X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element_subalgebra.py;h=7edf1df940fb3e16435eb7d232fea0ceb58eff7a;hb=3c7644ecfe369b6f83aa707b87d7a1f9aa246e27;hp=a4d7d1f3de4c43347b43451be15cc38b5e1c9556;hpb=58c6ee3de5d4ee8f8349011c789132d7800b34e9;p=sage.d.git diff --git a/mjo/eja/eja_element_subalgebra.py b/mjo/eja/eja_element_subalgebra.py index a4d7d1f..7edf1df 100644 --- a/mjo/eja/eja_element_subalgebra.py +++ b/mjo/eja/eja_element_subalgebra.py @@ -1,4 +1,6 @@ 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 @@ -18,6 +20,19 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide P = matrix(field, power_vectors) if orthonormalize_basis == False: + # 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 field 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 @@ -28,11 +43,6 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # 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) else: # If we're going to orthonormalize the basis anyway, we # might as well just do Gram-Schmidt on the whole list of @@ -47,13 +57,11 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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 ) - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) fdeja.__init__(self._superalgebra, superalgebra_basis, category=category, - check=False) + check_axioms=False) # The rank is the highest possible degree of a minimal # polynomial, and is bounded above by the dimension. We know @@ -61,33 +69,10 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # polynomial has the same degree as the space's dimension # (remember how we constructed the space?), so that must be # its rank too. - self.rank.set_cache(W.dimension()) - - - 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. @@ -95,7 +80,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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:: @@ -120,7 +105,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide The identity element acts like the identity over the rationals:: sage: set_random_seed() - sage: x = random_eja(field=QQ).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 @@ -140,7 +125,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide the rationals:: sage: set_random_seed() - sage: x = random_eja(field=QQ).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()) @@ -161,12 +146,6 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide """ if self.dimension() == 0: return self.zero() - else: - sa_one = self.superalgebra().one().to_vector() - # The extra hackery is because foo.to_vector() might not - # live in foo.parent().vector_space()! - coords = sum( a*b for (a,b) - in zip(sa_one, - self.superalgebra().vector_space().basis()) ) - return self.from_vector(self.vector_space().coordinate_vector(coords)) + + return self(self.superalgebra().one())