I'm struggling with the following task in my thesis: given $\mathbf{M}\in\mathbb{R}^{n\times n}$ and some vector $\mathbf{w}\in \mathbb{R}^n$, I would like to find some vector $\mathbf{v}\in\mathbb{R}^n$ that would fulfil: $$ \mathbf{v}\circ(\mathbf{Mv})=\mathbf{w} $$ where $\circ$ denotes elementwise (Haramard) product. In other words, this can be rewritten as: $$ w_i=v_i [\mathbf{Mv}]_i=v_i \sum_j M_{i, j}v_j $$ Even though $\mathbf{M}$ has all the nice properties in the world (strictly diagonally dominant $\rightarrow$ invertible, symmetric, positive semi-definite) I cannot solve this analytically (thankfully, I have already written a numerical algorithm). Is it known to be a sort of problem that cannot be reduced to canned linear algebra algorithms, like inverting or eigenvectors? Or am I missing something? Every time I go down the rabbit hole of rewriting this, I end up with some other matrix $\mathbf{M}'$ and another vector $\mathbf{w}'$ and all I need to do is to find some $\mathbf{v}'$ such that $\mathbf{v}'\circ(\mathbf{M}'\mathbf{v}')=\mathbf{w}'$...
EDIT: Maybe it's worth noting, that I have already reduced this to solving $\mathbf{v}\circ(\mathbf{Mv})=\vec{1}$.
Thanks!
There may be some better solution methods, but you can always cast the problem into an unconstrained minimisation of the degree-4 polynomial $f(v)=\sum_{i=1}^n\left(v_i\sum_{j=1}^nm_{ij}v_j-w_i\right)^2$ over $\mathbb R^n$. The original equation is considered solvable if the minimum value of $f$ is close to zero (within a pre-specified tolerance). Just use any unconstrained optimisation package to your liking.