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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.

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  • $\begingroup$ did you try for small $N$? how did you implement the logarithm of a vector? without more details on what specifically is going wrong it is unlikely a helpful answer can be obtained. $\endgroup$ Jul 18, 2019 at 13:48
  • $\begingroup$ Yes I have tried for N=1 and 2 and it doesn't converge. The logarithm of a vector I have just taken to be the natural logarithm of each component of the vector $\pmb{\Psi}$. $\endgroup$ Jul 18, 2019 at 16:14
  • $\begingroup$ so is this the set of equations you want to solve for $\Psi$, in order to find the critical point? (your element-wise evaluation of the logarithm is unusual, this is why I ask) $$\psi_n=\exp\left[-\frac{1}{\tau}\left(\sum_{i=1}^N|\psi_i|\right)^{-1}\sum_{j=1}^N u_{nj}\psi_j\right],\;\;n=1,2,\ldots N$$ $\endgroup$ Jul 18, 2019 at 18:37
  • $\begingroup$ Yes that what I am trying to do. $\endgroup$ Jul 18, 2019 at 18:42
  • $\begingroup$ you say the $N=1$ solution fails, but it's a trivial $\psi_1=e^{1/\tau}$ --- how can that fail? $\endgroup$ Jul 18, 2019 at 18:45

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