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