Python MatrixSpace.matrix方法代码示例

# 需要导入模块: from sage.matrix.matrix_space import MatrixSpace [as 别名]
# 或者: from sage.matrix.matrix_space.MatrixSpace import matrix [as 别名]
def algebraic_topological_model_delta_complex(K, base_ring=None):
    r"""
    Algebraic topological model for cell complex ``K``
    with coefficients in the field ``base_ring``.

    This has the same basic functionality as
    :func:`algebraic_topological_model`, but it also works for
    `\Delta`-complexes. For simplicial and cubical complexes it is
    somewhat slower, though.

    INPUT:

    - ``K`` -- a simplicial complex, a cubical complex, or a
      `\Delta`-complex
    - ``base_ring`` -- coefficient ring; must be a field

    OUTPUT: a pair ``(phi, M)`` consisting of

    - chain contraction ``phi``
    - chain complex `M`

    See :func:`algebraic_topological_model` for the main
    documentation. The difference in implementation between the two:
    this uses matrix and vector algebra. The other function does more
    of the computations "by hand" and uses cells (given as simplices
    or cubes) to index various dictionaries. Since the cells in
    `\Delta`-complexes are not as nice, the other function does not
    work for them, while this function relies almost entirely on the
    structure of the associated chain complex.

    EXAMPLES::

        sage: from sage.homology.algebraic_topological_model import algebraic_topological_model_delta_complex as AT_model
        sage: RP2 = simplicial_complexes.RealProjectivePlane()
        sage: phi, M = AT_model(RP2, GF(2))
        sage: M.homology()
        {0: Vector space of dimension 1 over Finite Field of size 2,
         1: Vector space of dimension 1 over Finite Field of size 2,
         2: Vector space of dimension 1 over Finite Field of size 2}
        sage: T = delta_complexes.Torus()
        sage: phi, M = AT_model(T, QQ)
        sage: M.homology()
        {0: Vector space of dimension 1 over Rational Field,
         1: Vector space of dimension 2 over Rational Field,
         2: Vector space of dimension 1 over Rational Field}

    If you want to work with cohomology rather than homology, just
    dualize the outputs of this function::

        sage: M.dual().homology()
        {0: Vector space of dimension 1 over Rational Field,
         1: Vector space of dimension 2 over Rational Field,
         2: Vector space of dimension 1 over Rational Field}
        sage: M.dual().degree_of_differential()
        1
        sage: phi.dual()
        Chain homotopy between:
          Chain complex endomorphism of Chain complex with at most 3 nonzero terms over Rational Field
          and Chain complex morphism:
            From: Chain complex with at most 3 nonzero terms over Rational Field
            To:   Chain complex with at most 3 nonzero terms over Rational Field

    In degree 0, the inclusion of the homology `M` into the chain
    complex `C` sends the homology generator to a single vertex::

        sage: K = delta_complexes.Simplex(2)
        sage: phi, M = AT_model(K, QQ)
        sage: phi.iota().in_degree(0)
        [0]
        [0]
        [1]

    In cohomology, though, one needs the dual of every degree 0 cell
    to detect the degree 0 cohomology generator::

        sage: phi.dual().iota().in_degree(0)
        [1]
        [1]
        [1]

    TESTS::

        sage: T = cubical_complexes.Torus()
        sage: C = T.chain_complex()
        sage: H, M = AT_model(T, QQ)
        sage: C.differential(1) * H.iota().in_degree(1).column(0) == 0
        True
        sage: C.differential(1) * H.iota().in_degree(1).column(1) == 0
        True
        sage: coC = T.chain_complex(cochain=True)
        sage: coC.differential(1) * H.dual().iota().in_degree(1).column(0) == 0
        True
        sage: coC.differential(1) * H.dual().iota().in_degree(1).column(1) == 0
        True
    """
    def conditionally_sparse(m):
        """
        Return a sparse matrix if the characteristic is zero.

        Multiplication of matrices with low density seems to be quicker
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