X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element_subalgebra.py;h=dceb3b405a4c5a663c61a966993ce890ed516b49;hb=4c8f9aac69d1cb4097b60b10e5b198b6372ec55e;hp=c058613e1b650a3c3007ad8700d1c36d0b2567c9;hpb=f9690e43873907af4da7a9ccd6d74c6937b7cdf8;p=sage.d.git diff --git a/mjo/eja/eja_element_subalgebra.py b/mjo/eja/eja_element_subalgebra.py index c058613..dceb3b4 100644 --- a/mjo/eja/eja_element_subalgebra.py +++ b/mjo/eja/eja_element_subalgebra.py @@ -1,53 +1,50 @@ 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 @@ -55,39 +52,10 @@ 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. - 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. @@ -95,17 +63,17 @@ 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:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES:: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 sage: x = sum(J.gens()) @@ -120,7 +88,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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 @@ -130,7 +98,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide reals with an orthonormal basis:: sage: set_random_seed() - sage: x = random_eja(AA).random_element() + sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: x = A.random_element() sage: A.one()*x == x and x*A.one() == x @@ -140,7 +108,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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()) @@ -151,7 +119,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide the algebraic reals with an orthonormal basis:: sage: set_random_seed() - sage: x = random_eja(AA).random_element() + sage: x = random_eja().random_element() sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) sage: actual = A.one().operator().matrix() sage: expected = matrix.identity(A.base_ring(), A.dimension()) @@ -161,7 +129,6 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide """ 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()) +