Conversion to float taking forever

[Nico Van Cleemput]

Today I came across the following problem. I start by computing the Gutman energy for a graph:

sage: c26 = Graph('YsP@H?PC?O`?@@?_?G?@??CC?G??GG?E??@_??K???W???W???H???E_')

sage: ge = sum(lam for lam in c26.adjacency_matrix().eigenvalues() if lam > 0)

sage: ge

19.87092248988078?

So far everything seems to be OK, but then I try

sage: float(ge)

Now you can wait for a very long time. (I actually didn't test to see how long it would exactly take) The type of ge is sage.rings.qqbar.AlgebraicNumber. Looking at the implementation of the float conversion, the problem seems to come from the following:

sage: AA(ge)

This indeed also seems to take forever. At that point I'm lost. I don't really understand much of that part of Sage, so I don't know how to look further for a solution to this problem. Is there another way to get a good float approximation of that value?

Cheers
Nico

[Marc Mezzarobba]

Nico Van Cleemput wrote:
> Looking at the implementation of the
> float conversion, the problem seems to come from the following:
>
> sage: AA(ge)

So apparently Sage is trying to prove that the imaginary part of the
algebraic result is exactly zero.

> Is there another way to get a good float approximation of that value?

Yes:

sage: float(CDF(ge))
19.870922489880776

--
Marc

[Volker Braun]

If you don't need provable correct answers then just change to RDF/CDF for fast computation:

sage: c26 = Graph('YsP@H?PC?O`?@@?_?G?@??CC?G??GG?E??@_??K???W???W???H???E_')

sage: sum(lam for lam in c26.adjacency_matrix().change_ring(RDF).eigenvalues() if lam > 0)

19.870922489880787

[John Cremona]

What people have not been saying explicitly is this: if you ask for
the eigenvalues of an integer matrix then they will be returned as
elements of QQbar, i.e. as algebraic numbers with a specific embedding
into CC, which display looking like approximate floating points
numbers but are not. These are then very inefficient to do a lot of
comutations with, as you discovered. To avoid this, explicitly change
your matrix's base ring to something approximate like RDF and then
the eigenvalue computation will be done there, with all the usual
warnings from numerical analysis about precision issues.

John

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