I have a matrix $A$ and a column vector $\vec{x}$. The number of columns of $A$ is much larger than the number of rows. All entries $a_{ij}$ of $A$ are non-negative. I know there is a vector $\vec{s}$ with scalar elements $s_i>=0$ so that: $$ A*\vec{s}=\vec{x} $$ I compute $\vec{s}$ by using a non-negative least squares solver (from Scipy).
My goal is to compute at least one more vector $s'$ that fullfills the equation $$ A*\vec{s'}=\vec{x} $$ with $s_i'>=0$.
For this I compute the Nullspace of $A$. Let $B$ be its base and $b_i$ the columns of $B$. I can now produce $s'$ $$ s'=s+\sum \lambda_ib_i $$ I now struggle with the problem of finding appropriate $\lambda_i$s (besides all being 0) so that the non-negativity constraint on the entries of $s'$ is fullfilled.
Thank you for your suggestions.