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 268 articles , 1 standard article )

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  1. Clempner, Julio B.; Poznyak, Alexander S.: A Tikhonov regularized penalty function approach for solving polylinear programming problems (2018)
  2. Rao, Kaustubh; Malan, Paul; Perot, J.Blair: A stopping criterion for the iterative solution of partial differential equations (2018)
  3. Ahmadi-Asl, Salman; Beik, Fatemeh Panjeh Ali: Iterative algorithms for least-squares solutions of a quaternion matrix equation (2017)
  4. Bellis, Cédric; Trabelsi, Manel; Frémy, Flavien: Reconstructing material properties by deconvolution of full-field measurement images: the conductivity case (2017)
  5. Bentbib, A.H.; El Guide, M.; Jbilou, K.: The block Lanczos algorithm for linear ill-posed problems (2017)
  6. Borges, Carlos; Gillman, Adrianna; Greengard, Leslie: High resolution inverse scattering in two dimensions using recursive linearization (2017)
  7. Breedveld, Sebastiaan; van den Berg, Bas; Heijmen, Ben: An interior-point implementation developed and tuned for radiation therapy treatment planning (2017)
  8. Chung, Julianne; Chung, Matthias: Optimal regularized inverse matrices for inverse problems (2017)
  9. Diamond, Steven; Boyd, Stephen: Stochastic matrix-free equilibration (2017)
  10. Hnětynková, Iveta; Kubínová, Marie; Plešinger, Martin: Noise representation in residuals of LSQR, LSMR, and CRAIG regularization (2017)
  11. Ji, Hao; Li, Yaohang: Block conjugate gradient algorithms for least squares problems (2017)
  12. Ling, Si-Tao; Jia, Zhi-Gang; Jiang, Tong-Song: LSQR algorithm with structured preconditioner for the least squares problem in quaternionic quantum theory (2017)
  13. Mojarrab, M.; Toutounian, F.: Global LSMR(Gl-LSMR) method for solving general linear systems with several right-hand sides (2017)
  14. Morikuni, Keiichi: Multistep matrix splitting iteration preconditioning for singular linear systems (2017)
  15. Novati, Paolo: Some properties of the Arnoldi-based methods for linear ill-posed problems (2017)
  16. 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)
  17. Renaut, Rosemary A.; Vatankhah, Saeed; Ardestani, Vahid E.: Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems (2017)
  18. Scott, Jennifer: On using Cholesky-based factorizations and regularization for solving rank-deficient sparse linear least-squares problems (2017)
  19. Voronin, Sergey; Zaroli, Christophe; Cuntoor, Naresh P.: Conjugate gradient based acceleration for inverse problems (2017)
  20. Zhang, Huamin; Yin, Hongcai: Conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations (2017)

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