eja: use fuzzy equality test with inexact base rings.

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

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index f327bf51aada40b33fd05fb0d1c38c124d4b545e..7f7b1f9f1c09703a2d8f48e295b203604fa5556c 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -61,7 +61,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         """
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
+            sage: from mjo.eja.eja_algebra import (
+            ....:   FiniteDimensionalEuclideanJordanAlgebra,
+            ....:   JordanSpinEJA,
+            ....:   random_eja)
 
         EXAMPLES:
 
@@ -75,13 +78,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
 
         TESTS:
 
-        The ``field`` we're given must be real::
+        The ``field`` we're given must be real with ``check=True``::
 
             sage: JordanSpinEJA(2,QQbar)
             Traceback (most recent call last):
             ...
             ValueError: field is not real
 
+        The multiplication table must be square with ``check=True``::
+
+            sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
+            Traceback (most recent call last):
+            ...
+            ValueError: multiplication table is not square
+
         """
         if check:
             if not field.is_subring(RR):
@@ -96,9 +106,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             category = MagmaticAlgebras(field).FiniteDimensional()
             category = category.WithBasis().Unital()
 
+        # The multiplication table had better be square
+        n = len(mult_table)
+        if check:
+            if not all( len(l) == n for l in mult_table ):
+                raise ValueError("multiplication table is not square")
+
         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
         fda.__init__(field,
-                     range(len(mult_table)),
+                     range(n),
                      prefix=prefix,
                      category=category)
         self.print_options(bracket='')
@@ -114,6 +130,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             for ls in mult_table
         ]
 
+        if check:
+            if not self._is_commutative():
+                raise ValueError("algebra is not commutative")
+            if not self._is_jordanian():
+                raise ValueError("Jordan identity does not hold")
+            if not self._inner_product_is_associative():
+                raise ValueError("inner product is not associative")
 
     def _element_constructor_(self, elt):
         """
@@ -235,6 +258,67 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
     def product_on_basis(self, i, j):
         return self._multiplication_table[i][j]
 
+    def _is_commutative(self):
+        r"""
+        Whether or not this algebra's multiplication table is commutative.
+
+        This method should of course always return ``True``, unless
+        this algebra was constructed with ``check=False`` and passed
+        an invalid multiplication table.
+        """
+        return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+                    for i in range(self.dimension())
+                    for j in range(self.dimension()) )
+
+    def _is_jordanian(self):
+        r"""
+        Whether or not this algebra's multiplication table respects the
+        Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+
+        We only check one arrangement of `x` and `y`, so for a
+        ``True`` result to be truly true, you should also check
+        :meth:`_is_commutative`. This method should of course always
+        return ``True``, unless this algebra was constructed with
+        ``check=False`` and passed an invalid multiplication table.
+        """
+        return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
+                    ==
+                    (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+                    for i in range(self.dimension())
+                    for j in range(self.dimension()) )
+
+    def _inner_product_is_associative(self):
+        r"""
+        Return whether or not this algebra's inner product `B` is
+        associative; that is, whether or not `B(xy,z) = B(x,yz)`.
+
+        This method should of course always return ``True``, unless
+        this algebra was constructed with ``check=False`` and passed
+        an invalid multiplication table.
+        """
+
+        # Used to check whether or not something is zero in an inexact
+        # ring. This number is sufficient to allow the construction of
+        # QuaternionHermitianEJA(2, RDF) with check=True.
+        epsilon = 1e-16
+
+        for i in range(self.dimension()):
+            for j in range(self.dimension()):
+                for k in range(self.dimension()):
+                    x = self.monomial(i)
+                    y = self.monomial(j)
+                    z = self.monomial(k)
+                    diff = (x*y).inner_product(z) - x.inner_product(y*z)
+
+                    if self.base_ring().is_exact():
+                        if diff != 0:
+                            return False
+                    else:
+                        if diff.abs() > epsilon:
+                            return False
+
+        return True
+
     @cached_method
     def characteristic_polynomial_of(self):
         """