On Fri, Apr 11, 2008 at 8:00 PM, Alex Ghitza [EMAIL PROTECTED] wrote:
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Hi,
There are some inconsistencies in linear algebra over fields like CDF.
Some trouble was already reported in #2256. The following is another issue:
sage: M = matrix(CDF, 2, 2, [(-1 - 2*I, 5 - 6*I), (-2 - 4*I, 10 - 12*I)])
sage: M.is_invertible()
True
sage: M.determinant()
5.3290705182e-15 + 1.7763568394e-15*I
sage: M.inverse()
[ 1.01330991616e+15 - 2.58956978574e+15*I -5.06654958079e+14 +
1.29478489287e+15*I]
[ 5.62949953421e+14 + 5.62949953421e+14*I -2.81474976711e+14 -
2.81474976711e+14*I]
So because of roundoff errors, Sage thinks that we have an invertible
matrix. But the code for echelon_form knows that it's not invertible:
sage: M.echelon_form()
[1.0 1.4
+ 3.2*I]
[-2.22044604925e-16 - 4.4408920985e-16*I
~ 0]
sage: M.rank()
1
This behavior is inconsistent. Either there is enough precision to
decide that the matrix is invertible, or there isn't. Doing this in two
different ways should not yield two mutually exclusive answers.
Similar problems occur over CC.
I hope it's clear by now that I have no idea how CDF and friends
actually work, so I don't know what would be a good solution or even
whether such a solution should exist. My initial guess would be that
finding the determinant requires more computations so there is more
precision loss than in computing the echelon form; in that case, maybe
over such fields we should not base is_invertible() and inverse() on
determinant() but rather on echelon_form().
This is now #2932: http://trac.sagemath.org/sage_trac/ticket/2932:
Just looking at the value of M.rank() should be enough, I think. Also,
M.rank() seems much faster than computing the determinant:
{{{
sage: M = matrix(CDF, 2, 2, [(-1 - 2*I, 5 - 6*I), (-2 - 4*I, 10 - 12*I)])
sage: %timeit M.rank()
100 loops, best of 3: 676 ns per loop
sage: %timeit M.det()
10 loops, best of 3: 2.81 µs per loop
}}}
Timings are comparable over ZZ,QQ,RR.
didier
Note also that this problem does not seem to occur over RR or RDF.
Best,
Alex
- --
Alexandru Ghitza
Assistant Professor
Department of Mathematics
Colby College
Waterville, ME 04901
http://bayes.colby.edu/~ghitza/
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