eja: fix an implicit TODO by eliminating lazy_import.

[sage.d.git] / mjo / eja / eja_element.py

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index da4b12335cdffbf046c6ce100b88deea9b997458..120870b7fa08b1ee0fcb936b92c62a9470a27754 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -1,14 +1,7 @@
-# -*- coding: utf-8 -*-
-

 from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 
 from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 

-# TODO: make this unnecessary somehow.
-from sage.misc.lazy_import import lazy_import
-lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
-lazy_import('mjo.eja.eja_element_subalgebra',
-            'FiniteDimensionalEuclideanJordanElementSubalgebra')

 from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
 from mjo.eja.eja_utils import _mat2vec
 
 from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
 from mjo.eja.eja_utils import _mat2vec
 


@@ -183,7 +176,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         """
             True
 
         """

-        p = self.parent().characteristic_polynomial()
+        p = self.parent().characteristic_polynomial_of()

         return p(*self.to_vector())
 
 
         return p(*self.to_vector())
 
 


@@ -346,6 +339,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         SETUP::
 
             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
         SETUP::
 
             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,

+            ....:                                  TrivialEJA,

             ....:                                  random_eja)
 
         EXAMPLES::
             ....:                                  random_eja)
 
         EXAMPLES::


@@ -364,6 +358,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x.det()
             -1
 
             sage: x.det()
             -1
 

+        The determinant of the sole element in the rank-zero trivial
+        algebra is ``1``, by three paths of reasoning. First, its
+        characteristic polynomial is a constant ``1``, so the constant
+        term in that polynomial is ``1``. Second, the characteristic
+        polynomial evaluated at zero is again ``1``. And finally, the
+        (empty) product of its eigenvalues is likewise just unity::
+
+            sage: J = TrivialEJA()
+            sage: J.zero().det()
+            1
+

         TESTS:
 
         An element is invertible if and only if its determinant is
         TESTS:
 
         An element is invertible if and only if its determinant is


@@ -382,15 +387,21 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x,y = J.random_elements(2)
             sage: (x*y).det() == x.det()*y.det()
             True
             sage: x,y = J.random_elements(2)
             sage: (x*y).det() == x.det()*y.det()
             True

-

         """
         P = self.parent()
         r = P.rank()
         """
         P = self.parent()
         r = P.rank()

-        p = P._charpoly_coeff(0)
-        # The _charpoly_coeff function already adds the factor of
-        # -1 to ensure that _charpoly_coeff(0) is really what
-        # appears in front of t^{0} in the charpoly. However,
-        # we want (-1)^r times THAT for the determinant.
+
+        if r == 0:
+            # Special case, since we don't get the a0=1
+            # coefficient when the rank of the algebra
+            # is zero.
+            return P.base_ring().one()
+
+        p = P._charpoly_coefficients()[0]
+        # The _charpoly_coeff function already adds the factor of -1
+        # to ensure that _charpoly_coefficients()[0] is really what
+        # appears in front of t^{0} in the charpoly. However, we want
+        # (-1)^r times THAT for the determinant.

         return ((-1)**r)*p(*self.to_vector())
 
 
         return ((-1)**r)*p(*self.to_vector())
 
 


@@ -420,11 +431,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: while not x.is_invertible():
             ....:     x = J.random_element()
             sage: x_vec = x.to_vector()
             sage: while not x.is_invertible():
             ....:     x = J.random_element()
             sage: x_vec = x.to_vector()

-            sage: x0 = x_vec[0]
+            sage: x0 = x_vec[:1]

             sage: x_bar = x_vec[1:]
             sage: x_bar = x_vec[1:]

-            sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
-            sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
-            sage: x_inverse = coeff*inv_vec
+            sage: coeff = x0.inner_product(x0) - x_bar.inner_product(x_bar)
+            sage: x_inverse = x_vec.parent()(x0.list() + (-x_bar).list())
+            sage: if not coeff.is_zero(): x_inverse = x_inverse/coeff

             sage: x.inverse() == J.from_vector(x_inverse)
             True
 
             sage: x.inverse() == J.from_vector(x_inverse)
             True
 


@@ -491,6 +502,14 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         if not self.is_invertible():
             raise ValueError("element is not invertible")
 
         if not self.is_invertible():
             raise ValueError("element is not invertible")
 

+        if self.parent()._charpoly_coefficients.is_in_cache():
+            # We can invert using our charpoly if it will be fast to
+            # compute. If the coefficients are cached, our rank had
+            # better be too!
+            r = self.parent().rank()
+            a = self.characteristic_polynomial().coefficients(sparse=False)
+            return (-1)**(r+1)*sum(a[i+1]*self**i for i in range(r))/self.det()
+

         return (~self.quadratic_representation())(self)
 
 
         return (~self.quadratic_representation())(self)
 
 


@@ -507,6 +526,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         whether or not the paren't algebra's zero element is a root
         of this element's minimal polynomial.
 
         whether or not the paren't algebra's zero element is a root
         of this element's minimal polynomial.
 

