+from sage.functions.other import sqrt
+from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
+def _all2list(x):
+ r"""
+ Flatten a vector, matrix, or cartesian product of those things
+ into a long list.
+ """
+ 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]
+ ]
+
+ 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()
+
+ # Define a scaling operation that can be used on tuples.
+ # Oh and our "zero" needs to belong to the right space.
+ scale = lambda x,alpha: x*alpha
+ zero = v[0].parent().zero()
+ if hasattr(v[0], 'cartesian_factors'):
+ P = v[0].parent()
+ scale = lambda x,alpha: P(tuple( x_i*alpha
+ for x_i in x.cartesian_factors() ))
+
+
+ def proj(x,y):
+ return scale(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] = scale(v[i], ~norm(v[i]))
+
+ return v
projects / sage.d.git / blobdiff