What is v0? When I run the above it isn't defined.
On Monday, January 31, 2022 at 11:19:49 PM UTC Emmanuel Charpentier wrote:
> As advertised, an atempt at a minimal (non-)working example :
>
> # Reproducible minimal example
> with seed(0): M = matrix(AA, 3, 3, lambda u,v: AA.random_element())
> # Working ring
> WR = M.base_ring().algebraic_closure()
> # A variable to carry the eigenvalues
> l = SR.var("l")
> # Vector of unknowns for the eigenvectors
> V =vector(list(var("v", n=2))+[SR(1)])
> # M.eigenvalues does not return. Get them by hand
> # Equation determining the eigenspace
> ESE = M-l*identity_matrix(WR, 3)
> # Eigenvalues
> EVa = ESE.det().roots(ring=WR)
> # M.eigenvectors_left() doesn't return.
> # Set up a system of equations whose roots define the eigenvectors.
> Sys = ESE*V
> # Solve for each eigenvalue
> # solve([u.subs(l==EVa[0][0]) for u in Sys],list(V[:2])) fails.
> # Try to solve manually by substitution
> # First equation, first variable.
> # solve(Sys[0].subs(l==EVa[0][0]), v0) fails.
> # Try to solve by explicit algebraic manipulation
> E00 = Sys[0].subs(l==EVa[0][0])
> # Isolate v0
> V00=E0/E00.coefficient(v0) # fast
>
> At this point, tryoing to proint V00‘s value “never returns”. However, a
> forst interruption (C-C) prints :
>
> sage: V00
> C-c
> C-c---------------------------------------------------------------------------
> AttributeError Traceback (most recent call last)
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in minpoly(self)
> 4492 try:
> -> 4493 return self._minimal_polynomial
> 4494 except AttributeError:
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/structure/element.pyx
> in sage.structure.element.Element.__getattr__
> (build/cythonized/sage/structure/element.c:4754)()
> 493 """
> --> 494 return self.getattr_from_category(name)
> 495
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/structure/element.pyx
> in sage.structure.element.Element.getattr_from_category
> (build/cythonized/sage/structure/element.c:4866)()
> 506 cls = P._abstract_element_class
> --> 507 return getattr_from_other_class(self, cls, name)
> 508
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/cpython/getattr.pyx
> in sage.cpython.getattr.getattr_from_other_class
> (build/cythonized/sage/cpython/getattr.c:2633)()
> 360 dummy_error_message.name = name
> --> 361 raise AttributeError(dummy_error_message)
> 362 attribute = <object>attr
>
> AttributeError: 'AlgebraicNumber' object has no attribute
> '_minimal_polynomial'
>
> During handling of the above exception, another exception occurred:
>
> KeyboardInterrupt Traceback (most recent call last)
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/structure/element.pyx
> in sage.structure.element.Element.__richcmp__
> (build/cythonized/sage/structure/element.c:10418)()
> 1110 return (<Element>self)._richcmp_(other, op)
> 1111 else:
> -> 1112 return coercion_model.richcmp(self, other, op)
> 1113
> 1114 cpdef _richcmp_(left, right, int op):
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/structure/coerce.pyx
> in sage.structure.coerce.CoercionModel.richcmp
> (build/cythonized/sage/structure/coerce.c:20625)()
> 1979 assert not (isinstance(x, Element) and
> 1980
> (<Element>x)._parent.get_flag(Parent_richcmp_element_without_coercion))
> -> 1981 return PyObject_RichCompare(x, y, op)
> 1982
> 1983 # Comparing with coercion didn't work, try something else.
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/structure/element.pyx
> in sage.structure.element.Element.__richcmp__
> (build/cythonized/sage/structure/element.c:10394)()
> 1108 # an instance of Element. The explicit casts below make
> 1109 # Cython generate optimized code for this call.
