Systems Optimization Laboratory
Stanford, CA 94305-4121 USA
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LNLQ: Sparse Equations and Least Squares
- AUTHORS:
Ron Estrin,
Dominique Orban.
Michael Saunders.
- CONTRIBUTORS:
- CONTENTS: Implementation of a conjugate-gradient type method
for solving the least-norm 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 Golub-Kahan 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 ill-conditioned.
LNLQ reduces the error \(\|x - x_k\|\) monotonically.
If \(A\) is symmetric, use
SYMMLQ,
MINRES,
or MINRES-QLP.
- 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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