Definition of LU descomposition 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.