X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element_subalgebra.py;h=dceb3b405a4c5a663c61a966993ce890ed516b49;hb=4c8f9aac69d1cb4097b60b10e5b198b6372ec55e;hp=2a82940f51c47c44aea0efdc47a202c243319ba6;hpb=d6d7353ccdcd1ab58c5f0e4621cb114e4a8c65ed;p=sage.d.git diff --git a/mjo/eja/eja_element_subalgebra.py b/mjo/eja/eja_element_subalgebra.py index 2a82940..dceb3b4 100644 --- a/mjo/eja/eja_element_subalgebra.py +++ b/mjo/eja/eja_element_subalgebra.py @@ -1,181 +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_algebra import FiniteDimensionalEuclideanJordanAlgebra -from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement +from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra -class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): - """ - SETUP:: - sage: from mjo.eja.eja_algebra import random_eja +class FiniteDimensionalEJAElementSubalgebra(FiniteDimensionalEJASubalgebra): + def __init__(self, elt, orthonormalize=True, **kwargs): + superalgebra = elt.parent() - TESTS:: + 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) - The natural representation of an element in the subalgebra is - the same as its natural representation in the superalgebra:: - - sage: set_random_seed() - sage: A = random_eja().random_element().subalgebra_generated_by() - sage: y = A.random_element() - sage: actual = y.natural_representation() - sage: expected = y.superalgebra_element().natural_representation() - sage: actual == expected - True - - The left-multiplication-by operator for elements in the subalgebra - works like it does in the superalgebra, even if we orthonormalize - our basis:: - - sage: set_random_seed() - sage: x = random_eja(AA).random_element() - sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) - sage: y = A.random_element() - sage: y.operator()(A.one()) == y - True - - """ - - def superalgebra_element(self): - """ - Return the object in our algebra's superalgebra that corresponds - to myself. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, - ....: random_eja) - - EXAMPLES:: - - sage: J = RealSymmetricEJA(3) - sage: x = sum(J.gens()) - sage: x - e0 + e1 + e2 + e3 + e4 + e5 - sage: A = x.subalgebra_generated_by() - sage: A(x) - f1 - sage: A(x).superalgebra_element() - e0 + e1 + e2 + e3 + e4 + e5 - - TESTS: - - We can convert back and forth faithfully:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: A = x.subalgebra_generated_by() - sage: A(x).superalgebra_element() == x - True - sage: y = A.random_element() - sage: A(y.superalgebra_element()) == y - True - - """ - return self.parent().superalgebra().linear_combination( - zip(self.parent()._superalgebra_basis, self.to_vector()) ) - - - - -class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): - """ - The subalgebra of an EJA generated by a single element. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, - ....: JordanSpinEJA) - - TESTS: - - Ensure that our generator names don't conflict with the superalgebra:: - - sage: J = JordanSpinEJA(3) - sage: J.one().subalgebra_generated_by().gens() - (f0,) - sage: J = JordanSpinEJA(3, prefix='f') - sage: J.one().subalgebra_generated_by().gens() - (g0,) - sage: J = JordanSpinEJA(3, prefix='b') - sage: J.one().subalgebra_generated_by().gens() - (c0,) - - Ensure that we can find subalgebras of subalgebras:: - - sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by() - sage: B = A.one().subalgebra_generated_by() - sage: B.dimension() - 1 - - """ - 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() - - # A half-assed attempt to ensure that we don't collide with - # the superalgebra's prefix (ignoring the fact that there - # could be super-superelgrbas in scope). If possible, we - # try to "increment" the parent algebra's prefix, although - # this idea goes out the window fast because some prefixen - # are off-limits. - prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ] - try: - prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] - except ValueError: - prefix = prefixen[0] - - # 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) - - 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 ) - n = len(superalgebra_basis) - mult_table = [[W.zero() for i in range(n)] for j in range(n)] - for i in range(n): - for j in range(n): - product = superalgebra_basis[i]*superalgebra_basis[j] - # product.to_vector() might live in a vector subspace - # if our parent algebra is already a subalgebra. We - # use V.from_vector() to make it "the right size" in - # that case. - product_vector = V.from_vector(product.to_vector()) - mult_table[i][j] = W.coordinate_vector(product_vector) + # 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 @@ -183,83 +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() - - natural_basis = tuple( b.natural_representation() - for b in superalgebra_basis ) - - - self._vector_space = W - self._superalgebra_basis = superalgebra_basis - - - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) - - - 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) - - - def _element_constructor_(self, elt): - """ - Construct an element of this subalgebra from the given one. - The only valid arguments are elements of the parent algebra - that happen to live in this subalgebra. - - SETUP:: - - sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra - - EXAMPLES:: - - sage: J = RealSymmetricEJA(3) - sage: x = sum( i*J.gens()[i] for i in range(6) ) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False) - sage: [ K(x^k) for k in range(J.rank()) ] - [f0, f1, f2] - - :: - - """ - if elt == 0: - # Just as in the superalgebra class, we need to hack - # this special case to ensure that random_element() can - # coerce a ring zero into the algebra. - return self.zero() - - if elt in self.superalgebra(): - coords = self.vector_space().coordinate_vector(elt.to_vector()) - return self.from_vector(coords) - + self.rank.set_cache(self.dimension()) + @cached_method def one(self): """ Return the multiplicative identity element of this algebra. @@ -267,17 +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()) @@ -292,7 +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 @@ -302,7 +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 @@ -312,7 +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()) @@ -323,7 +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()) @@ -333,58 +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) - - - def natural_basis_space(self): - """ - Return the natural basis space of this algebra, which is identical - to that of its superalgebra. - - This is correct "by definition," and avoids a mismatch when the - subalgebra is trivial (with no natural basis to infer anything - from) and the parent is not. - """ - return self.superalgebra().natural_basis_space() - - - def superalgebra(self): - """ - Return the superalgebra that this algebra was generated from. - """ - return self._superalgebra - - - def vector_space(self): - """ - SETUP:: - - sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra - - EXAMPLES:: - - sage: J = RealSymmetricEJA(3) - sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False) - sage: K.vector_space() - Vector space of degree 6 and dimension 3 over... - User basis matrix: - [ 1 0 1 0 0 1] - [ 1 0 2 0 0 5] - [ 1 0 4 0 0 25] - sage: (x^0).to_vector() - (1, 0, 1, 0, 0, 1) - sage: (x^1).to_vector() - (1, 0, 2, 0, 0, 5) - sage: (x^2).to_vector() - (1, 0, 4, 0, 0, 25) - - """ - return self._vector_space + return self(self.superalgebra().one()) - Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement