jun-Liang; Gao, Junbin; Ju, Fujiao; Shen, Jinghua: Modulus methods for nonnegatively constrained image restoration (2016)
  16. Gazzola, Silvia; Novati, Paolo: Inheritance of the discrete Picard condition in Krylov subspace methods (2016)
  17. Greif, C.; Paige, C.C.; Titley-Peloquin, D.; Varah, J.M.: Numerical equivalences among Krylov subspace algorithms for skew-symmetric matrices (2016)
  18. Hajarian, Masoud: Least squares solution of the linear operator equation (2016)
  19. Hannukainen, A.; Harhanen, L.; Hyvönen, N.; Majander, H.: Edge-promoting reconstruction of absorption and diffusivity in optical tomography (2016)
  20. Hannukainen, A.; Hyvönen, N.; Majander, H.; Tarvainen, T.: Efficient inclusion of total variation type priors in quantitative photoacoustic tomography (2016)

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