+from sage.functions.other import sqrt
+from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
-from sage.rings.number_field.number_field import NumberField
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.real_lazy import RLF
+
+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()
+ if hasattr(x, 'cartesian_factors'):
+ # If it's a formal cartesian product space element, then
+ # we also know what to do...
+ return sum(( x_i.list() for x_i in x ), [])
+ else:
+ # But what if it's a tuple or something else?
+ return sum( map(_all2list,x), [] )
def _mat2vec(m):
return vector(m.base_ring(), m.list())
-def gram_schmidt(v):
+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
sage: from mjo.eja.eja_utils import gram_schmidt
- EXAMPLES::
+ 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: [ u_i.inner_product(u_i).sqrt() == 1 for u_i in u ]
+ sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
True
- sage: u[0].inner_product(u[1]) == 0
+ sage: bool(u[0].inner_product(u[1]) == 0)
True
- sage: u[0].inner_product(u[2]) == 0
+ sage: bool(u[0].inner_product(u[2]) == 0)
True
- sage: u[1].inner_product(u[2]) == 0
+ 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::
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 (y.inner_product(x)/x.inner_product(x))*x
+ return (inner_product(x,y)/inner_product(x,x))*x
v = list(v) # make a copy, don't clobber the input
R = v[0].base_ring()
# First orthogonalize...
- for i in xrange(1,len(v)):
+ 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() )
# 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() ]
- # Now pretend to normalize, building a new ring R that contains
- # all of the necessary square roots.
- norms_squared = [0]*len(v)
-
- for i in xrange(len(v)):
- norms_squared[i] = v[i].inner_product(v[i])
- ns = [norms_squared[i].numerator(), norms_squared[i].denominator()]
-
- # Do the numerator and denominator separately so that we
- # adjoin e.g. sqrt(2) and sqrt(3) instead of sqrt(2/3).
- for j in xrange(len(ns)):
- PR = PolynomialRing(R, 'z')
- z = PR.gen()
- p = z**2 - ns[j]
- if p.is_irreducible():
- R = NumberField(p,
- 'sqrt' + str(ns[j]),
- embedding=RLF(ns[j]).sqrt())
-
- # When we're done, we have to change every element's ring to the
- # extension that we wound up with, and then normalize it (which
- # should work, since "R" contains its norm now).
- for i in xrange(len(v)):
- v[i] = v[i].change_ring(R) / R(norms_squared[i]).sqrt()
+ # 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] = v[i] / norm(v[i])
return v
projects / sage.d.git / blobdiff