GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
GMRES On (Nearly) Singular Systems
SIAM Journal on Matrix Analysis and Applications
A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems
SIAM Journal on Scientific Computing
Approximate Inverse Preconditioners via Sparse-Sparse Iterations
SIAM Journal on Scientific Computing
The Moore--Penrose Generalized Inverse for Sums of Matrices
SIAM Journal on Matrix Analysis and Applications
On the Relations between ILUs and Factored Approximate Inverses
SIAM Journal on Matrix Analysis and Applications
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
Preconditioning Sparse Nonsymmetric Linear Systems with the Sherman--Morrison Formula
SIAM Journal on Scientific Computing
Balanced Incomplete Factorization
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
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
GMRES Methods for Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
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In this paper, we propose a preconditioning algorithm for least squares problems $\displaystyle{\min_{x\in{{\mathbb{R}}}^n}}\|b-Ax\|_2$ , where A can be matrices with any shape or rank. The preconditioner is constructed to be a sparse approximation to the Moore---Penrose inverse of the coefficient matrix A. For this preconditioner, we provide theoretical analysis to show that under our assumption, the least squares problem preconditioned by this preconditioner is equivalent to the original problem, and the GMRES method can determine a solution to the preconditioned problem before breakdown happens. In the end of this paper, we also give some numerical examples showing the performance of the method.