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