[WIP] Add implementation of partial sqrt algorithm

[wbhart]

This works ok for small degree, but for large degree and large input it can take too long to find a b(n) it can factor.

Another approach is needed.

[wbhart]

I am ready to merge this. If anyone has any comments on it, now is the time.

[fredrik-johansson]

Perhaps document that this isn't any faster than actually computing the square root (as presently implemented).

[wbhart]

I just added the WIP back as I have found another case that hangs. I have no idea why at this point.

[bobot]

Is there a description of the representation used by the library? It seems not to keep an interval or other way to distinguish the root of the polynomial. Does it make the naïve $P(X^2)=0$ polynomial for the square root impossible?

[wbhart]

That's correct. We implement number fields without embedding. If you wanted a \bar{Q} implementation you might look at Fredrik Johansson's Calcium library.

I'm not sure what you are asking about P(X^2). Could you be more specific?

[bobot]

Even if Calcium library is interesting (thank you for pointing it!) I'm not sure I'm looking at embedding, but I must look more into it (mainly the roots used seem fixed in a field). I tried this library libpoly, but I have efficiency problem since the polynomials used are not minimal so I was looking at other libraries. For P(X^2) it is just the simple way I used to implement positive root in the library: SRI-CSL/libpoly#57 .

EDIT: Calcium was the answer to my problem and it is a great library! Thanks a lot.