Topics in matrix analysis
Matrix computations (3rd ed.)
Applied numerical linear algebra
Applied numerical linear algebra
Matrix market: a web resource for test matrix collections
Proceedings of the IFIP TC2/WG2.5 working conference on Quality of numerical software: assessment and enhancement
Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
Backward Perturbation Bounds for Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
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
Computational methods for least squares problems and clinical trials
Computational methods for least squares problems and clinical trials
Stopping Criteria for the Iterative Solution of Linear 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
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems
SIAM Journal on Scientific Computing
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We propose practical stopping criteria for the iterative solution of sparse linear least squares (LS) problems. Although we focus our discussion on the algorithm LSQR of Paige and Saunders, the ideas discussed here may also be applicable to other algorithms. We review why the 2-norm of the projection of the residual vector onto the range of $A$ is a useful measure of convergence, and we show how this projection can be estimated efficiently at every iteration of LSQR. We also give practical and cheaply computable estimates of the backward error for the LS problem.