Fix few bugs in ClusterAlgebra discovered while working on #26771

[etn40ff]

As pointed out by the people at #26771, ClusterAlgebra has few bugs.

• Default arguments do not play nicely with UniqueRepresentation

  sage: A1 = ClusterAlgebra(['A',2])
  sage: A2 = ClusterAlgebra(['A',2], cluster_variable_prefix='x')
  sage: A1 is A2
  False

• mutate_initial keeps the same variable names; In theory this is not a problem but may create confusion. For example suppose we are looking at the Grassmannian of planes in a 4 dimensional vector space. Then the associate cluster algebra is

  sage: A = ClusterAlgebra( matrix(5,[0,1,-1,1,-1]), cluster_variable_names=['p13'], coefficient_names=['p12','p23','p34','p41'], scalars=QQbar); A
  A Cluster Algebra with cluster variable p13 and coefficients p12, p23, p34, p41 over Algebraic Field

Swapping out p13 from the initial seed should give p24 or, at the very least *not* p13. Getting p24 requires some understanding of what the ring in question really is that goes beyond the scope of this class. Currently we get

  sage: A.mutate_initial(0)
  A Cluster Algebra with cluster variable p13 and coefficients p12, p23, p34, p41 over Algebraic Field

• mutate_initial does not compute all the required F-polynomials and it does not enforce the fact that they are not rational expressions

  sage: A1 = ClusterAlgebra(['A',[2,1],1])
  sage: A2 = A1.mutate_initial([0,1,0])
  sage: len(A2.g_vectors_so_far()) == len(A2.F_polynomials_so_far())
  False
  sage: all(parent(f) == A2._U for f in A2.F_polynomials_so_far())
  False

• finally mutate_initial does not enforce the fact that initial g-vectors belong to the initial cluster but rather prepend the performed mutation sequence to all the paths already known.

  sage: A2.find_g_vector((0,0,1))
  [0, 1, 0]

This, while technically not false, may generate confusion.

• _mutated_F is doing / rather than // resulting in slower computations and parent issues.

This ticket deals with all these issues.

[etn40ff]

New commits:

​d3cd59d

Fix 28176

[chapoton]

bot is not happy:

sage -t --long src/sage/algebras/cluster_algebra.py  # Tab character found

[etn40ff]

Oops. Fixed, sorry Frédéric!

With the occasion I also got some dramatic speedup in mutate_initial by not mutating F-polynomials a quadratic number of times.

Also I changed a division into a polynomial division in _mutated_F; this enforces F-polynomials to live in the correct ambient and yields a decent speedup in mutate.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​9fc136c	Fix 28176
​33f4fdd	Fixed tabs and significant speedup

[chapoton]

doc does not build

+[dochtml] OSError: /home/chapoton/sage/local/lib/python2.7/site-packages/sage/algebras/cluster_algebra.py:docstring of sage.algebras.cluster_algebra.ClusterAlgebra.mutate_initial:4: WARNING: Content block expected for the "WARNING" directive; none found.
+Makefile:2036: recipe for target 'doc-html' failed

and missing :: here

+        A faster example without recomputing F-polynomials

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​0b22eb7

Doctest

[etn40ff]

Replying to chapoton:

doc does not build

+[dochtml] OSError: /home/chapoton/sage/local/lib/python2.7/site-packages/sage/algebras/cluster_algebra.py:docstring of sage.algebras.cluster_algebra.ClusterAlgebra.mutate_initial:4: WARNING: Content block expected for the "WARNING" directive; none found.

I have never seen this before, is it just a missing indentation? In case it is fixed now.

[tscrim]

Replying to etn40ff:

Replying to chapoton:

doc does not build

+[dochtml] OSError: /home/chapoton/sage/local/lib/python2.7/site-packages/sage/algebras/cluster_algebra.py:docstring of sage.algebras.cluster_algebra.ClusterAlgebra.mutate_initial:4: WARNING: Content block expected for the "WARNING" directive; none found.

I have never seen this before, is it just a missing indentation? In case it is fixed now.

Yes, that is correct.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​dccaa2c

B to B0 in class_call and added a few doctests

[gh-kelleye]

It would be helpful to have a comment explaining a little bit about how lines 1285 - 1287 work; without explanation, understanding it takes some non-trivial thought.

[etn40ff]

Done

---

New commits:

​6d7ce8d

Comments as requested

[gh-kelleye]

All the changes (and additional comments, thank you!) look good to me.