eja: more tests/examples.

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

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 803ec636520515543c873ecc59669475a0048a3c..832dcef1fac0baa573b4883bc4e2ddd3fbfd55a8 100644 (file)
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -2,10 +2,69 @@ from sage.functions.other import sqrt
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
 
+def _scale(x, alpha):
+    r"""
+    Scale the vector, matrix, or cartesian-product-of-those-things
+    ``x`` by ``alpha``.
+
+    This works around the inability to scale certain elements of
+    Cartesian product spaces, as reported in
+
+      https://trac.sagemath.org/ticket/31435
+
+    ..WARNING:
+
+        This will do the wrong thing if you feed it a tuple or list.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_utils import _scale
+
+    EXAMPLES::
+
+        sage: v = vector(QQ, (1,2,3))
+        sage: _scale(v,2)
+        (2, 4, 6)
+        sage: m = matrix(QQ, [[1,2],[3,4]])
+        sage: M = cartesian_product([m.parent(), m.parent()])
+        sage: _scale(M((m,m)), 2)
+        ([2 4]
+        [6 8], [2 4]
+        [6 8])
+
+    """
+    if hasattr(x, 'cartesian_factors'):
+        P = x.parent()
+        return P(tuple( _scale(x_i, alpha)
+                        for x_i in x.cartesian_factors() ))
+    else:
+        return x*alpha
+
+
 def _all2list(x):
     r"""
     Flatten a vector, matrix, or cartesian product of those things
     into a long list.
+
+    EXAMPLES::
+
+        sage: from mjo.eja.eja_utils import _all2list
+        sage: V1 = VectorSpace(QQ,2)
+        sage: V2 = MatrixSpace(QQ,2)
+        sage: x1 = V1([1,1])
+        sage: x2 = V1([1,-1])
+        sage: y1 = V2.one()
+        sage: y2 = V2([0,1,1,0])
+        sage: _all2list((x1,y1))
+        [1, 1, 1, 0, 0, 1]
+        sage: _all2list((x2,y2))
+        [1, -1, 0, 1, 1, 0]
+        sage: M = cartesian_product([V1,V2])
+        sage: _all2list(M((x1,y1)))
+        [1, 1, 1, 0, 0, 1]
+        sage: _all2list(M((x2,y2)))
+        [1, -1, 0, 1, 1, 0]
+
     """
     if hasattr(x, 'list'):
         # Easy case...
@@ -92,6 +151,28 @@ def gram_schmidt(v, inner_product=None):
         [0 0], [1/2*sqrt(2)           0], [0 1]
         ]
 
+    It even works on Cartesian product spaces whose factors are vector
+    or matrix spaces::
+
+        sage: V1 = VectorSpace(AA,2)
+        sage: V2 = MatrixSpace(AA,2)
+        sage: M = cartesian_product([V1,V2])
+        sage: x1 = V1([1,1])
+        sage: x2 = V1([1,-1])
+        sage: y1 = V2.one()
+        sage: y2 = V2([0,1,1,0])
+        sage: z1 = M((x1,y1))
+        sage: z2 = M((x2,y2))
+        sage: def ip(a,b):
+        ....:     return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
+        sage: U = gram_schmidt([z1,z2], inner_product=ip)
+        sage: ip(U[0],U[1])
+        0
+        sage: ip(U[0],U[0])
+        1
+        sage: ip(U[1],U[1])
+        1
+
     TESTS:
 
     Ensure that zero vectors don't get in the way::
@@ -102,7 +183,6 @@ def gram_schmidt(v, inner_product=None):
         sage: v = [v1,v2,v3]
         sage: len(gram_schmidt(v)) == 2
         True
-
     """
     if inner_product is None:
         inner_product = lambda x,y: x.inner_product(y)
@@ -119,18 +199,16 @@ def gram_schmidt(v, inner_product=None):
 
     R = v[0].base_ring()
 
-    # Define a scaling operation that can be used on tuples.
-    # Oh and our "zero" needs to belong to the right space.
-    scale = lambda x,alpha: x*alpha
+    # Our "zero" needs to belong to the right space for sum() to work.
     zero = v[0].parent().zero()
-    if hasattr(v[0], 'cartesian_factors'):
-        P = v[0].parent()
-        scale = lambda x,alpha: P(tuple( x_i*alpha
-                                         for x_i in x.cartesian_factors() ))
 
+    sc = lambda x,a: a*x
+    if hasattr(v[0], 'cartesian_factors'):
+        # Only use the slow implementation if necessary.
+        sc = _scale
 
     def proj(x,y):
-        return scale(x, (inner_product(x,y)/inner_product(x,x)))
+        return sc(x, (inner_product(x,y)/inner_product(x,x)))
 
     # First orthogonalize...
     for i in range(1,len(v)):
@@ -147,6 +225,6 @@ def gram_schmidt(v, inner_product=None):
     # them here because then our subalgebra would have a bigger field
     # than the superalgebra.
     for i in range(len(v)):
-        v[i] = scale(v[i], ~norm(v[i]))
+        v[i] = sc(v[i], ~norm(v[i]))
 
     return v