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
+ ``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.
+
+ EXAMPLES::
+
+ sage: from mjo.eja.eja_utils import _all2list
+ 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]
+
+ """
+ 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), [] )
+
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
-
v = list(v) # make a copy, don't clobber the input
# Drop all zero vectors before we start.
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