from sage.matrix.constructor import matrix
+from sage.misc.cachefunc import cached_method
+from sage.rings.all import QQ
-from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
-class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra):
- def __init__(self, elt, orthonormalize_basis):
- self._superalgebra = elt.parent()
- category = self._superalgebra.category().Associative()
- V = self._superalgebra.vector_space()
- field = self._superalgebra.base_ring()
+class FiniteDimensionalEJAElementSubalgebra(FiniteDimensionalEJASubalgebra):
+ def __init__(self, elt, orthonormalize=True, **kwargs):
+ superalgebra = elt.parent()
- # This list is guaranteed to contain all independent powers,
- # because it's the maximal set of powers that could possibly
- # be independent (by a dimension argument).
- powers = [ elt**k for k in range(V.dimension()) ]
- power_vectors = [ p.to_vector() for p in powers ]
- P = matrix(field, power_vectors)
+ powers = tuple( elt**k for k in range(superalgebra.dimension()) )
+ power_vectors = ( p.to_vector() for p in powers )
+ P = matrix(superalgebra.base_ring(), power_vectors)
- if orthonormalize_basis == False:
- # In this case, we just need to figure out which elements
- # of the "powers" list are redundant... First compute the
- # vector subspace spanned by the powers of the given
- # element.
-
- # Figure out which powers form a linearly-independent set.
- ind_rows = P.pivot_rows()
-
- # Pick those out of the list of all powers.
- superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
-
- # If our superalgebra is a subalgebra of something else, then
- # these vectors won't have the right coordinates for
- # V.span_of_basis() unless we use V.from_vector() on them.
- basis_vectors = map(power_vectors.__getitem__, ind_rows)
+ if orthonormalize:
+ basis = powers # let god sort 'em out
else:
- # If we're going to orthonormalize the basis anyway, we
- # might as well just do Gram-Schmidt on the whole list of
- # powers. The redundant ones will get zero'd out. If this
- # looks like a roundabout way to orthonormalize, it is.
- # But converting everything from algebra elements to vectors
- # to matrices and then back again turns out to be about
- # as fast as reimplementing our own Gram-Schmidt that
- # works in an EJA.
- G,_ = P.gram_schmidt(orthonormal=True)
- basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
- superalgebra_basis = [ self._superalgebra.from_vector(b)
- for b in basis_vectors ]
-
- W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
+ # Echelonize the matrix ourselves, because otherwise the
+ # call to P.pivot_rows() below can choose a non-optimal
+ # row-reduction algorithm. In particular, scaling can
+ # help over AA because it avoids the RecursionError that
+ # gets thrown when we have to look too hard for a root.
+ #
+ # Beware: QQ supports an entirely different set of "algorithm"
+ # keywords than do AA and RR.
+ algo = None
+ if superalgebra.base_ring() is not QQ:
+ algo = "scaled_partial_pivoting"
+ P.echelonize(algorithm=algo)
+
+ # In this case, we just need to figure out which elements
+ # of the "powers" list are redundant... First compute the
+ # vector subspace spanned by the powers of the given
+ # element.
+
+ # Figure out which powers form a linearly-independent set.
+ ind_rows = P.pivot_rows()
+
+ # Pick those out of the list of all powers.
+ basis = tuple(map(powers.__getitem__, ind_rows))
+
+
+ super().__init__(superalgebra,
+ basis,
+ associative=True,
+ **kwargs)
# The rank is the highest possible degree of a minimal
# polynomial, and is bounded above by the dimension. We know
# polynomial has the same degree as the space's dimension
# (remember how we constructed the space?), so that must be
# its rank too.
- rank = W.dimension()
-
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
- return fdeja.__init__(self._superalgebra,
- superalgebra_basis,
- rank=rank,
- category=category)
-
-
- def _a_regular_element(self):
- """
- Override the superalgebra method to return the one
- regular element that is sure to exist in this
- subalgebra, namely the element that generated it.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import random_eja
-
- TESTS::
-
- sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: J._a_regular_element().is_regular()
- True
-
- """
- if self.dimension() == 0:
- return self.zero()
- else:
- return self.monomial(1)
+ self.rank.set_cache(self.dimension())
+ @cached_method
def one(self):
"""
Return the multiplicative identity element of this algebra.
The superclass method computes the identity element, which is
beyond overkill in this case: the superalgebra identity
restricted to this algebra is its identity. Note that we can't
- count on the first basis element being the identity -- it migth
+ count on the first basis element being the identity -- it might
have been scaled if we orthonormalized the basis.
SETUP::
The identity element acts like the identity over the rationals::
sage: set_random_seed()
- sage: x = random_eja().random_element()
+ sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
sage: A = x.subalgebra_generated_by()
sage: x = A.random_element()
sage: A.one()*x == x and x*A.one() == x
the rationals::
sage: set_random_seed()
- sage: x = random_eja().random_element()
+ sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
sage: A = x.subalgebra_generated_by()
sage: actual = A.one().operator().matrix()
sage: expected = matrix.identity(A.base_ring(), A.dimension())
"""
if self.dimension() == 0:
return self.zero()
- else:
- sa_one = self.superalgebra().one().to_vector()
- sa_coords = self.vector_space().coordinate_vector(sa_one)
- return self.from_vector(sa_coords)
+
+ return self(self.superalgebra().one())
+
projects / sage.d.git / blobdiff