eja: make random_element() more random.

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

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 9aa40eeacdacbb54c792158b1462d2f36d3dc4d6..91088a2eb799ce616bfffc5ee3e3980afe17f4ac 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -581,6 +581,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             Vector space of degree 6 and dimension 2...
             sage: J1
             Euclidean Jordan algebra of dimension 3...
+            sage: J0.one().natural_representation()
+            [0 0 0]
+            [0 0 0]
+            [0 0 1]
+            sage: orig_df = AA.options.display_format
+            sage: AA.options.display_format = 'radical'
+            sage: J.from_vector(J5.basis()[0]).natural_representation()
+            [          0           0 1/2*sqrt(2)]
+            [          0           0           0]
+            [1/2*sqrt(2)           0           0]
+            sage: J.from_vector(J5.basis()[1]).natural_representation()
+            [          0           0           0]
+            [          0           0 1/2*sqrt(2)]
+            [          0 1/2*sqrt(2)           0]
+            sage: AA.options.display_format = orig_df
+            sage: J1.one().natural_representation()
+            [1 0 0]
+            [0 1 0]
+            [0 0 0]
 
         TESTS:
 
@@ -595,9 +614,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
             True
 
-        The identity elements in the two subalgebras are the
-        projections onto their respective subspaces of the
-        superalgebra's identity element::
+        The decomposition is into eigenspaces, and its components are
+        therefore necessarily orthogonal. Moreover, the identity
+        elements in the two subalgebras are the projections onto their
+        respective subspaces of the superalgebra's identity element::
 
             sage: set_random_seed()
             sage: J = random_eja()
@@ -607,6 +627,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             ....:         x = J.random_element()
             sage: c = x.subalgebra_idempotent()
             sage: J0,J5,J1 = J.peirce_decomposition(c)
+            sage: ipsum = 0
+            sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
+            ....:     w = w.superalgebra_element()
+            ....:     y = J.from_vector(y)
+            ....:     z = z.superalgebra_element()
+            ....:     ipsum += w.inner_product(y).abs()
+            ....:     ipsum += w.inner_product(z).abs()
+            ....:     ipsum += y.inner_product(z).abs()
+            sage: ipsum
+            0
             sage: J1(c) == J1.one()
             True
             sage: J0(J.one() - c) == J0.one()
@@ -642,10 +672,57 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         return (J0, J5, J1)
 
 
-    def random_elements(self, count):
+    def random_element(self, thorough=False):
+        r"""
+        Return a random element of this algebra.
+
+        Our algebra superclass method only returns a linear
+        combination of at most two basis elements. We instead
+        want the vector space "random element" method that
+        returns a more diverse selection.
+
+        INPUT:
+
+        - ``thorough`` -- (boolean; default False) whether or not we
+          should generate irrational coefficients for the random
+          element when our base ring is irrational; this slows the
+          algebra operations to a crawl, but any truly random method
+          should include them
+
+        """
+        # For a general base ring... maybe we can trust this to do the
+        # right thing? Unlikely, but.
+        V = self.vector_space()
+        v = V.random_element()
+
+        if self.base_ring() is AA:
+            # The "random element" method of the algebraic reals is
+            # stupid at the moment, and only returns integers between
+            # -2 and 2, inclusive. Instead, we implement our own
+            # "random vector" method, and then coerce that into the
+            # algebra. We use the vector space degree here instead of
+            # the dimension because a subalgebra could (for example) be
+            # spanned by only two vectors, each with five coordinates.
+            # We need to generate all five coordinates.
+            if thorough:
+                v *= QQbar.random_element().real()
+            else:
+                v *= QQ.random_element()
+
+        return self.from_vector(V.coordinate_vector(v))
+
+    def random_elements(self, count, thorough=False):
         """
         Return ``count`` random elements as a tuple.
 
+        INPUT:
+
+        - ``thorough`` -- (boolean; default False) whether or not we
+          should generate irrational coefficients for the random
+          elements when our base ring is irrational; this slows the
+          algebra operations to a crawl, but any truly random method
+          should include them
+
         SETUP::
 
             sage: from mjo.eja.eja_algebra import JordanSpinEJA
@@ -660,7 +737,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             True
 
         """
-        return tuple( self.random_element() for idx in range(count) )
+        return tuple( self.random_element(thorough)
+                      for idx in range(count) )
 
     @classmethod
     def random_instance(cls, field=AA, **kwargs):