The promlem seems tolie wit (my use of) polynomial rings to compute the
eigenvectors. Manually, after executing
dims = M.dimensions()
if dims[0] != dims[1]: raise DomainError("Not a square matrix !")
dim = dims[0]
BR = M.base_ring()
try:
WR = BR.algebraic_closure()
except AttributeError:
WR = BR
WPR = WR["t"]
t = WPR.gens()[0]
# CP = WPR(M.charpoly(var="t"))
# Eigenvalues
CP = M.charpoly(var="t")
lambdas = CP.roots(ring=WR)
# Eigenvectors
Id = identity_matrix(WR,dim)
WM = M.apply_map(lambda u:WR(u))
PR = PolynomialRing(WR,list(("v%d"%u for u in range(dim))))
J=PR.ideal(*(WM-Id*lambdas[0][0])*vector(PR.gens()))
neither J.dimension() nor J.variety() returns ; the interruption gives :
sage: J.dimension()
C-c
C-c---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
<ipython-input-684-2e62bddf16b7> in <module>
----> 1 J.dimension()
/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/polynomial/multi_polynomial_ideal.py
in __call__(self, *args, **kwds)
295 if not R.base_ring().is_field():
296 raise ValueError("Coefficient ring must be a field for
function '%s'."%(self.f.__name__))
--> 297 return self.f(self._instance, *args, **kwds)
298
299 require_field = RequireField
/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar_decorators.py
in wrapper(*args, **kwds)
111 # same_field=True might trigger an exception otherwise.
112
--> 113 numfield, new_elems, morphism =
number_field_elements_from_algebraics(orig_elems, same_field=True, minimal=True)
114
115 elem_dict = dict(zip(orig_elems, new_elems))
/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
in number_field_elements_from_algebraics(numbers, minimal, same_field,
embedded, prec)
2759 real_numbers = []
2760 for v in numbers:
-> 2761 if v._exact_field().is_complex() and real_case:
2762 # the number comes from a complex algebraic number field
2763 embedded_rt = v.interval_fast(RealIntervalField(prec))
/usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
in _exact_field(self)
4428 if isinstance(sd, (ANRational, ANExtensionElement)):
4429 return sd.generator()
-> 4430 self.exactify()
4431 return self._exact_field()
4432
/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)
8180
8181 if op == 'real':
-> 8182 arg.exactify()
8183 rv = (arg + arg.conjugate()) / 2
8184 rv.exactify()
/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()
KeyboardInterrupt:
Suggestions ?
Le samedi 29 janvier 2022 à 15:34:05 UTC+1, Emmanuel Charpentier a écrit :
> Setup : Sage 9.5.rc1 running in Debian testing on core i7 + 16 GB RAM.
>
> def test(Size=2, Ring=QQ):
> from time import time as stime
> with seed(0):
> M = matrix(Ring, Size, Size, lambda u, v:Ring.random_element())
> t0 = stime()
> SL = M.eigenspaces_left(algebraic_multiplicity=True)
> t1 = stime()
> VL = M.eigenvectors_left()
> t2 = stime()
> return t1-t0, SL, t2-t1, VL
>
> test() runs as expected, as well as test(Size=5) and test(Ring=AA), with
> very reasonable runtimes (under a second, IIRC). But test(Size=3, Ring=AA)
> “never returns” (meaning that it hadn’t returned when I interrupted it
> after tens of minutes). Further exploration showec that neither
> M.eigenspaces_left(algebraic_multiplicity=True) nor M.eigenvectors_left()
> return.
>
> Is that an expected behaviour ?
