eja: rename operator_inner_product -> operator_trace inner_product.

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

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 8422fbff3c3a3f1523a84708ee659bd605da7ffe..a8abeff6be073b2b0aea3dd4f5af933c251666a5 100644 (file)
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -244,53 +244,39 @@ 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)
-    def norm(x):
-        ip = inner_product(x,x)
-        # Don't expand the given field; the inner-product's codomain
-        # is already correct. For example QQ(2).sqrt() returns sqrt(2)
-        # in SR, and that will give you weird errors about symbolics
-        # when what's really going wrong is that you're trying to
-        # orthonormalize in QQ.
-        return ip.parent()(ip.sqrt())
-
-    v = list(v) # make a copy, don't clobber the input
-
-    # Drop all zero vectors before we start.
-    v = [ v_i for v_i in v if not v_i.is_zero() ]
 
+    """
     if len(v) == 0:
         # cool
         return v
 
-    # Our "zero" needs to belong to the right space for sum() to work.
-    zero = v[0].parent().zero()
+    V = v[0].parent()
+
+    if inner_product is None:
+        inner_product = lambda x,y: x.inner_product(y)
 
     sc = lambda x,a: a*x
-    if hasattr(v[0], 'cartesian_factors'):
+    if hasattr(V, 'cartesian_factors'):
         # Only use the slow implementation if necessary.
         sc = _scale
 
     def proj(x,y):
+        # project y onto the span of {x}
         return sc(x, (inner_product(x,y)/inner_product(x,x)))
 
-    # First orthogonalize...
-    for i in range(1,len(v)):
-        # Earlier vectors can be made into zero so we have to ignore them.
-        v[i] -= sum( (proj(v[j],v[i])
-                      for j in range(i)
-                      if not v[j].is_zero() ),
-                     zero )
+    def normalize(x):
+        # Don't extend the given field with the necessary
+        # square roots. This will probably throw weird
+        # errors about the symbolic ring if you e.g. try
+        # to use it on a set of rational vectors that isn't
+        # already orthonormalized.
+        return sc(x, ~inner_product(x,x).sqrt())
 
-    # And now drop all zero vectors again if they were "orthogonalized out."
-    v = [ v_i for v_i in v if not v_i.is_zero() ]
+    v_out = [] # make a copy, don't clobber the input
 
-    # Just normalize. If the algebra is missing the roots, we can't add
-    # them here because then our subalgebra would have a bigger field
-    # than the superalgebra.
-    for i in range(len(v)):
-        v[i] = sc(v[i], ~norm(v[i]))
+    for (i, v_i) in enumerate(v):
+        ortho_v_i = v_i - V.sum( proj(v_out[j],v_i) for j in range(i) )
+        if not ortho_v_i.is_zero():
+            v_out.append(normalize(ortho_v_i))
 
-    return v
+    return v_out