Definition of LU decomposition of a matrix depends on the base ring

[tmonteil]

As reported on ​this ask question, there is an inconsistency in the definition of the LU decomposition of a matrix depending on its base ring. If a matrix A belongs to ZZ, QQ, AA, QQbar, A.LU() returns a triple (P,L,U) such that A=PLU. If a matrix A belongs to RDF, A.LU() returns a triple (P,L,U) such that PA=LU. For example:

sage: A = random_matrix(ZZ,4)
sage: A.LU()
(
[0 1 0 0]  [    1     0     0     0]  [   72    -4   -38     0]
[0 0 1 0]  [    0     1     0     0]  [    0    -1    -2    10]
[1 0 0 0]  [ 1/36   8/9     1     0]  [    0     0  17/6 -80/9]
[0 0 0 1], [-1/36   1/9    -1     1], [    0     0     0   -17]
)
sage: B = A.change_ring(RDF)
sage: B.LU()
(
[0.0 0.0 1.0 0.0]
[1.0 0.0 0.0 0.0]
[0.0 1.0 0.0 0.0]
[0.0 0.0 0.0 1.0],

[             1.0              0.0              0.0              0.0]
[             0.0              1.0              0.0              0.0]
[ 0.0277777777778   0.888888888889              1.0              0.0]
[-0.0277777777778   0.111111111111             -1.0              1.0],

[          72.0           -4.0          -38.0            0.0]
[           0.0           -1.0           -2.0           10.0]
[           0.0            0.0  2.83333333333 -8.88888888889]
[           0.0            0.0            0.0          -17.0]
)


sage: B.LU()[0] == A.LU()[0]
False
sage: B.LU()[0] == A.LU()[0].transpose()
True
sage: 

The aim of this ticket is to fix this inconsistency and choose a common definition for all rings.

[tmonteil]

It is worth noticing that the source code for matrices over RDF calls scipy.linalg.lu which returns a triple (P,L,U) such that A=PLU and then transpose P with the documentation:

# Numpy has a different convention than we had with GSL
# So we invert (transpose) the P to match our prior behavior
# TODO: It's an awful waste to store a huge matrix for P, which
# is just a simple permutation, really.

So, i guess this extra transposition should be reverted, so that it becomes consistent with both scipy and the implementation for exact rings.

[kcrisman]

Probably in principle the old behavior should be deprecated ... but the inconsistency is annoying, agreed.

[gh-ChamanAgrawal]

New commits:

​2512540

Reverted transpose to maintain consistency with scipy, numpy

[gh-ChamanAgrawal]

I have removed to the extra copy of the matrix and also reverted the transposing step to maintain consistent behavior with scipy and also with LU() in matrices of other class in sagemath.

[SimonKing]

Replying to kcrisman:

Probably in principle the old behavior should be deprecated ... but the inconsistency is annoying, agreed.

The proposed change does not change the syntax of using the .LU() method, but changes the semantic of the output. How would it be possible to deprecate it?

Do you say that if the .LU() method is used for a matrix over RDF then a deprecation warning should be used? That warning would appear also in the case that the user expects the new (not the old) behaviour, i.e., when a warning is inappropriate.

[kcrisman]

The proposed change does not change the syntax of using the .LU() method, but changes the semantic of the output. How would it be possible to deprecate it?

Good point. Perhaps we need to just put a visible note in the documentation that might help clarify if someone is perplexed by the output changing - I guess that was my point, that it is a "silent" change. But they should agree over all rings, yes.

[gh-ChamanAgrawal]

Replying to kcrisman:

The proposed change does not change the syntax of using the .LU() method, but changes the semantic of the output. How would it be possible to deprecate it?

Good point. Perhaps we need to just put a visible note in the documentation that might help clarify if someone is perplexed by the output changing - I guess that was my point, that it is a "silent" change. But they should agree over all rings, yes.

I think along with a note in the documentation LU() should also give a warning on using for this behavior change, what do you think?

[SimonKing]

Replying to gh-ChamanAgrawal:

I think along with a note in the documentation LU() should also give a warning on using for this behavior change, what do you think?

I don't think so. A warning (in the sense of: "If M.LU() is called then a warning is printed") is disturbing, because the mere fact of calling .LU() is not a problem. What is a potential problem is whether or not the user expects the output in a particular form. And we cannot guess on the expectation of the user. Hence, certainly it should be written in the documentation that there has been a change, give an example for which that change happens, and insert a link to this ticket. The syntax for such link in the docstrings is

See :trac:`18365`.

But there should be no warning inserted into the code.

[git]

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

​6fdae81

Added Note for behaviour change

[gh-ChamanAgrawal]

I have added a note for the change as below, any suggestion for change?

.. NOTE::
            The behaviour of ``LU()`` has been changed. Earlier ``LU()`` 
            returned ``P,L,U`` such that ``P*A=L*U``, where ``P`` represents
            the permutation and is the matrix inverse of the ``P`` returned
            by this method. The computation of this matrix inverse can be accomplished
            quickly with just a transpose as the matrix is orthogonal/unitary.
                
            For Details See :trac:`18365`.

        EXAMPLES::

            sage: m = matrix(RDF,4,range(16))
            sage: P,L,U = m.LU()
            sage: P*L*U # rel tol 2e-16
            [ 0.0  1.0  2.0  3.0]
            [ 4.0  5.0  6.0  7.0]
            [ 8.0  9.0 10.0 11.0]
            [12.0 13.0 14.0 15.0]
            
        Below example illustrate the change in behaviour of ``LU()``. ::

            sage: m == P*L*U
            True
            sage: P*m == L*U
            False

[gh-mwageringel]

I suggest that this gets merged now, despite the backwards incompatibility.