Well, applying a naive exact linear algebra routine to inexact data,
and that's what Sage is doing here, is prone to errors.
sage: A=m.augment(identity_matrix(RR,2))
sage: A
[-6.12323399573677e-17 -1.72508242466029 1.00000000000000
0.000000000000000]
[ 0.579682446302195 6.12323399573677e-17 0.000000000000000
1.00000000000000]
sage: A.echelonize(algorithm="classical");A # OOOPS! - that's the default here
[ 1.00000000000000 0.000000000000000 4.00000000000000
1.72508242466029]
[ 0.000000000000000 1.00000000000000 -0.579682446302195
-6.12323399573676e-17]
sage: A=m.augment(identity_matrix(RR,2))
sage: A.echelonize(algorithm='scaled_partial_pivoting');A # that's how
it should be
[ 1.00000000000000 0.000000000000000 6.12323399573677e-17
1.72508242466029]
[ 0.000000000000000 1.00000000000000 -0.579682446302195
-6.12323399573677e-17]
On Sat, Nov 5, 2022 at 10:21 AM Emmanuel Charpentier
<emanuel.charpent...@gmail.com> wrote:
>
> something is definitely unhinged here : On 9.8.beta3 running on Debian
> testing on core i7 + 16 GB RAM, after running :
>
> a = RR(-4967757600021511 / 2**106)
> b = RR(-7769080564883485 / 2**52)
> c = RR( 5221315298319565 / 2**53)
> m = matrix([[a, b], [c, -a]])
> M = matrix([[var("p%d%d"%(u, v), latex_name="p_{%s,%d}"%(u, v))
> for v in range(2)]
> for u in range(2)])
> S = dict(zip(M.list(), [a, b, c, -a]))
> MN = M.apply_map(lambda u:u.subs(S))
>
> one gets :
>
> sage: m.parent()
> Full MatrixSpace of 2 by 2 dense matrices over Real Field with 53 bits of
> precision
> sage: m*~m
> [ 1.00000000000000 -1.23259516440783e-32]
> [ 2.31872978520878 1.00000000000000]
> sage: (~m)*m
> [ 1.00000000000000 -6.90032969864117]
> [6.16297582203915e-33 1.00000000000000]
> sage: MN.parent()
> Full MatrixSpace of 2 by 2 dense matrices over Symbolic Ring
> sage: MN*~MN
> [ 1.00000000000000 -1.23259516440783e-32]
> [ 2.31872978520878 1.00000000000000]
> sage: (~MN)*MN
> [ 1.00000000000000 -6.90032969864117]
> [6.16297582203915e-33 1.00000000000000]
>
> all being wrong, wrong, wrong…
>
> However :
>
> sage: (M*~M).apply_map(lambda u:u.subs(S))
> [ 1.00000000000000 0]
> [-3.54953126192945e-17 1.00000000000000]
> sage: ((~M)*M).apply_map(lambda u:u.subs(S))
> [ 1.00000000000000 1.05630833481279e-16]
> [ 0 1.00000000000000]
>
> both being acceptable.
>
> One also notes that the form of :
>
> sage: ~M
> [1/p00 - p01*p10/(p00^2*(p01*p10/p00 - p11))
> p01/(p00*(p01*p10/p00 - p11))]
> [ p10/(p00*(p01*p10/p00 - p11))
> -1/(p01*p10/p00 - p11)]
> sage: (~M).apply_map(simplify)
> [1/p00 - p01*p10/(p00^2*(p01*p10/p00 - p11))
> p01/(p00*(p01*p10/p00 - p11))]
> [ p10/(p00*(p01*p10/p00 - p11))
> -1/(p01*p10/p00 - p11)]
>
> is somewhat unexpected ; one expects :
>
> sage: (~M).apply_map(lambda u:u.simplify_full())
> [-p11/(p01*p10 - p00*p11) p01/(p01*p10 - p00*p11)]
> [ p10/(p01*p10 - p00*p11) -p00/(p01*p10 - p00*p11)]
>
> which is also the form returned by maxima :
>
> sage: maxima_calculus.invert(M).sage()
> [-p11/(p01*p10 - p00*p11) p01/(p01*p10 - p00*p11)]
> [ p10/(p01*p10 - p00*p11) -p00/(p01*p10 - p00*p11)]
>
> giac :
>
> sage: giac.inverse(giac(M)).sage()
> [[-p11/(p01*p10 - p00*p11), p01/(p01*p10 - p00*p11)],
> [p10/(p01*p10 - p00*p11), -p00/(p01*p10 - p00*p11)]]
>
> fricas :
>
> sage: fricas.inverse(M._fricas_()).sage()
> [-p11/(p01*p10 - p00*p11) p01/(p01*p10 - p00*p11)]
> [ p10/(p01*p10 - p00*p11) -p00/(p01*p10 - p00*p11)]
>
> mathematica :
>
> sage: mathematica.Inverse(M).sage()
> [[-p11/(p01*p10 - p00*p11), p01/(p01*p10 - p00*p11)],
> [p10/(p01*p10 - p00*p11), -p00/(p01*p10 - p00*p11)]]
>
> and (somewhat un-backconvertible) :
>
> sage: sympy.sympify(M)^-1._sage_()
> Matrix([
> [ p11/(p00*p11 - p01*p10), -p01/(p00*p11 - p01*p10)],
> [-p10/(p00*p11 - p01*p10), p00/(p00*p11 - p01*p10)]])
>
> This is, IMNSHO, a critical bug. Could you open a tichet for this, and mark
> it as such ?
>
> Le samedi 5 novembre 2022 à 07:59:27 UTC+1, Håkan Granath a écrit :
>>
>> Hi, there seems to be a problem with inverses of matrices with elements in
>> RR. It only occurs very sporadically for me, but here is an example:
>>
>> a = RR(-4967757600021511 / 2**106)
>> b = RR(-7769080564883485 / 2**52)
>> c = RR( 5221315298319565 / 2**53)
>>
>> m = matrix([[a, b], [c, -a]])
>>
>> print(m)
>> print()
>> print(~m)
>>
>> On my machines it produces the output
>>
>> [-6.12323399573677e-17 -1.72508242466029]
>> [ 0.579682446302195 6.12323399573677e-17]
>>
>> [ 4.00000000000000 1.72508242466029]
>> [ -0.579682446302195 -6.12323399573676e-17]
>>
>> Clearly the element 4 is wrong (the correct inverse is -m). Is this a known
>> bug?
>>
>> Some system information:
>>
>> SageMath version 9.7, using Python 3.10.5
>> OS: Ubuntu 20.04.5 LTS
>> CPU: Intel(R) Core(TM) i7-7700 CPU @ 3.60GHz
>>
>> Best regards,
>>
>> Håkan Granath
>
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