CONTENTS: Implementation of a conjugate-gradient type method
for solving sparse linear equations and
sparse least-squares problems:
\begin{align*}
\text{Solve } & Ax=b
\\ \text{or minimize } & \|Ax-b\|^2
\\ \text{or minimize } & \|Ax-b\|^2 + \lambda^2 \|x\|^2,
\end{align*}
where the matrix \(A\) may be square or rectangular
(over-determined or under-determined), and may have any rank.
It is represented by a routine for computing \(Av\) and \(A^T u\)
for given vectors \(v\) and \(u\).
The scalar \(\lambda\) is a damping parameter.
If \(\lambda > 0\), the solution is "regularized" in the sense that
a unique solution always exists, and \(\|x\|\) is bounded.
The method is based on the Golub-Kahan bidiagonalization
process. It is algebraically equivalent to applying SYMMLQ
to the normal equation
\(
(A^T A + \lambda^2 I) x = A^T b,
\)
but has better numerical properties, especially if
\(A\) is ill-conditioned.
LSLQ reduces the error \(\|x - x_k\|\) monotonically.
If \(A\) is symmetric, use
SYMMLQ,
MINRES,
or MINRES-QLP.
If \(A\) has low column rank
and \(\lambda=0\), the solution is not unique.
LSLQ returns the solution of minimum length.
Thus for under-determined systems, it solves the problem
\(\min \|x\| \text{ subject to } Ax=b\).
More generally, it solves the problem
\(\min \|x\| \text{ subject to } A^T Ax=A^T b\),
where \(A\) may have any shape or rank.
For \(\min \|x\| \text{ subject to } Ax=b\),
consider using CRAIG
or LNLQ.