+        That is... unless the coefficients of our algebra's
+        "characteristic polynomial of" function are already cached!
+        In that case, we just use the determinant (which will be fast
+        as a result).
+

         Beware that we can't use the superclass method, because it
         relies on the algebra being associative.
 
         Beware that we can't use the superclass method, because it
         relies on the algebra being associative.
 


@@ -537,6 +561,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             else:
                 return False
 
             else:
                 return False
 

+        if self.parent()._charpoly_coefficients.is_in_cache():
+            # The determinant will be quicker than computing the minimal
+            # polynomial from scratch, most likely.
+            return (not self.det().is_zero())
+

         # In fact, we only need to know if the constant term is non-zero,
         # so we can pass in the field's zero element instead.
         zero = self.base_ring().zero()
         # In fact, we only need to know if the constant term is non-zero,
         # so we can pass in the field's zero element instead.
         zero = self.base_ring().zero()


@@ -778,8 +807,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = JordanSpinEJA.random_instance()
 
             sage: set_random_seed()
             sage: J = JordanSpinEJA.random_instance()

+            sage: n = J.dimension()

             sage: x = J.random_element()
             sage: x = J.random_element()

-            sage: x == x.coefficient(0)*J.one() or x.degree() == 2
+            sage: x.degree() == min(n,2) or (x == x.coefficient(0)*J.one())

             True
 
         TESTS:
             True
 
         TESTS:


@@ -857,13 +887,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         TESTS:
 
         The minimal polynomial of the identity and zero elements are
         TESTS:
 
         The minimal polynomial of the identity and zero elements are

-        always the same::
+        always the same, except in trivial algebras where the minimal
+        polynomial of the unit/zero element is ``1``::

 
             sage: set_random_seed()
 
             sage: set_random_seed()

-            sage: J = random_eja(nontrivial=True)
-            sage: J.one().minimal_polynomial()
+            sage: J = random_eja()
+            sage: mu = J.one().minimal_polynomial()
+            sage: t = mu.parent().gen()
+            sage: mu + int(J.is_trivial())*(t-2)

             t - 1
             t - 1

-            sage: J.zero().minimal_polynomial()
+            sage: mu = J.zero().minimal_polynomial()
+            sage: t = mu.parent().gen()
+            sage: mu + int(J.is_trivial())*(t-1)

             t
 
         The degree of an element is (by one definition) the degree
             t
 
         The degree of an element is (by one definition) the degree


@@ -881,7 +916,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         two here so that said elements actually exist::
 
             sage: set_random_seed()
         two here so that said elements actually exist::
 
             sage: set_random_seed()

-            sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
+            sage: n_max = max(2, JordanSpinEJA._max_random_instance_size())

             sage: n = ZZ.random_element(2, n_max)
             sage: J = JordanSpinEJA(n)
             sage: y = J.random_element()
             sage: n = ZZ.random_element(2, n_max)
             sage: J = JordanSpinEJA(n)
             sage: y = J.random_element()


@@ -907,7 +942,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         and in particular, a re-scaling of the basis::
 
             sage: set_random_seed()
         and in particular, a re-scaling of the basis::
 
             sage: set_random_seed()

-            sage: n_max = RealSymmetricEJA._max_test_case_size()
+            sage: n_max = RealSymmetricEJA._max_random_instance_size()

             sage: n = ZZ.random_element(1, n_max)
             sage: J1 = RealSymmetricEJA(n)
             sage: J2 = RealSymmetricEJA(n,normalize_basis=False)
             sage: n = ZZ.random_element(1, n_max)
             sage: J1 = RealSymmetricEJA(n)
             sage: J2 = RealSymmetricEJA(n,normalize_basis=False)


@@ -931,7 +966,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
                 # in the "normal" case without us having to think about it.
                 return self.operator().minimal_polynomial()
 
                 # in the "normal" case without us having to think about it.
                 return self.operator().minimal_polynomial()
 

-        A = self.subalgebra_generated_by()
+        A = self.subalgebra_generated_by(orthonormalize_basis=False)

         return A(self).operator().minimal_polynomial()
 
 
         return A(self).operator().minimal_polynomial()
 
 