>
> FWIW, stack trace at interruption :
>
> sage: AA3=test(Size=3, Ring=AA)
> AA3=test(Size=3, Ring=AA) 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: 'sage.rings.real_mpfi.RealIntervalFieldElement' object has no
> attribute '_evaluate_polynomial'
>
> During handling of the above exception, another exception occurred:
>
> KeyboardInterrupt Traceback (most recent call last)
> <ipython-input-344-ff7c46010ad2> in <module>
> ----> 1 AA3=test(Size=Integer(3), Ring=AA)
>
> <string> in test(Size, Ring)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix.eigenspaces_left
> (build/cythonized/sage/matrix/matrix2.c:43588)()
> 6270 self = self.change_ring(F)
> 6271 A = self - alpha
> -> 6272 W = A.kernel()
> 6273 V.append((alpha,
> W.ambient_module().span_of_basis(W.basis()), e))
> 6274 else:
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix.left_kernel
> (build/cythonized/sage/matrix/matrix2.c:32440)()
> 4947
> 4948 tm = verbose("computing left kernel for %sx%s matrix" %
> (self.nrows(), self.ncols()),level=1)
> -> 4949 K = self.transpose().right_kernel(*args, **kwds)
> 4950 self.cache('left_kernel', K)
> 4951 verbose("done computing left kernel for %sx%s matrix" %
> (self.nrows(), self.ncols()),level=1,t=tm)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix.right_kernel
> (build/cythonized/sage/matrix/matrix2.c:31909)()
> 4785
> 4786 # Go get the kernel matrix, this is where it all happens
> -> 4787 M = self.right_kernel_matrix(*args, **kwds)
> 4788
> 4789 ambient = R**self.ncols()
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix.right_kernel_matrix
> (build/cythonized/sage/matrix/matrix2.c:30829)()
> 4394
> 4395 if M is None and R in _Fields:
> -> 4396 format, M = self._right_kernel_matrix_over_field()
> 4397
> 4398 if M is None and R.is_integral_domain():
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix._right_kernel_matrix_over_field
> (build/cythonized/sage/matrix/matrix2.c:27969)()
> 3686 from sage.matrix.matrix_space import MatrixSpace
> 3687 tm = verbose("computing right kernel matrix over an arbitrary
> field for %sx%s matrix" % (self.nrows(), self.ncols()),level=1)
> -> 3688 E = self.echelon_form(*args, **kwds)
> 3689 pivots = E.pivots()
> 3690 pivots_set = set(pivots)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix.echelon_form
> (build/cythonized/sage/matrix/matrix2.c:52440)()
> 7686 E = self.__copy__()
> 7687 if algorithm == 'default':
> -> 7688 v = E.echelonize(cutoff=cutoff, **kwds)
> 7689 else:
> 7690 v = E.echelonize(algorithm = algorithm, cutoff=cutoff,
> **kwds)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix.echelonize
> (build/cythonized/sage/matrix/matrix2.c:51817)()
> 7583 if self.base_ring() in _Fields:
> 7584 if algorithm in ['classical', 'partial_pivoting',
> 'scaled_partial_pivoting', 'scaled_partial_pivoting_valuation']:
> -> 7585 self._echelon_in_place(algorithm)
> 7586 elif algorithm == 'strassen':
> 7587 self._echelon_strassen(cutoff)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/matrix/matrix2.pyx
> in sage.matrix.matrix2.Matrix._echelon_in_place
> (build/cythonized/sage/matrix/matrix2.c:53308)()
> 7848 sig_check()
> 7849 for r in range(start_row, nr):
> -> 7850 if A.get_unsafe(r, c):
> 7851 pivots.append(c)
> 7852 a_inverse = ~A.get_unsafe(r, c)
>
> /usr/local/sage-9/local/var/lib/sage/venv-python3.9/lib/python3.9/site-packages/sage/rings/qqbar.py
> in __bool__(self)
> 4043 right = sd._right if op is operator.sub else -sd._right
> 4044
> -> 4045 lp = left.minpoly()
> 4046 rp = right.minpoly()
> 4047 if lp != rp:
>
> /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)
> 8430 right = self._right
> 8431 left.exactify()
> -> 8432 right.exactify()
> 8433 gen = left._exact_field().union(right._exact_field())
> 8434 left_value = gen(left._exact_value())
>
> /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)
> 8430 right = self._right
> 8431 left.exactify()
> -> 8432 right.exactify()
> 8433 gen = left._exact_field().union(right._exact_field())
> 8434 left_value = gen(left._exact_value())
>
> /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)
> 8430 right = self._right
> 8431 left.exactify()
> -> 8432 right.exactify()
> 8433 gen = left._exact_field().union(right._exact_field())
> 8434 left_value = gen(left._exact_value())
>
> /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)
> 8176
> 8177 if op == '~':
> -> 8178 arg.exactify()
> 8179 return arg._descr.invert(None)
> 8180
>
> /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()
> KeyboardInterrupt:
>
> Any hint appreciated…
>
>
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