X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=d4e9990ecc6749057905d3b1d5ac700bd34cdc71;hb=8f3ba1d9473195c1cbafe967a127800d90fde3ee;hp=4d70e062c44e98ba90584fb0f97d3bbfef91e491;hpb=3c1c9170143a6412b050602cd79a383e3da6c821;p=sage.d.git

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 4d70e06..d4e9990 100644
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -1,9 +1,123 @@
 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 _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.
+
+    If the entries of the matrix themselves belong to a real vector
+    space (such as the complex numbers which can be thought of as
+    pairs of real numbers), they will also be expanded in vector form
+    and flattened into the list.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_utils import _all2list
+        sage: from mjo.octonions import Octonions, OctonionMatrixAlgebra
+
+    EXAMPLES::
+
+        sage: _all2list([[1]])
+        [1]
+
+    ::
+
+        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]
+
+    ::
+
+        sage: _all2list(Octonions().one())
+        [1, 0, 0, 0, 0, 0, 0, 0]
+        sage: _all2list(OctonionMatrixAlgebra(1).one())
+        [1, 0, 0, 0, 0, 0, 0, 0]
+
+    ::
+
+        sage: V1 = VectorSpace(QQ,2)
+        sage: V2 = OctonionMatrixAlgebra(1,field=QQ)
+        sage: C = cartesian_product([V1,V2])
+        sage: x1 = V1([3,4])
+        sage: y1 = V2.one()
+        sage: _all2list(C( (x1,y1) ))
+        [3, 4, 1, 0, 0, 0, 0, 0, 0, 0]
+
+    """
+    if hasattr(x, 'to_vector'):
+        # This works on matrices of e.g. octonions directly, without
+        # first needing to convert them to a list of octonions and
+        # then recursing down into the list. It also avoids the wonky
+        # list(x) when x is an element of a CFM. I don't know what it
+        # returns but it aint the coordinates. This will fall through
+        # to the iterable case the next time around.
+        return _all2list(x.to_vector())
+
+    try:
+        xl = list(x)
+    except TypeError: # x is not iterable
+        return [x]
+
+    if xl == [x]:
+        # Avoid the retardation of list(QQ(1)) == [1].
+        return [x]
+
+    return sum(list( map(_all2list, xl) ), [])
+
+
 
 def _mat2vec(m):
         return vector(m.base_ring(), m.list())
@@ -64,6 +178,45 @@ def gram_schmidt(v, inner_product=None):
         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]
+        ]
+
+    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::
@@ -74,14 +227,17 @@ 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)
-    norm = lambda x: inner_product(x,x).sqrt()
-
-    def proj(x,y):
-        return (inner_product(x,y)/inner_product(x,x))*x
+    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
 
@@ -94,10 +250,24 @@ def gram_schmidt(v, inner_product=None):
 
     R = v[0].base_ring()
 
+    # Our "zero" needs to belong to the right space for sum() to work.
+    zero = v[0].parent().zero()
+
+    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 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() )
+        v[i] -= sum( (proj(v[j],v[i])
+                      for j in range(i)
+                      if not v[j].is_zero() ),
+                     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() ]
@@ -106,6 +276,6 @@ 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] = v[i] / norm(v[i])
+        v[i] = sc(v[i], ~norm(v[i]))
 
     return v