# LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

@article{Paige1982LSQRAA, title={LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, author={C. Paige and M. Saunders}, journal={ACM Trans. Math. Softw.}, year={1982}, volume={8}, pages={43-71} }

An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method… Expand

#### 3,758 Citations

LSMR: An iterative algorithm for sparse least-squares problems

- Computer Science, Mathematics
- ArXiv
- 2010

An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problems min ‖Ax−b‖2, with A being sparse or a fast linear operator, and it is observed in practice that ‖rk‖ also decreases monotonically, so that compared to LSQR it is safer to terminate L SMR early. Expand

Solving Generalized Least-Squares Problems with LSQR

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 1999

Numerical experiments comparing LSQR(A-1)with similar preconditioned Krylov methods are described which demonstrate that the new method exhibits superior numerical properties when the Schur complement BTA-1B is ill conditioned. Expand

An inexact Levenberg-Marquardt method for large sparse nonlinear least squres

- Mathematics
- 1985

A method for solving problems of the form is presented. The approach of Levenberg and Marquardt is used, except that the linear least squares subproblem arising at each iteration is not solved… Expand

Extending Lsqr to Generalised Least-squares and Schur Complement Problems without Resorting to Cholesky Decompositions

- 1997

An iterative method for solving problems of the form A B B T 0 x y = b 0 in a generalised least-squares sense is given, where the n n symmetric positive-deenite matrix A and the n m full column rank… Expand

Some properties of LSQR for large sparse linear least squares problems

- Mathematics, Computer Science
- J. Syst. Sci. Complex.
- 2010

This paper derives an analogously elegant formula for residual norms of LSQR and its mathematically equivalent CGLS. Expand

On an iterative method for solving the least squares problem of rank-deficient systems

- Mathematics, Computer Science
- Int. J. Comput. Math.
- 2015

An iterative method for finding the least squares (LS) solution to the inconsistent system Ax=b, where A is an m×n matrix of rank r, and a numerical test to find E without using decomposition methods is proposed. Expand

Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems

- Mathematics
- 1998

{The conjugate gradient method applied to the normal equations ATAx=ATb (CGLS) is often used for solving large sparse linear least squares problems. The mathematically equivalent algorithm LSQR based… Expand

A matrix LSQR algorithm for solving constrained linear operator equations

- Mathematics
- 2014

In this work, an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear opera- tor equation A( X) = B and the minimum Frobenius norm residual problem jjA( X)… Expand

IMPLICITLY RESTARTING THE LSQR ALGORITHM

- Mathematics
- 2014

The LSQR algorithm is a popular method for solving least-squares problems. For some matrices, LSQR may require a prohibitively large number of iterations to determine an approximate solution within a… Expand

A PRECONDITIONER FOR THE LSQR ALGORITHM

- Mathematics
- 2008

Iterative methods are often suitable for solving least squares problems minkAx bk2, where A 2 m◊n is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these… Expand

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