From e4d568e25c62d79a2dbe34b81ee4dc21edf09316 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Tue, 8 Dec 2020 19:11:04 -0500 Subject: [PATCH] eja: eliminate the special element subalgebra class. --- mjo/eja/TODO | 5 +- mjo/eja/eja_algebra.py | 70 ++++++++++++++++++- mjo/eja/eja_element.py | 10 ++- mjo/eja/eja_element_subalgebra.py | 109 ------------------------------ 4 files changed, 76 insertions(+), 118 deletions(-) delete mode 100644 mjo/eja/eja_element_subalgebra.py diff --git a/mjo/eja/TODO b/mjo/eja/TODO index 0e969cd..25ef99d 100644 --- a/mjo/eja/TODO +++ b/mjo/eja/TODO @@ -20,7 +20,4 @@ sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6 6. Profile the construction of "large" matrix algebras (like the 15-dimensional QuaternionHermitianAlgebra(3)) to find out why - they're so slow. - -7. Drop the element-subalgebra in favor of a regular subalgebra. The - cached "one" can be set in the method. + they're so slow. \ No newline at end of file diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 3bc2964..83ec50e 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -750,23 +750,57 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) - EXAMPLES:: + EXAMPLES: + + We can compute unit element in the Hadamard EJA:: + + sage: J = HadamardEJA(5) + sage: J.one() + e0 + e1 + e2 + e3 + e4 + + The unit element in the Hadamard EJA is inherited in the + subalgebras generated by its elements:: sage: J = HadamardEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A.one() + f0 + sage: A.one().superalgebra_element() + e0 + e1 + e2 + e3 + e4 TESTS: - The identity element acts like the identity:: + The identity element acts like the identity, regardless of + whether or not we orthonormalize:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() sage: J.one()*x == x and x*J.one() == x True + sage: A = x.subalgebra_generated_by() + sage: y = A.random_element() + sage: A.one()*y == y and y*A.one() == y + True + + :: + + sage: set_random_seed() + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: x = J.random_element() + sage: J.one()*x == x and x*J.one() == x + True + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: y = A.random_element() + sage: A.one()*y == y and y*A.one() == y + True - The matrix of the unit element's operator is the identity:: + The matrix of the unit element's operator is the identity, + regardless of the base field and whether or not we + orthonormalize:: sage: set_random_seed() sage: J = random_eja() @@ -774,6 +808,27 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): sage: expected = matrix.identity(J.base_ring(), J.dimension()) sage: actual == expected True + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by() + sage: actual = A.one().operator().matrix() + sage: expected = matrix.identity(A.base_ring(), A.dimension()) + sage: actual == expected + True + + :: + + sage: set_random_seed() + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: actual = J.one().operator().matrix() + sage: expected = matrix.identity(J.base_ring(), J.dimension()) + sage: actual == expected + True + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: actual = A.one().operator().matrix() + sage: expected = matrix.identity(A.base_ring(), A.dimension()) + sage: actual == expected + True Ensure that the cached unit element (often precomputed by hand) agrees with the computed one:: @@ -785,6 +840,15 @@ class FiniteDimensionalEJA(CombinatorialFreeModule): sage: J.one() == cached True + :: + + sage: set_random_seed() + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: cached = J.one() + sage: J.one.clear_cache() + sage: J.one() == cached + True + """ # We can brute-force compute the matrices of the operators # that correspond to the basis elements of this algebra. diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index d3e9a33..e30dbb1 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -1411,8 +1411,14 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): True """ - from mjo.eja.eja_element_subalgebra import FiniteDimensionalEJAElementSubalgebra - return FiniteDimensionalEJAElementSubalgebra(self, **kwargs) + from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + powers = tuple( self**k for k in range(self.degree()) ) + A = FiniteDimensionalEJASubalgebra(self.parent(), + powers, + associative=True, + **kwargs) + A.one.set_cache(A(self.parent().one())) + return A def subalgebra_idempotent(self): diff --git a/mjo/eja/eja_element_subalgebra.py b/mjo/eja/eja_element_subalgebra.py deleted file mode 100644 index 34a63af..0000000 --- a/mjo/eja/eja_element_subalgebra.py +++ /dev/null @@ -1,109 +0,0 @@ -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 FiniteDimensionalEJASubalgebra - - -class FiniteDimensionalEJAElementSubalgebra(FiniteDimensionalEJASubalgebra): - def __init__(self, elt, **kwargs): - superalgebra = elt.parent() - - # TODO: going up to the superalgebra dimension here is - # overkill. We should append p vectors as rows to a matrix - # and continually rref() it until the rank stops going - # up. When n=10 but the dimension of the algebra is 1, that - # can save a shitload of time (especially over AA). - powers = tuple( elt**k for k in range(elt.degree()) ) - - super().__init__(superalgebra, - powers, - associative=True, - **kwargs) - - # The rank is the highest possible degree of a minimal - # polynomial, and is bounded above by the dimension. We know - # in this case that there's an element whose minimal - # 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(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 might - have been scaled if we orthonormalized the basis. - - SETUP:: - - sage: from mjo.eja.eja_algebra import (HadamardEJA, - ....: random_eja) - - EXAMPLES:: - - sage: J = HadamardEJA(5) - sage: J.one() - e0 + e1 + e2 + e3 + e4 - sage: x = sum(J.gens()) - sage: A = x.subalgebra_generated_by(orthonormalize=False) - sage: A.one() - f0 - sage: A.one().superalgebra_element() - e0 + e1 + e2 + e3 + e4 - - TESTS: - - The identity element acts like the identity over the rationals:: - - sage: set_random_seed() - 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 - True - - The identity element acts like the identity over the algebraic - reals with an orthonormal basis:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: A = x.subalgebra_generated_by() - sage: x = A.random_element() - sage: A.one()*x == x and x*A.one() == x - True - - The matrix of the unit element's operator is the identity over - the rationals:: - - sage: set_random_seed() - sage: x = random_eja(field=QQ,orthonormalize=False).random_element() - sage: A = x.subalgebra_generated_by(orthonormalize=False) - sage: actual = A.one().operator().matrix() - sage: expected = matrix.identity(A.base_ring(), A.dimension()) - sage: actual == expected - True - - The matrix of the unit element's operator is the identity over - the algebraic reals with an orthonormal basis:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: A = x.subalgebra_generated_by() - sage: actual = A.one().operator().matrix() - sage: expected = matrix.identity(A.base_ring(), A.dimension()) - sage: actual == expected - True - - """ - if self.dimension() == 0: - return self.zero() - - return self(self.superalgebra().one()) - -- 2.44.2