Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the following one:

Let $W$ be a subspace of $\mathbb{R}^{n}$ and $(e_1,\dots,e_n)$ be the standard basis of $\mathbb{R}^{n}$. Find all $F\subseteq\{1,\dots,n\}$ such that $W_{F}:=W\cap \left\langle e_{i}|i\in F\right\rangle$ is 1-dimensional and intersects non-trivially the cone of vectors with non-negative entries (let $u_{F}$ be such non-trivial vector). Then our desired set is generated (as a cone) by all such $u_{F}$ 's for appropriate $F$ 's.

Thanks.