# Solving a large system of nonlinear equations, where timeseries are the unknown

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

• * means, all variables for column $i$. Those vector $x$ are parts of $X$. Typical values for alpha range from 10 to 25. – CTZStef May 17 '16 at 15:19
• Does the * subscript in $x_{*,i}$ signify anything? Also, I'm confused about the span of $X$, which looks like it should have the same span as $x_{*,i}$. If you can give any more information about the parameters $r,s,\alpha$, this may be helpful also. e.g. Might $s$ be negative? Or are you interested in positive $s$ only? - modified this since I messed up formatting. – Charles May 17 '16 at 15:24
• $r,s$ are both positive. – CTZStef May 17 '16 at 16:37
• Instead for trying to find the zeros, it may be easier to try to minimize $F(X)$ like you have done with simulated annealing. Have you tried simultaneous perturbation stochastic descent? – Biswajit Banerjee May 21 '16 at 5:37
• Minimizing is actually what I am doing, yes. Thanks for the suggestion, seems interesting! – CTZStef May 21 '16 at 15:10