> -> 1110 return (<Element>self)._richcmp_(other, op)
> 1111 else:
> 1112 return coercion_model.richcmp(self, other, op)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/structure/element.pyx
> in sage.structure.element.Element._richcmp_
> (build/cythonized/sage/structure/element.c:10500)()
> 1112 return coercion_model.richcmp(self, other, op)
> 1113
> -> 1114 cpdef _richcmp_(left, right, int op):
> 1115 r"""
> 1116 Basic default implementation of rich comparisons for elements
> with
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in _richcmp_(self, other, op)
> 4897 ci1 = self._value.imag().abs()
> 4898 ci2 = other._value.imag().abs()
> -> 4899 if ci1.overlaps(ci2) and self.minpoly() == other.minpoly():
> 4900 c = cmp_elements_with_same_minpoly(self, other,
> self.minpoly())
> 4901 if c is not None:
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in minpoly(self)
> 4493 return self._minimal_polynomial
> 4494 except AttributeError:
> -> 4495 self.exactify()
> 4496 self._minimal_polynomial = self._descr.minpoly()
> 4497 return self._minimal_polynomial
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in exactify(self)
> 4363 if isinstance(od, (ANRational, ANExtensionElement)):
> 4364 return
> -> 4365 self._set_descr(self._descr.exactify())
> 4366
> 4367 def _set_descr(self, new_descr):
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in exactify(self)
> 8429 left = self._left
> 8430 right = self._right
> -> 8431 left.exactify()
> 8432 right.exactify()
> 8433 gen = left._exact_field().union(right._exact_field())
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in exactify(self)
> 4363 if isinstance(od, (ANRational, ANExtensionElement)):
> 4364 return
> -> 4365 self._set_descr(self._descr.exactify())
> 4366
> 4367 def _set_descr(self, new_descr):
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in exactify(self)
> 8429 left = self._left
> 8430 right = self._right
> -> 8431 left.exactify()
> 8432 right.exactify()
> 8433 gen = left._exact_field().union(right._exact_field())
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in exactify(self)
> 4363 if isinstance(od, (ANRational, ANExtensionElement)):
> 4364 return
> -> 4365 self._set_descr(self._descr.exactify())
> 4366
> 4367 def _set_descr(self, new_descr):
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in exactify(self)
> 8431 left.exactify()
> 8432 right.exactify()
> -> 8433 gen = left._exact_field().union(right._exact_field())
> 8434 left_value = gen(left._exact_value())
> 8435 right_value = gen(right._exact_value())
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in union(self, other)
> 3230 op = QQx(op)
> 3231 # pari_nf = self._field.pari_nf()
> -> 3232 pari_nf = self.pari_field()
> 3233 factors = list(pari_nf.nffactor(op).lift())[0]
> 3234 x, y = QQxy.gens()
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in pari_field(self)
> 3134 if self._pari_field is None:
> 3135 pari_pol = self._field.pari_polynomial("y")
> -> 3136 self._pari_field = pari_pol.nfinit(1)
> 3137 return self._pari_field
> 3138
>
> cypari2/auto_gen.pxi in cypari2.gen.Gen_base.nfinit()
>
> and a *second* interrupt prints :
>
> KeyboardInterrupt:
> Exception ignored in: 'sage.symbolic.expression.py_is_integer'
> Traceback (most recent call last):
> File "sage/structure/parent.pyx", line 1163, in
> sage.structure.parent.Parent.__contains__
> (build/cythonized/sage/structure/parent.c:10408)
> File "sage/structure/parent.pyx", line 898, in
> sage.structure.parent.Parent.__call__
> (build/cythonized/sage/structure/parent.c:9388)
> File "sage/structure/coerce_maps.pyx", line 287, in
> sage.structure.coerce_maps.NamedConvertMap._call_
> (build/cythonized/sage/structure/coerce_maps.c:6085)
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 5011, in _integer_
> return AA(self)._integer_(ZZ)
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 5485, in _integer_
> self.exactify()
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 4365, in exactify
> self._set_descr(self._descr.exactify())
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 8182, in exactify
> arg.exactify()
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 4365, in exactify
> self._set_descr(self._descr.exactify())
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 8431, in exactify
> left.exactify()
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 4365, in exactify
> self._set_descr(self._descr.exactify())
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 8431, in exactify
> left.exactify()
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 4365, in exactify
> self._set_descr(self._descr.exactify())
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 8433, in exactify
> gen = left._exact_field().union(right._exact_field())
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 3232, in union
> pari_nf = self.pari_field()
> File
> "/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py",
> line 3136, in pari_field
> self._pari_field = pari_pol.nfinit(1)
> File "cypari2/auto_gen.pxi", line 23641, in cypari2.gen.Gen_base.nfinit
> KeyboardInterrupt:
> 1.000000000000000?*v0 - 35.57125011095806?*v1 - 492.8896998473554?
>
> Similarly, V00.coefficient(v0).is_one() “never returns”.
> The current aritmetic on algebraics is therefore problematic for this kind
> of problems.
> Suggestions ?
>
> Le lundi 31 janvier 2022 à 17:59:34 UTC+1, Emmanuel Charpentier a écrit :
>
>> Thank you, Nils !
>> Trying to pinpoint things by running a manual solution step by step, it
>> seems that the stumbling oint is the division on two algebraics. I'll try
>> to make a clear minimal example and report it.
>> Le lundi 31 janvier 2022 à 00:49:22 UTC+1, Nils Bruin a écrit :
>>
>>> On Saturday, 29 January 2022 at 13:51:14 UTC-8 Emmanuel Charpentier
>>> wrote:
>>>
>>>> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
>>>>
>>>> in pari_field(self)
>>>>
>>>>> 3134 if self._pari_field is None:
>>>>> 3135 pari_pol = self._field.pari_polynomial("y")
>>>>> -> 3136 self._pari_field = pari_pol.nfinit(1)
>>>>> 3137 return self._pari_field
>>>>> 3138
>>>>>
>>>>> cypari2/auto_gen.pxi in cypari2.gen.Gen_base.nfinit()
>>>>> KeyboardInterrupt:
>>>>>
>>>>>
>>>>> nf_init is a perfectly respectable place to hang. After interruption,
>>> can you "%debug" and see what the value of pari_pol is? I'd expect that
>>> nfinit determines the ring of integers, which means factoring the
>>> discriminant. [QQbar shouldn't need the ring of integers of element
>>> arithmetic, but this has been stumbled on before: QQbar often tries to find
>>> an "optimized" form of the number field, which for small examples is often
>>> quite doable and, if you end up doing a LOT of arithmetic in the same
>>> field, is often worth the investment. However, because of factoring of the
>>> discriminant, it's fully subexponential in complexity, whereas all the
>>> things QQbar needs to do are polynomial time. So we already know that QQbar
>>> will do things with the wrong theoretical asymptotic complexity. That can
>>> come back and bite you, when reality and asymptotics start behaving
>>> similarly (which happens eventually ... asymptotically speakinh)
>>>
>>
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