from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
-def _change_ring(x, R):
- r"""
- Change the ring of a vector, matrix, or a cartesian product of
- those things.
-
- SETUP::
-
- sage: from mjo.eja.eja_utils import _change_ring
-
- EXAMPLES::
-
- sage: v = vector(QQ, (1,2,3))
- sage: m = matrix(QQ, [[1,2],[3,4]])
- sage: _change_ring(v, RDF)
- (1.0, 2.0, 3.0)
- sage: _change_ring(m, RDF)
- [1.0 2.0]
- [3.0 4.0]
- sage: _change_ring((v,m), RDF)
- (
- [1.0 2.0]
- (1.0, 2.0, 3.0), [3.0 4.0]
- )
- sage: V1 = cartesian_product([v.parent(), v.parent()])
- sage: V = cartesian_product([v.parent(), V1])
- sage: V((v, (v, v)))
- ((1, 2, 3), ((1, 2, 3), (1, 2, 3)))
- sage: _change_ring(V((v, (v, v))), RDF)
- ((1.0, 2.0, 3.0), ((1.0, 2.0, 3.0), (1.0, 2.0, 3.0)))
-
- """
- try:
- return x.change_ring(R)
- except AttributeError:
- try:
- from sage.categories.sets_cat import cartesian_product
- if hasattr(x, 'element_class'):
- # x is a parent and we're in a recursive call.
- return cartesian_product( [_change_ring(x_i, R)
- for x_i in x.cartesian_factors()] )
- else:
- # x is an element, and we want to change the ring
- # of its parent.
- P = x.parent()
- Q = cartesian_product( [_change_ring(P_i, R)
- for P_i in P.cartesian_factors()] )
- return Q(x)
- except AttributeError:
- # No parent for x
- return x.__class__( _change_ring(x_i, R) for x_i in x )
-
def _scale(x, alpha):
r"""
Scale the vector, matrix, or cartesian-product-of-those-things
Flatten a vector, matrix, or cartesian product of those things
into a long list.
- EXAMPLES::
+ 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: _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, 'list'):
- # Easy case...
- return x.list()
- else:
- # But what if it's a tuple or something else? This has to
- # handle cartesian products of cartesian products, too; that's
- # why it's recursive.
- return sum( map(_all2list,x), [] )
+ 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())
"""
if inner_product is None:
inner_product = lambda x,y: x.inner_product(y)
- norm = lambda x: inner_product(x,x).sqrt()
+ 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
projects / sage.d.git / blobdiff