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# 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:

1. Can I use the fmin_tnc function to find solutions with higher precision?
2. I there another way in sage to find real solutions to polynomial systems locally (e.g. with the pari/gp)?
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## Comments

Could you provide the system? I do not know whether pari/gp can help, but it is not hard to setup a Newton method. From what I saw the method fmin_trunc looks much more powerful than what you need.

( 2014-06-29 05:26:32 -0500 )edit

@vdelecroix: I figured out how to do this, see my answer below

( 2014-07-03 02:07:10 -0500 )edit

@mf: cool! Could you select your answer (that way your question will be seen as answered in the list).

( 2014-07-03 03:00:36 -0500 )edit

@vdelecroix: will do as soon as I have enough karma to do so..

( 2014-07-03 03:14:08 -0500 )edit

## 1 answer

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One easy way to this is to use mpmath.findroot!

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## Comments

Nice! I've added a link to the documentation.

( 2014-07-03 07:22:08 -0500 )edit
( 2018-11-23 11:14:35 -0500 )edit

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Asked: 2014-05-20 11:25:59 -0500

Seen: 552 times

Last updated: Jul 03 '14