GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
The QLP Approximation to the Singular Value Decomposition
SIAM Journal on Scientific Computing
Solving Generalized Least-Squares Problems with LSQR
SIAM Journal on Matrix Analysis and Applications
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
An inverse perturbation theorem for the linear least squares problem
ACM SIGNUM Newsletter
Computational methods for least squares problems and clinical trials
Computational methods for least squares problems and clinical trials
Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES
SIAM Journal on Matrix Analysis and Applications
Stopping Criteria for the Iterative Solution of Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
Estimating the Backward Error in LSQR
SIAM Journal on Matrix Analysis and Applications
The university of Florida sparse matrix collection
ACM Transactions on Mathematical Software (TOMS)
GMRES Methods for Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems
SIAM Journal on Scientific Computing
Backward perturbation analysis of least squares problems
Backward perturbation analysis of least squares problems
Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems
ACM Transactions on Mathematical Software (TOMS)
Numerical Algorithms
A matching pursuit approach to solenoidal filtering of three-dimensional velocity measurements
Journal of Computational Physics
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An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares problems $\min \|Ax-b\|_2$, with $A$ being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation $A^T\! Ax = A^T\! b$, so that the quantities $\|A^T\! r_k\|$ are monotonically decreasing (where $r_k = b - Ax_k$ is the residual for the current iterate $x_k$). We observe in practice that $\|r_k\|$ also decreases monotonically, so that compared to LSQR (for which only $\|r_k\|$ is monotonic) it is safer to terminate LSMR early. We also report some experiments with reorthogonalization.