When does fsolve use Levenberg-Marquardt algorithm?

Hi,

when does Matlabs fsolve method uses the Levenberg-Marquardt algorithm? The warning tells me, that the trust-region-dogleg algorithm cannot handle non-square systems.

What are the features/characteristics of a non-square system? Do I have more unknowns than equations? What else can be the 'problem'?

Thanks!

Answer 1

[John D'Errico]

With fewer unknowns than equations, nothing you do can "solve" the problem, because there will be normally be infinitely many solutions (if any exist). At best, you can find a completely arbitrary solution among many. The solution you obtain will be completely dependent on your start point. Change the start point, and you will find some completely different solution.

With more unknowns than equations, you will have no EXACT solution in general. At best, you will need to find a set of parameters that minimizes the sum of squares of residuals. This is what algorithms like Levenberg-Marquardt are designed to solve, and why that algorithm must be invoked. At that point, you might as well have decided to used lsqnonlin instead of fsolve, since lsqnonlin is designed to solve that class of problem anyway.

As for what else can be the problem, it tells you exactly what the problem was! You have a non-square system.

[Manuela Gräfe]

Ok, thank you. I am wondering because I built a script to solve a equation system with some fixed parameters. And with an for-loop I am solving this equation system for about 1000 times with different fixed parameters. The initial values for the fsolve method remain the same.

And so sometimes the fsolve uses the Levenberg-Marquadt. But I can't specify when it does so. Most of the time the trust-region does it's job.

And that is why I am wondering what is typical for a non-square system...

[John D'Errico]

If you change the number of fixed parameters, fsolve must change the way it solves the problem. Not all such problems will have an exact solution.