Infinite loop in the normalizer

[magv]

Infinite loop in the normalizer

[magv]

OK, I think I know what the problem is: you do need to change < 0.5 into < -0.5 (there is a typo in Lee's paper); the test suite is failing after that because algorithm 3 goes like this: if there are points that can be mutually balanced, do that, otherwise try something else; in contrast, when find_balances finds no pair of points to mutually balance, it gives up.

With < 0.5 in place, lots of point pairs which are not mutually balanceable where still passing the conditions for such, thus masking the problem in find_balances.

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Additionally, find_balances never tries to mutually balance points which are not apparent singularities (that is, is_point_fuchsian is false). This is also a problem.

[gituliar]

At first sight it seems for me that it is not possible to normalize the matrix you wrote. The reason is that any balance increases one's point eigenvalue and decreases another's one by 1. See beggining of the Section 6 on p.14:

A necessary condition of existance of the eps-form is that eigenvalues of all matrix residues have the form `n+alpha*eps".

What we should do hence is to terminate the loop if this condition does not hold.

[magv]

A matrix can be normalizable, but not have an eps-form. These are separate things generally. It would probably be good to give users the ability to normalize matrices, even if they won't be able to factor epsilon later on. With that said, it is true that this particular matrix can not be normalized (without introducing additional singularities at least), because it has two singularities: one at `0` with eigenvalue `1/2` and one at `oo` with eigenvalue `-1/2`; shifting one of them will just swap `1/2` and `-1/2` places, thus leaving the matrix unnormalized. The problem is that current `find_balances` code thinks that this matrix has valid balances. It has not. This is a symptom of a bigger problem: `find_balances` does not follow Lee's algorithm 3. Blacklisting matrices with non-integer eigenvalues will not solve this problem: it'll just make me come up with another matrix where `find_balances` misbehaves.

[magv]

Also note that algorithm 3 can produce a matrix that is normalized in all but one point (which is the case for the matrix posted above). The first paragraph of 6-th section describes how to normalize the remaining point too (this is done by introducing new singularities). I think we want that part implemented too. (This bug report is about a separate problem though).

[magv]

Good job with 7f3250c.

BTW, do you remember reference to Bolibrukh's unnormalizable matrix? It turns out that his matrix too has 1/2 and -1/2 eigenvalues; normalize now refuses to normalize it as well.

[magv]

Correction: Bolibrukh's matrix does not in fact have 1/2 eigenvalues. It's eigenvalues (after fuchsification) are all in (-1, 0, 1). My mistake here was caused by a bug in matrix_import, which is fixed in 325da62.

In any case, normalize refuses to normalize the corrected Bolibrukh's matrix, and I've added a test case for this in 918f74e.

[gituliar]

Concerning non-integer eigenvalues, J.Henn gives an example of such a matrix in [1], eq. (4.13) (you will find this matrix is in the examples dir). As we discussed some time ago, balance method does not work in this case, so Henn suggest a transformation of variables which is not rational (BTW, R.Lee also mentions this solution. but with no explicit examples). Maybe it is worth to make some research in this direction and implement it in Fuchsia at some point, e.g. as experimental feature.

[1] http://arxiv.org/abs/1412.2296