Hey Stefan,
From just looking at the function, there is only one case in the for
loop that is non-trivial, so I'm thinking the function could be
restructured as:
cdef bint __exchange(self, long x, long y):
"""
Put element indexed by ``x`` into basis, taking out element ``y``.
Assumptions are that this is a valid basis exchange.
.. NOTE::
Safe for noncommutative rings.
"""
cdef long px, py, r
cdef LeanMatrix A = self._A
px = self._prow[x]
py = self._prow[y]
piv = A.get_unsafe(px, py)
pivi = piv ** (-1)
# This would be a simple modification of the previous, removing the
for loop
#a = pivi + self._one
#A.row_scale(px, pivi)
#A.set_unsafe(px, py, a) # pivoting without column scaling.
Add extra so column does not need adjusting
# if A and A' are the matrices before and after pivoting, then
# ker[I A] equals ker[I A'] except for the labelling of the columns
#if a:
# A.row_add(px, px, -a)
#A.set_unsafe(px, py, pivi)
# This is with some other (possible) (micro) optimizations
if pivi + self._one != 0:
A.row_scale(px, -pivi * pivi)
else:
A.row_scale(px, pivi)
A.set_unsafe(px, py, pivi)
self._prow[y] = px
self._prow[x] = py
BasisExchangeMatroid.__exchange(self, x, y)
For the second one, I use the fact that
pivi * A_ij - (pivi + 1) * pivi * A_ij = -pivi^2 * A_ij
All I've done is unraveled the outcome of the code by following the logic
(interpret that as I haven't tested it), but this should net (again,
provided I didn't make any mistakes) a nice speed increase.
Also, if you're not really using matrices, such as doing a significant
number of multiplications/using category model/etc., but instead as a nice
data structure, I'd be more inclined to say to strip out the non
speed-essential methods, making it as lightweight and specific as possible
and converting to proper matrices when needed, and include it into sage
(after perhaps changing the name). How does this sound to everyone?
Best,
Travis
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