I am using the Newton's method to solve $3\times3$ systems.
For some particular cases, it turns out that at a given iteration, the Jacobian matrix cannot be inverted and that its determinant is very close to zero (looking at the matrix, there are terms that are around
1e+0 and others that are
After investigations, it is clear that one variable has no influence on the system when it is close to the solution.
What is the most clever thing to do with such an issue ?
I would like to have an algorithm that can adapt itself to such situations when they happen.
It is about an optical optimisation problem. The point is to add a surface to an optical system so that it fits some optical properties. The Newton's method finds the roots of a function that takes as input the parameters of the surface and outputs the differences between the optical properties computed and the targeted properties.
I noticed that if the system is complex enough, then we have convergence. But if the system is too simple, the surface is more spherical and the Jacobian goes to very small values, because some parameters like astigmatism axis become influence-less.