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 _mat2vec(m):
return vector(m.base_ring(), m.list())
-
-def gram_schmidt(v):
- """
- 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::
-
- 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: u[0].inner_product(u[1]) == 0
- True
- sage: u[0].inner_product(u[2]) == 0
- True
- sage: u[1].inner_product(u[2]) == 0
- True
-
- 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
-
- """
- def proj(x,y):
- return (y.inner_product(x)/x.inner_product(x))*x
-
- 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()
-
- # First orthogonalize...
- for i in xrange(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()
-
- return v