[with patch, positive review] Improve accuracy of graph eigenvalues

[rbeezer]

Eigenspaces and eigenvalues of graphs are computed by converting the adjacency matrix to have RDF as the base ring, but there are now better routines in place for eigenvalues of integer matrices, so the eigenspaces() and eigenvalues() methods should be using those.

At present, the approximate values of the eigenvalues lead to eigenspaces "splitting" into pieces (i.e. several eigenspaces that should all be one), so in that regard current results are inaccurate.

[rbeezer]

The patch generally improves graph eigenvalues by not altering the adjacency matrix and therefore allowing new routines to take advantage of the adjacency matrix being a matrix of integers. It also corrects a serious bug for eigenvalues of digraphs. More specifically

1. The adjacency matrix is no longer converted to a matrix of reals or complexes.

2. Eigenspaces are now more abstract (but are exact). More numerical results come from the new eigenvectors() method.

3. Any complex part of an eigenvalue was previously being stripped, as if a graph could never be a digraph. This has been corrected and a simple doctest added.

4. While in the neighborhood, the characteristic_polynomial() got some cosmetic changes in its docstring.

5. Long-term, the spectrum() command should return some sort of object, like a Factorization object, as discussed on sage-devel. Then the current spectrum() could be renamed as eigenvalues().

[saliola]

Looks good. And great job with the documentation!

Patch applies cleanly to 4.1.alpha2; testing in progress on sage-math. I'll report back soon.

[saliola]

No new doctest failures are introduced by this patch on 4.1.alpha2.

Positive review.