-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"""
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.hurwitz import (QuaternionMatrixAlgebra,
+ ....: 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: _all2list(QuaternionAlgebra(QQ, -1, -1).one())
+ [1, 0, 0, 0]
+ sage: _all2list(QuaternionMatrixAlgebra(1).one())
+ [1, 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...
+ 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. 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()
- 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), [] )
-def _mat2vec(m):
- return vector(m.base_ring(), m.list())
+ 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( map(_all2list, xl) , [])
-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()
-
- v = list(v) # make a copy, don't clobber the input
-
- # Drop all zero vectors before we start.
- v = [ v_i for v_i in v if not v_i.is_zero() ]
+ """
if len(v) == 0:
# cool
return v
- R = v[0].base_ring()
+ V = v[0].parent()
- # Our "zero" needs to belong to the right space for sum() to work.
- zero = v[0].parent().zero()
+ if inner_product is None:
+ inner_product = lambda x,y: x.inner_product(y)
sc = lambda x,a: a*x
- if hasattr(v[0], 'cartesian_factors'):
+ if hasattr(V, 'cartesian_factors'):
# Only use the slow implementation if necessary.
sc = _scale
def proj(x,y):
+ # project y onto the span of {x}
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() ),
- zero )
+ def normalize(x):
+ # Don't extend the given field with the necessary
+ # square roots. This will probably throw weird
+ # errors about the symbolic ring if you e.g. try
+ # to use it on a set of rational vectors that isn't
+ # already orthonormalized.
+ return sc(x, ~inner_product(x,x).sqrt())
- # 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() ]
+ v_out = [] # make a copy, don't clobber the input
- # 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] = sc(v[i], ~norm(v[i]))
+ for (i, v_i) in enumerate(v):
+ ortho_v_i = v_i - V.sum( proj(v_out[j],v_i) for j in range(i) )
+ if not ortho_v_i.is_zero():
+ v_out.append(normalize(ortho_v_i))
- return v
+ return v_out
projects / sage.d.git / blobdiff