GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Preconditioning techniques for nonsymmetric and indefinite linear systems
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
Journal of Computational Physics
CIMGS: An Incomplete Orthogonal Factorization Preconditioner
SIAM Journal on Scientific Computing
GMRES On (Nearly) Singular Systems
SIAM Journal on Matrix Analysis and Applications
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems
SIAM Journal on Scientific Computing
Breakdown-free GMRES for Singular Systems
SIAM Journal on Matrix Analysis and Applications
Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES
SIAM Journal on Matrix Analysis and Applications
Greville's method for preconditioning least squares problems
Advances in Computational Mathematics
The university of Florida sparse matrix collection
ACM Transactions on Mathematical Software (TOMS)
Greville's method for preconditioning least squares problems
Advances in Computational Mathematics
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems
SIAM Journal on Scientific Computing
Numerical Algorithms
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The standard iterative method for solving large sparse least squares problems $\min\|\mbox{\boldmath$b$}-A\mbox{\boldmath$x$}\|_2$, $A\in\mathbf{R}^{m\times n}$, is the CGLS method, or its stabilized version, LSQR, which is mathematically equivalent to applying the conjugate gradient method to the normal equation $A^{\mbox{\tiny T}}A\mbox{\boldmath$x$}=A^{\mbox{\tiny T}}\mbox{\boldmath$b$}$. We consider alternative methods using a matrix $B\in\mathbf{R}^{n\times m}$ and applying the generalized minimal residual (GMRES) method to $\min\|\mbox{\boldmath$b$}-AB\mbox{\boldmath$z$}\|_2$ or $\min\|B\mbox{\boldmath$b$}-BA\mbox{\boldmath$x$}\|_2$. We give a sufficient condition concerning $B$ for the GMRES methods to give a least squares solution without breakdown for arbitrary $\mbox{\boldmath$b$}$, for overdetermined, underdetermined, and possibly rank-deficient problems. We then give a convergence analysis of the GMRES methods as well as the CGLS method. Then, we propose using the robust incomplete factorization (RIF) for $B$. Finally, we show by numerical experiments on overdetermined and underdetermined problems that, for ill-conditioned problems, the GMRES methods with RIF give least squares solutions faster than the CGLS and LSQR methods with RIF, and are similar in performance to the reorthogonalized CGLS with RIF.