I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations:$$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{bmatrix}=0,X\in{}R^{N\times{}M}$$With:$$f_i\left(X\right)=x_{*,i}-F^{-1}\left[k\left(\omega{}\right)+\sum_{j=1}^Mq\left(\omega{}\right)\dot{}F\left[H^{\left(j\right)}\left(x_{*,j}\right)\right]\right]=0$$Where $F$ is the Fourier transform and $H$ a nonlinear function, $k$ and $q$ are constants. $H$ has the following form:$$H\left(x\right)=r{\left(\frac{x}{s}\right)}^{\alpha{}},\s,\ r,\ x\in{}R^N$$The highest $\alpha$ is, the harder it is to find a solution. I tried both simulated annealing and Jacobian-Free Newton Krylov numerical methods, without much success. I believe an issue here might be the way I state the problem.Should JFKN or SA compute a perturbation/update on each matrix element separately? It does not sound reasonable to me, as I am dealing with timeseries, each column of $X$ being a timeseries of length $N$; I guess perturbation should be flexible yet consistent from one element to the next. Ha! A bit of insight or advice would be very much appreciated here!Thank you
Solving a large system of nonlinear equations, where timeseries are the unknown
nonlinear equations;nonlinear programming;newton method