X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=29edf5b8a339b073e1426a12a98a6143e7af5069;hb=667e0df9c07589c03616ad8cf42eebe5c86de50b;hp=8f2d8f32b0dd2e2cf6e693166688d6be1f3ea999;hpb=f72c84ce3d46f2611a65417c72e9017754ec156f;p=sage.d.git

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 8f2d8f3..29edf5b 100644
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -1,12 +1,30 @@
+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 _all2list(x):
+    r"""
+    Flatten a vector, matrix, or cartesian product of those things
+    into a long list.
+    """
+    if hasattr(x, 'list'):
+        # Easy case...
+        return x.list()
+    if hasattr(x, 'cartesian_factors'):
+        # If it's a formal cartesian product space element, then
+        # we also know what to do...
+        return sum(( x_i.list() for x_i in x ), [])
+    else:
+        # But what if it's a tuple or something else?
+        return sum( map(_all2list,x), [] )
 
 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
@@ -17,22 +35,65 @@ 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))
         sage: v3 = vector(QQ,(2,1,-1))
         sage: v = [v1,v2,v3]
         sage: u = gram_schmidt(v)
-        sage: [ u_i.inner_product(u_i).sqrt() == 1 for u_i in u ]
+        sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
         True
-        sage: u[0].inner_product(u[1]) == 0
+        sage: bool(u[0].inner_product(u[1]) == 0)
         True
-        sage: u[0].inner_product(u[2]) == 0
+        sage: bool(u[0].inner_product(u[2]) == 0)
         True
-        sage: u[1].inner_product(u[2]) == 0
+        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
+
+    This Gram-Schmidt routine can be used on matrices as well, so long
+    as an appropriate inner-product is provided::
+
+        sage: E11 = matrix(QQ, [ [1,0],
+        ....:                    [0,0] ])
+        sage: E12 = matrix(QQ, [ [0,1],
+        ....:                    [1,0] ])
+        sage: E22 = matrix(QQ, [ [0,0],
+        ....:                    [0,1] ])
+        sage: I = matrix.identity(QQ,2)
+        sage: trace_ip = lambda X,Y: (X*Y).trace()
+        sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
+        [
+        [1 0]  [          0 1/2*sqrt(2)]  [0 0]
+        [0 0], [1/2*sqrt(2)           0], [0 1]
+        ]
+
     TESTS:
 
     Ensure that zero vectors don't get in the way::
@@ -45,8 +106,12 @@ 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
 
@@ -60,36 +125,17 @@ 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() )
 
     # 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() ]
 
-    # Now pretend to normalize, building a new ring R that contains
-    # all of the necessary square roots.
-    norms_squared = [0]*len(v)
-
-    for i in xrange(len(v)):
-        norms_squared[i] = v[i].inner_product(v[i])
-        ns = [norms_squared[i].numerator(), norms_squared[i].denominator()]
-
-        # Do the numerator and denominator separately so that we
-        # adjoin e.g. sqrt(2) and sqrt(3) instead of sqrt(2/3).
-        for j in xrange(len(ns)):
-            PR = PolynomialRing(R, 'z')
-            z = PR.gen()
-            p = z**2 - ns[j]
-            if p.is_irreducible():
-                R = NumberField(p,
-                                'sqrt' + str(ns[j]),
-                                embedding=RLF(ns[j]).sqrt())
-
-    # When we're done, we have to change every element's ring to the
-    # extension that we wound up with, and then normalize it (which
-    # should work, since "R" contains its norm now).
-    for i in xrange(len(v)):
-        v[i] = v[i].change_ring(R) / R(norms_squared[i]).sqrt()
+    # 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] = v[i] / norm(v[i])
 
     return v