  We propose a simple projection and rescaling algorithm that finds maximum
support solutions to the pair of feasibility problems \[ \text{find} \; x\in
L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat
x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ is a linear subspace of
$\mathbb{R}^n$ and $L^\perp$ is its orthogonal complement. The algorithm
complements a basic procedure that involves only projections onto $L$ and
$L^\perp$ with a periodic rescaling step. The number of rescaling steps and
thus overall computational work performed by the algorithm are bounded above in
terms of a condition measure of the above pair of problems.
  Our algorithm is a natural but significant extension of a previous projection
and rescaling algorithm that finds a solution to the problem \[ \text{find} \;
x\in L\cap\mathbb{R}^n_{++} \] when this problem is feasible. As a byproduct of
our new developments, we obtain a sharper analysis of the projection and
rescaling algorithm in the latter special case.
