from sage.functions.other import sqrt
from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector

def _scale(x, alpha):
    r"""
    Scale the vector, matrix, or cartesian-product-of-those-things
    ``x`` by ``alpha``.

    This works around the inability to scale certain elements of
    Cartesian product spaces, as reported in

      https://trac.sagemath.org/ticket/31435

    ..WARNING:

        This will do the wrong thing if you feed it a tuple or list.

    SETUP::

        sage: from mjo.eja.eja_utils import _scale

    EXAMPLES::

        sage: v = vector(QQ, (1,2,3))
        sage: _scale(v,2)
        (2, 4, 6)
        sage: m = matrix(QQ, [[1,2],[3,4]])
        sage: M = cartesian_product([m.parent(), m.parent()])
        sage: _scale(M((m,m)), 2)
        ([2 4]
        [6 8], [2 4]
        [6 8])

    """
    if hasattr(x, 'cartesian_factors'):
        P = x.parent()
        return P(tuple( _scale(x_i, alpha)
                        for x_i in x.cartesian_factors() ))
    else:
        return x*alpha


def _all2list(x):
    r"""
    Flatten a vector, matrix, or cartesian product of those things
    into a long list.

    EXAMPLES::

        sage: from mjo.eja.eja_utils import _all2list
        sage: V1 = VectorSpace(QQ,2)
        sage: V2 = MatrixSpace(QQ,2)
        sage: x1 = V1([1,1])
        sage: x2 = V1([1,-1])
        sage: y1 = V2.one()
        sage: y2 = V2([0,1,1,0])
        sage: _all2list((x1,y1))
        [1, 1, 1, 0, 0, 1]
        sage: _all2list((x2,y2))
        [1, -1, 0, 1, 1, 0]
        sage: M = cartesian_product([V1,V2])
        sage: _all2list(M((x1,y1)))
        [1, 1, 1, 0, 0, 1]
        sage: _all2list(M((x2,y2)))
        [1, -1, 0, 1, 1, 0]

    """
    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]
        ]

    It even works on Cartesian product spaces whose factors are vector
    or matrix spaces::

        sage: V1 = VectorSpace(AA,2)
        sage: V2 = MatrixSpace(AA,2)
        sage: M = cartesian_product([V1,V2])
        sage: x1 = V1([1,1])
        sage: x2 = V1([1,-1])
        sage: y1 = V2.one()
        sage: y2 = V2([0,1,1,0])
        sage: z1 = M((x1,y1))
        sage: z2 = M((x2,y2))
        sage: def ip(a,b):
        ....:     return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
        sage: U = gram_schmidt([z1,z2], inner_product=ip)
        sage: ip(U[0],U[1])
        0
        sage: ip(U[0],U[0])
        1
        sage: ip(U[1],U[1])
        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()

    # Our "zero" needs to belong to the right space for sum() to work.
    zero = v[0].parent().zero()

    sc = lambda x,a: a*x
    if hasattr(v[0], 'cartesian_factors'):
        # Only use the slow implementation if necessary.
        sc = _scale

    def proj(x,y):
        return sc(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] = sc(v[i], ~norm(v[i]))

    return v