Mathematica FindRoot exploration of parameter space

[NRedacted]

I am solving three non-linear equations in three variables (H0D,H0S and H1S) using FindRoot. In addition to the three variables of interest, there are four parameters in these equations that I would like to be able to vary. My parameters and the range in which I want to vary them are as follows:

$CF \in \{0,15\}$,$CR \in \{0,8\}$,$T \in \{0,0.35\}$,$H1R \in \{40,79\}$

The problem is that my non-linear system may not have any solutions for part of this parameter range. What I basically want to ask is if there is a smart way to find out exactly what part of my parameter range admits real solutions.

I could run a FindRoot inside a loop but because of non-linearity, FindRoot is very sensitive to initial conditions so frequently error messages could be because of bad initial conditions rather than absence of a solution.

Is there a way for me to find out what parameter space works, short of plugging 10^4 combinations of parameter values by hand and playing around with the initial conditions and hoping that FindRoot gives me a solution?

Thanks a lot,

 

[mmwave]

Hello,

Thank you for sharing your problem. Finding solutions to non-linear equations can be a challenging task, especially when there are multiple variables and parameters involved. One approach you could take is to use a numerical method such as a root-finding algorithm to systematically search for solutions within your parameter range.

One possible solution is to use a grid search method, where you discretize your parameter space into a grid and then systematically test different combinations of parameter values. This approach can be computationally intensive, but it can provide a comprehensive overview of the parameter space and identify regions where solutions exist.

Another option is to use a global optimization algorithm, such as genetic algorithms or simulated annealing, which are specifically designed to handle non-linear problems with multiple variables and parameters. These algorithms search for an optimal solution by exploring different regions of the parameter space and can often find solutions even in cases where traditional root-finding methods fail.

Additionally, you could try varying the initial conditions for your FindRoot algorithm to see if that helps to identify solutions. You could also consider using a different root-finding algorithm, as some may be more robust or better suited for your specific problem.

In summary, there are several potential approaches you could take to find solutions to your non-linear system within your parameter range. I would recommend exploring different options and seeing which one works best for your specific problem. Good luck!