from sage.matrix.constructor import matrix
from sage.misc.cachefunc import cached_method
from sage.rings.all import QQ

from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra


class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra):
    def __init__(self, elt, orthonormalize_basis):
        self._superalgebra = elt.parent()
        category = self._superalgebra.category().Associative()
        V = self._superalgebra.vector_space()
        field = self._superalgebra.base_ring()

        # This list is guaranteed to contain all independent powers,
        # because it's the maximal set of powers that could possibly
        # be independent (by a dimension argument).
        powers = [ elt**k for k in range(V.dimension()) ]
        power_vectors = [ p.to_vector() for p in powers ]
        P = matrix(field, power_vectors)

        if orthonormalize_basis == False:
            # Echelonize the matrix ourselves, because otherwise the
            # call to P.pivot_rows() below can choose a non-optimal
            # row-reduction algorithm. In particular, scaling can
            # help over AA because it avoids the RecursionError that
            # gets thrown when we have to look too hard for a root.
            #
            # Beware: QQ supports an entirely different set of "algorithm"
            # keywords than do AA and RR.
            algo = None
            if field is not QQ:
                algo = "scaled_partial_pivoting"
            P.echelonize(algorithm=algo)

            # In this case, we just need to figure out which elements
            # of the "powers" list are redundant... First compute the
            # vector subspace spanned by the powers of the given
            # element.

            # Figure out which powers form a linearly-independent set.
            ind_rows = P.pivot_rows()

            # Pick those out of the list of all powers.
            superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
        else:
            # If we're going to orthonormalize the basis anyway, we
            # might as well just do Gram-Schmidt on the whole list of
            # powers. The redundant ones will get zero'd out. If this
            # looks like a roundabout way to orthonormalize, it is.
            # But converting everything from algebra elements to vectors
            # to matrices and then back again turns out to be about
            # as fast as reimplementing our own Gram-Schmidt that
            # works in an EJA.
            G,_ = P.gram_schmidt(orthonormal=True)
            basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
            superalgebra_basis = [ self._superalgebra.from_vector(b)
                                   for b in basis_vectors ]

        fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
        fdeja.__init__(self._superalgebra,
                       superalgebra_basis,
                       category=category,
                       check_axioms=False)

        # The rank is the highest possible degree of a minimal
        # polynomial, and is bounded above by the dimension. We know
        # in this case that there's an element whose minimal
        # polynomial has the same degree as the space's dimension
        # (remember how we constructed the space?), so that must be
        # its rank too.
        self.rank.set_cache(self.dimension())


    @cached_method
    def one(self):
        """
        Return the multiplicative identity element of this algebra.

        The superclass method computes the identity element, which is
        beyond overkill in this case: the superalgebra identity
        restricted to this algebra is its identity. Note that we can't
        count on the first basis element being the identity -- it might
        have been scaled if we orthonormalized the basis.

        SETUP::

            sage: from mjo.eja.eja_algebra import (HadamardEJA,
            ....:                                  random_eja)

        EXAMPLES::

            sage: J = HadamardEJA(5)
            sage: J.one()
            e0 + e1 + e2 + e3 + e4
            sage: x = sum(J.gens())
            sage: A = x.subalgebra_generated_by()
            sage: A.one()
            f0
            sage: A.one().superalgebra_element()
            e0 + e1 + e2 + e3 + e4

        TESTS:

        The identity element acts like the identity over the rationals::

            sage: set_random_seed()
            sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
            sage: A = x.subalgebra_generated_by()
            sage: x = A.random_element()
            sage: A.one()*x == x and x*A.one() == x
            True

        The identity element acts like the identity over the algebraic
        reals with an orthonormal basis::

            sage: set_random_seed()
            sage: x = random_eja().random_element()
            sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
            sage: x = A.random_element()
            sage: A.one()*x == x and x*A.one() == x
            True

        The matrix of the unit element's operator is the identity over
        the rationals::

            sage: set_random_seed()
            sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
            sage: A = x.subalgebra_generated_by()
            sage: actual = A.one().operator().matrix()
            sage: expected = matrix.identity(A.base_ring(), A.dimension())
            sage: actual == expected
            True

        The matrix of the unit element's operator is the identity over
        the algebraic reals with an orthonormal basis::

            sage: set_random_seed()
            sage: x = random_eja().random_element()
            sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
            sage: actual = A.one().operator().matrix()
            sage: expected = matrix.identity(A.base_ring(), A.dimension())
            sage: actual == expected
            True

        """
        if self.dimension() == 0:
            return self.zero()

        return self(self.superalgebra().one())