-from sage.functions.other import sqrt
-from sage.matrix.constructor import matrix
-from sage.modules.free_module_element import vector
+from sage.structure.element import is_Matrix
def _scale(x, alpha):
r"""
SETUP::
sage: from mjo.eja.eja_utils import _all2list
- sage: from mjo.octonions import Octonions, OctonionMatrixAlgebra
+ sage: from mjo.hurwitz import (QuaternionMatrixAlgebra,
+ ....: Octonions,
+ ....: OctonionMatrixAlgebra)
EXAMPLES::
sage: _all2list(OctonionMatrixAlgebra(1).one())
[1, 0, 0, 0, 0, 0, 0, 0]
+ ::
+
+ sage: _all2list(QuaternionAlgebra(QQ, -1, -1).one())
+ [1, 0, 0, 0]
+ sage: _all2list(QuaternionMatrixAlgebra(1).one())
+ [1, 0, 0, 0]
+
::
sage: V1 = VectorSpace(QQ,2)
# 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())
+ # returns but it aint the coordinates. We don't recurse
+ # because vectors can only contain ring elements as entries.
+ return x.to_vector().list()
+
+ if is_Matrix(x):
+ # This sucks, but for performance reasons we don't want to
+ # call _all2list recursively on the contents of a matrix
+ # when we don't have to (they only contain ring elements
+ # as entries)
+ return x.list()
try:
xl = list(x)
# Avoid the retardation of list(QQ(1)) == [1].
return [x]
- return sum(list( map(_all2list, xl) ), [])
-
-
+ return sum( map(_all2list, xl) , [])
-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, 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
- vectors over the smallest extention ring containing the necessary
- roots.
+ vectors over the same base ring, which means that your base ring
+ needs to contain the appropriate roots.
SETUP::
EXAMPLES:
+ If you start with an orthonormal set, you get it back. We can use
+ the rationals here because we don't need any square roots::
+
+ sage: v1 = vector(QQ, (1,0,0))
+ sage: v2 = vector(QQ, (0,1,0))
+ sage: v3 = vector(QQ, (0,0,1))
+ sage: v = [v1,v2,v3]
+ sage: gram_schmidt(v) == v
+ True
+
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: v1 = vector(AA,(1,2,3))
+ sage: v2 = vector(AA,(1,-1,6))
+ sage: v3 = vector(AA,(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 )
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: v1 = vector(AA,(1,2,3))
+ sage: v2 = vector(AA,(1,-1,6))
+ sage: v3 = vector(AA,(2,1,-1))
sage: v = [v1,v2,v3]
- sage: B = matrix(QQ, [ [6, 4, 2],
+ sage: B = matrix(AA, [ [6, 4, 2],
....: [4, 5, 4],
....: [2, 4, 9] ])
sage: ip = lambda x,y: (B*x).inner_product(y)
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],
+ sage: E11 = matrix(AA, [ [1,0],
....: [0,0] ])
- sage: E12 = matrix(QQ, [ [0,1],
+ sage: E12 = matrix(AA, [ [0,1],
....: [1,0] ])
- sage: E22 = matrix(QQ, [ [0,0],
+ sage: E22 = matrix(AA, [ [0,0],
....: [0,1] ])
- sage: I = matrix.identity(QQ,2)
+ sage: I = matrix.identity(AA,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]
+ [1 0] [ 0 0.7071067811865475?] [0 0]
+ [0 0], [0.7071067811865475? 0], [0 1]
]
It even works on Cartesian product spaces whose factors are vector
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: v1 = vector(AA,(1,2,3))
+ sage: v2 = vector(AA,(1,-1,6))
+ sage: v3 = vector(AA,(0,0,0))
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 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
# cool
return v
- R = v[0].base_ring()
-
# Our "zero" needs to belong to the right space for sum() to work.
zero = v[0].parent().zero()
projects / sage.d.git / blobdiff