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