from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
-def _change_ring(x, R):
+def _charpoly_sage_input(s):
r"""
- Change the ring of a vector, matrix, or a cartesian product of
- those things.
+ Helper function that you can use on the string output from sage
+ to convert a charpoly coefficient into the corresponding input
+ to be cached.
SETUP::
- sage: from mjo.eja.eja_utils import _change_ring
+ sage: from mjo.eja.eja_utils import _charpoly_sage_input
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)))
+ sage: J = JordanSpinEJA(4,QQ)
+ sage: J._charpoly_coefficients()[0]
+ X1^2 - X2^2 - X3^2 - X4^2
+ sage: _charpoly_sage_input("X1^2 - X2^2 - X3^2 - X4^2")
+ 'X[0]**2 - X[1]**2 - X[2]**2 - X[3]**2'
"""
- 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 )
+ import re
+
+ exponent_out = r"\^"
+ exponent_in = r"**"
+
+ digit_out = r"X([0-9]+)"
+
+ def replace_digit(m):
+ # m is a match object
+ return "X[" + str(int(m.group(1)) - 1) + "]"
+
+ s = re.sub(exponent_out, exponent_in, s)
+ return re.sub(digit_out, replace_digit, s)
+
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...
- 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())
"""
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
projects / sage.d.git / blobdiff