GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Preconditioned conjugate gradients for solving singular systems
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
SIAM Journal on Scientific and Statistical Computing
Solving positive (semi) definite linear systems by preconditioned iterative methods
Proceedings of a conference on Preconditioned conjugate gradient methods
Preconditioners for indefinite systems arising in optimization
SIAM Journal on Matrix Analysis and Applications
The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
Preconditioning of Indefinite and Almost Singular Finite Element Elliptic Equations
SIAM Journal on Scientific Computing
MINRES and MINERR Are Better than SYMMLQ in Eigenpair Computations
SIAM Journal on Scientific Computing
The QLP Approximation to the Singular Value Decomposition
SIAM Journal on Scientific Computing
Iterative Regularization and MINRES
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)
A Note on Preconditioning for Indefinite Linear Systems
SIAM Journal on Scientific Computing
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Iterative Methods for Nearly Singular Linear Systems
SIAM Journal on Scientific Computing
Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning
SIAM Journal on Matrix Analysis and Applications
Accurate Solution of Weighted Least Squares by Iterative Methods
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Algorithms for sparse matrix eigenvalue problems.
Algorithms for sparse matrix eigenvalue problems.
Ipsol: an interior point solver for nonconvex optimization problems
Ipsol: an interior point solver for nonconvex optimization problems
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
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems
SIAM Journal on Scientific Computing
Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems
ACM Transactions on Mathematical Software (TOMS)
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CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where $R$ is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce $R$ to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.