I am trying to find the critical points of the following non-linear system $$\pmb{F}(\pmb{\Psi}) = -\tau\lVert\pmb{\Psi}\rVert_{\ell^1}\ln\pmb{\Psi}-\pmb{U}\pmb{\Psi}$$ where $\tau$ is temperature$, \pmb{\Psi}\in\mathbb{R}^N$ and $\pmb{U}\in\mathbb{R}^{N\times N}$ is a matrix with entries $u_{i,j} = -\cos(\frac{4\pi}{N}(i-j))$, where N is the number of divisions of the interval $[0,2\pi]$. I have implemented Newtons method to solve this system but it is not converging.
I believe its to do with my initial conditions but I am unsure how I can go about determining them for this system.