newton's method for multiple variables / arbitrary precision
I am trying to find a numerical approximation with arbitrary precision to a real solution to a system of multivariate polynomial equations. I start out with an approximation which is somewhat close to solving the system, up to an precision of about 1e-05. (Meaning that the equations that I try to evaluate are not zero, but smaller than 1e-05 for my starting value)
In this question is it recommended to use scipy's fmin_tnc method, which is what I did. This works out very nicely and it quickly gave a new solution which now solves my system with precision 1e-07. In the Scipy doc it is stated that one can set the "epsilon" parameter, but not smaller than machine precision. So it seems like I can't get much more precision with this method?!
Let's say I want to solve my system with precision 1e-250. My questions are:
- Can I use the fmin_tnc function to find solutions with higher precision?
- I there another way in sage to find real solutions to polynomial systems locally (e.g. with the pari/gp)?