from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
+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())
[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::
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
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() ]
# 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
projects / sage.d.git / blobdiff