Systems Optimization Laboratory
Stanford, CA 943054121 USA

LNLQ: Sparse Equations and Least Squares
 AUTHORS:
Ron Estrin,
Dominique Orban.
Michael Saunders.
 CONTRIBUTORS:
 CONTENTS: Implementation of a conjugategradient type method
for solving the leastnorm problem
\begin{align*}
\text{minimize } & \x\^2 \text{ subject to } Ax = b,
\end{align*}
where the matrix \(A\) may be square or rectangular
with any rank, and the constraints are assumed to be consistent.
\(A\) is represented by a routine for computing \(Av\) and \(A^T u\)
for given vectors \(v\) and \(u\).
The method is based on the GolubKahan bidiagonalization
process. It is algebraically equivalent to applying SYMMLQ
to the normal equation of the second kind,
\(
AA^T y = b, \ x = A^T y,
\)
but has better numerical properties, especially if
\(A\) is illconditioned.
LNLQ reduces the error \(\x  x_k\\) monotonically.
If \(A\) is symmetric, use
SYMMLQ,
MINRES,
or MINRESQLP.
 REFERENCES:
R. Estrin, D. Orban, and M. A. Saunders,
LNLQ: An iterative method for linear least squares
with an error minimization property,
submitted to SIMAX (2019).
 RELEASE:
2019: Matlab version.
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