Revert "eja: drop custom gram_schmidt() routine that isn't noticeably faster."

This reverts commit 1e9700cdd04434465ffcad148d078f7fa361e426. We only
bring back the gram_schmidt() function (but don't use it yet) because
the plan is to modify it to take a custom inner-product.

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index cbbf1e3a7bb0f1f66c9820121831d09d76895751..49e3078709ef72084de02050ec57f7f1d84a823e 100644 (file)
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -1,9 +1,81 @@
 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
+from sage.rings.real_lazy import RLF
 
 def _mat2vec(m):
         return vector(m.base_ring(), m.list())
 
 def _vec2mat(v):
         return matrix(v.base_ring(), sqrt(v.degree()), v.list())
+
+def gram_schmidt(v):
+    """
+    Perform Gram-Schmidt on the list ``v`` which are assumed to be
+    vectors over the same base ring. Returns a list of orthonormalized
+    vectors over the smallest extention ring containing the necessary
+    roots.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_utils import gram_schmidt
+
+    EXAMPLES::
+
+        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: u = gram_schmidt(v)
+        sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
+        True
+        sage: bool(u[0].inner_product(u[1]) == 0)
+        True
+        sage: bool(u[0].inner_product(u[2]) == 0)
+        True
+        sage: bool(u[1].inner_product(u[2]) == 0)
+        True
+
+    TESTS:
+
+    Ensure that zero vectors don't get in the way::
+
+        sage: v1 = vector(QQ,(1,2,3))
+        sage: v2 = vector(QQ,(1,-1,6))
+        sage: v3 = vector(QQ,(0,0,0))
+        sage: v = [v1,v2,v3]
+        sage: len(gram_schmidt(v)) == 2
+        True
+
+    """
+    def proj(x,y):
+        return (y.inner_product(x)/x.inner_product(x))*x
+
+    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
+
+    R = v[0].base_ring()
+
+    # First orthogonalize...
+    for i in xrange(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() )
+
+    # 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() ]
+
+    # 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()
+
+    return v