@@ -986,6 +1021,10 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         """
         B = self.parent().natural_basis()
         W = self.parent().natural_basis_space()
         """
         B = self.parent().natural_basis()
         W = self.parent().natural_basis_space()

+
+        # This is just a manual "from_vector()", but of course
+        # matrix spaces aren't vector spaces in sage, so they
+        # don't have a from_vector() method.

         return W.linear_combination(zip(B,self.to_vector()))
 
 
         return W.linear_combination(zip(B,self.to_vector()))
 
 


@@ -1062,16 +1101,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: set_random_seed()
             sage: x = JordanSpinEJA.random_instance().random_element()
             sage: x_vec = x.to_vector()
             sage: set_random_seed()
             sage: x = JordanSpinEJA.random_instance().random_element()
             sage: x_vec = x.to_vector()

+            sage: Q = matrix.identity(x.base_ring(), 0)

             sage: n = x_vec.degree()
             sage: n = x_vec.degree()

-            sage: x0 = x_vec[0]
-            sage: x_bar = x_vec[1:]
-            sage: A = matrix(AA, 1, [x_vec.inner_product(x_vec)])
-            sage: B = 2*x0*x_bar.row()
-            sage: C = 2*x0*x_bar.column()
-            sage: D = matrix.identity(AA, n-1)
-            sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
-            sage: D = D + 2*x_bar.tensor_product(x_bar)
-            sage: Q = matrix.block(2,2,[A,B,C,D])
+            sage: if n > 0:
+            ....:     x0 = x_vec[0]
+            ....:     x_bar = x_vec[1:]
+            ....:     A = matrix(x.base_ring(), 1, [x_vec.inner_product(x_vec)])
+            ....:     B = 2*x0*x_bar.row()
+            ....:     C = 2*x0*x_bar.column()
+            ....:     D = matrix.identity(x.base_ring(), n-1)
+            ....:     D = (x0^2 - x_bar.inner_product(x_bar))*D
+            ....:     D = D + 2*x_bar.tensor_product(x_bar)
+            ....:     Q = matrix.block(2,2,[A,B,C,D])

             sage: Q == x.quadratic_representation().matrix()
             True
 
             sage: Q == x.quadratic_representation().matrix()
             True
 


@@ -1229,7 +1270,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: (J0, J5, J1) = J.peirce_decomposition(c1)
             sage: (f0, f1, f2) = J1.gens()
             sage: f0.spectral_decomposition()
             sage: (J0, J5, J1) = J.peirce_decomposition(c1)
             sage: (f0, f1, f2) = J1.gens()
             sage: f0.spectral_decomposition()

-            [(0, 1.000000000000000?*f2), (1, 1.000000000000000?*f0)]
+            [(0, f2), (1, f0)]

 
         """
         A = self.subalgebra_generated_by(orthonormalize_basis=True)
 
         """
         A = self.subalgebra_generated_by(orthonormalize_basis=True)


@@ -1285,6 +1326,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         """
             True
 
         """

+        from mjo.eja.eja_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra

         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
 
 
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
 
 


@@ -1300,12 +1342,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         TESTS:
 
         Ensure that we can find an idempotent in a non-trivial algebra
         TESTS:
 
         Ensure that we can find an idempotent in a non-trivial algebra

-        where there are non-nilpotent elements::
+        where there are non-nilpotent elements, or that we get the dumb
+        solution in the trivial algebra::

 
             sage: set_random_seed()
 
             sage: set_random_seed()

-            sage: J = random_eja(nontrivial=True)
+            sage: J = random_eja()

             sage: x = J.random_element()
             sage: x = J.random_element()

-            sage: while x.is_nilpotent():
+            sage: while x.is_nilpotent() and not J.is_trivial():

             ....:     x = J.random_element()
             sage: c = x.subalgebra_idempotent()
             sage: c^2 == c
             ....:     x = J.random_element()
             sage: c = x.subalgebra_idempotent()
             sage: c^2 == c


@@ -1400,7 +1443,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             # the trace is an empty sum.
             return P.base_ring().zero()
 
             # the trace is an empty sum.
             return P.base_ring().zero()
 

-        p = P._charpoly_coeff(r-1)
+        p = P._charpoly_coefficients()[r-1]

         # The _charpoly_coeff function already adds the factor of
         # -1 to ensure that _charpoly_coeff(r-1) is really what
         # appears in front of t^{r-1} in the charpoly. However,
         # The _charpoly_coeff function already adds the factor of
         # -1 to ensure that _charpoly_coeff(r-1) is really what
         # appears in front of t^{r-1} in the charpoly. However,