X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=4d70e062c44e98ba90584fb0f97d3bbfef91e491;hb=3c7644ecfe369b6f83aa707b87d7a1f9aa246e27;hp=cf75e325697dcefb3bf682b855f8d83e3e4f89e2;hpb=e529e0e2775cf50207c7d01d5907214d03cdff5c;p=sage.d.git

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index cf75e32..4d70e06 100644
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -1,3 +1,5 @@
+from sage.functions.other import sqrt
+from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
 from sage.rings.number_field.number_field import NumberField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -6,7 +8,10 @@ from sage.rings.real_lazy import RLF
 def _mat2vec(m):
         return vector(m.base_ring(), m.list())
 
-def gram_schmidt(v):
+def _vec2mat(v):
+        return matrix(v.base_ring(), sqrt(v.degree()), v.list())
+
+def gram_schmidt(v, inner_product=None):
     """
     Perform Gram-Schmidt on the list ``v`` which are assumed to be
     vectors over the same base ring. Returns a list of orthonormalized
@@ -17,7 +22,9 @@ def gram_schmidt(v):
 
         sage: from mjo.eja.eja_utils import gram_schmidt
 
-    EXAMPLES::
+    EXAMPLES:
+
+    The usual inner-product and norm are default::
 
         sage: v1 = vector(QQ,(1,2,3))
         sage: v2 = vector(QQ,(1,-1,6))
@@ -33,6 +40,30 @@ def gram_schmidt(v):
         sage: bool(u[1].inner_product(u[2]) == 0)
         True
 
+
+    But if you supply a custom inner product, the result is
+    orthonormal with respect to that (and not the usual inner
+    product)::
+
+        sage: v1 = vector(QQ,(1,2,3))
+        sage: v2 = vector(QQ,(1,-1,6))
+        sage: v3 = vector(QQ,(2,1,-1))
+        sage: v = [v1,v2,v3]
+        sage: B = matrix(QQ, [ [6, 4, 2],
+        ....:                  [4, 5, 4],
+        ....:                  [2, 4, 9] ])
+        sage: ip = lambda x,y: (B*x).inner_product(y)
+        sage: norm = lambda x: ip(x,x)
+        sage: u = gram_schmidt(v,ip)
+        sage: all( norm(u_i) == 1 for u_i in u )
+        True
+        sage: ip(u[0],u[1]).is_zero()
+        True
+        sage: ip(u[0],u[2]).is_zero()
+        True
+        sage: ip(u[1],u[2]).is_zero()
+        True
+
     TESTS:
 
     Ensure that zero vectors don't get in the way::
@@ -45,8 +76,12 @@ def gram_schmidt(v):
         True
 
     """
+    if inner_product is None:
+        inner_product = lambda x,y: x.inner_product(y)
+    norm = lambda x: inner_product(x,x).sqrt()
+
     def proj(x,y):
-        return (y.inner_product(x)/x.inner_product(x))*x
+        return (inner_product(x,y)/inner_product(x,x))*x
 
     v = list(v) # make a copy, don't clobber the input
 
@@ -60,7 +95,7 @@ def gram_schmidt(v):
     R = v[0].base_ring()
 
     # First orthogonalize...
-    for i in xrange(1,len(v)):
+    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() )
 
@@ -70,7 +105,7 @@ def gram_schmidt(v):
     # 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 xrange(len(v)):
-        v[i] = v[i] / v[i].norm()
+    for i in range(len(v)):
+        v[i] = v[i] / norm(v[i])
 
     return v