LSMR: An iterative algorithm for sparse least-squares problems

SIAM Journal on Scientific Computing

Volumn 33, Issue 5, 2011, Pages 2950-2971

  Fong, David Chin Lung ^a   Saunders, Michael ^b  

^a STANFORD UNIVERSITY   (United States)

^b Stanford University   (United States)

Author keywords

Conjugate gradient method; Golub Kahan process; Iterative method; Krylov subspace method; Least squares problem; LSQR; Minimum residual method; MINRES; Sparse matrix

Indexed keywords

LEAST SQUARES APPROXIMATIONS; LINEAR SYSTEMS; MATHEMATICAL OPERATORS; PROBLEM SOLVING;

CONJUGATE-GRADIENT METHOD; GOLUB-KAHAN PROCESS; ITERATIVE ALGORITHM; KRYLOV-SUBSPACE METHODS; LEAST SQUARE PROBLEMS; LSQR; MINIMUM-RESIDUAL METHOD; MINRES; RESIDUAL METHOD; SPARSE MATRICES;

CONJUGATE GRADIENT METHOD;

EID: 81555200713     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/10079687X     Document Type: Conference Paper

References (26)

1. M. Arioli and S. Gratton, Least-Squares Problems, Normal Equations, and Stopping Criteria for the Conjugate Gradient Method, Technical report RAL-TR-2008-008, Rutherford Appleton Laboratory, Oxfordshire, UK, 2008.
2. S. J. Benbow, Solving generalized least-squares problems with LSQR, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 166-177.
3. X.-W. Chang, C. C. Paige, and D. Titley-Peloquin, Stopping criteria for the iterative solution of linear least squares problems, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 831-852.
4. S.-C. Choi, C. C. Paige, and M. A. Saunders, MINRES-QLP: A Krylov subspace method for indefinite or singular symmetric systems, SIAM J. Sci. Comput., to appear.
5. T. A. Davis, University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/ research/sparse/matrices.
6. G. H. Golub and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp. 205-224.
7. S. Gratton, P. Jiránek, and D. Titley-Peloquin, On the Accuracy of the Karlson-Waldén Estimate of the Backward Error for Linear Least Squares Problems, CERFACS Technical report TR/PA/11/19, CERFACS, Toulouse, France, 2011.
8. J. F. Grcar, M. A. Saunders, and Z. Su, Estimates of Optimal Backward Perturbations for Linear Least Squares Problems, Report SOL 2007-1, Department of Management Science and Engineering, Stanford University, Stanford, CA, 2007.
9. K. Hayami, J.-F. Yin, and T. Ito, GMRES methods for least squares problems, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2400-2430.
10. N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
11. LSMR software for linear systems and least squares, http://www.stanford. edu/group/ SOL/software.html.
12. D. G. Luenberger, The conjugate residual method for constrained minimization problems, SIAM J. Numer. Anal., 7 (1970), pp. 390-398.
13. P. Jiránek and D. Titley-Peloquin, Estimating the backward error in LSQR, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2055-2074.
14. C. C. Paige, M. Rozložník, and Z. Strakoš, Modified Gram-Schmidt (MGS), least squares, and backward stability of MGS-GMRES, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 264-284. (Pubitemid 46277544)
15. C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
16. C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), pp. 43-71.
17. C. C. Paige and M. A. Saunders, Algorithm 583; LSQR: Sparse linear equations and leastsquares problems, ACM Trans. Math. Software, 8 (1982), pp. 195-209.
18. V. Pereyra, private communication, 2010.
19. PROPACK software for SVD of sparse matrices, http://soi.stanford.edu/ ~rmunk/ PROPACK/.
20. Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
21. G. W. Stewart, An inverse perturbation theorem for the linear least squares problem, SIGNUM Newsletter, 10 (1975), pp. 39-40.
22. G. W. Stewart, Research, development and LINPACK, in Mathematical Software III, J. R. Rice, ed., Academic Press, New York, 1977, pp. 1-14.
23. G. W. Stewart, The QLP approximation to the singular value decomposition, SIAM J. Sci. Comput., 20 (1999), pp. 1336-1348. (Pubitemid 32251220)
24. Z. Su, Computational Methods for Least Squares Problems and Clinical Trials, Ph.D. thesis, SCCM, Stanford University, Stanford, CA, 2005.
25. D. Titley-Peloquin, Backward Perturbation Analysis of Least Squares Problems, Ph.D. thesis, School of Computer Science, McGill University, Montreal, QC, Canada, 2010.
26. B. Waldén, R. Karlson, and J.-G. Sun, Optimal backward perturbation bounds for the linear least squares problem, Numer. Linear Algebra Appl., 2 (1995), pp. 271-286.