+from sage.functions.other import sqrt
from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
-from sage.functions.other import sqrt
+
+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())
+
+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.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_utils import gram_schmidt
+
+ EXAMPLES:
+
+ 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: v = [v1,v2,v3]
+ sage: u = gram_schmidt(v)
+ sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
+ True
+ sage: bool(u[0].inner_product(u[1]) == 0)
+ True
+ sage: bool(u[0].inner_product(u[2]) == 0)
+ True
+ sage: bool(u[1].inner_product(u[2]) == 0)
+ True
+
+
+ But if you supply a custom inner product, the result is
+ 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: v = [v1,v2,v3]
+ sage: B = matrix(QQ, [ [6, 4, 2],
+ ....: [4, 5, 4],
+ ....: [2, 4, 9] ])
+ sage: ip = lambda x,y: (B*x).inner_product(y)
+ sage: norm = lambda x: ip(x,x)
+ sage: u = gram_schmidt(v,ip)
+ sage: all( norm(u_i) == 1 for u_i in u )
+ True
+ sage: ip(u[0],u[1]).is_zero()
+ True
+ sage: ip(u[0],u[2]).is_zero()
+ True
+ sage: ip(u[1],u[2]).is_zero()
+ True
+
+ 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],
+ ....: [0,0] ])
+ sage: E12 = matrix(QQ, [ [0,1],
+ ....: [1,0] ])
+ sage: E22 = matrix(QQ, [ [0,0],
+ ....: [0,1] ])
+ sage: I = matrix.identity(QQ,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]
+ ]
+
+ 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: v1 = vector(QQ,(1,2,3))
+ sage: v2 = vector(QQ,(1,-1,6))
+ sage: v3 = vector(QQ,(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()
+
+ # 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() ),
+ 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() ]
+
+ # 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]))
+
+ return v