Python Matrix.pivots方法代码示例

# 需要导入模块: from sage.matrix.constructor import Matrix [as 别名]
# 或者: from sage.matrix.constructor.Matrix import pivots [as 别名]
def basis_of_short_vectors(self, show_lengths=False, safe_flag=None):
    """
    Return a basis for `ZZ^n` made of vectors with minimal lengths Q(`v`).

    OUTPUT: a tuple of vectors, and optionally a tuple of values for
    each vector.

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
        sage: Q.basis_of_short_vectors()
        ((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1))
        sage: Q.basis_of_short_vectors(True)
        (((1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)), (1, 3, 5, 7))

    The returned vectors are immutable::

        sage: v = Q.basis_of_short_vectors()[0]
        sage: v
        (1, 0, 0, 0)
        sage: v[0] = 0
        Traceback (most recent call last):
        ...
        ValueError: vector is immutable; please change a copy instead (use copy())
    """
    if safe_flag is not None:
        from sage.misc.superseded import deprecation
        deprecation(18673, "The safe_flag argument to basis_of_short_vectors() is deprecated and no longer used")

    ## Set an upper bound for the number of vectors to consider
    Max_number_of_vectors = 10000

    ## Generate a PARI matrix for the associated Hessian matrix
    M_pari = self.__pari__()

    ## Run through all possible minimal lengths to find a spanning set of vectors
    n = self.dim()
    M1 = Matrix([[0]])
    vec_len = 0
    while M1.rank() < n:
        vec_len += 1
        pari_mat = M_pari.qfminim(vec_len, Max_number_of_vectors)[2]
        number_of_vecs = ZZ(pari_mat.matsize()[1])
        vector_list = []
        for i in range(number_of_vecs):
            new_vec = vector([ZZ(x) for x in list(pari_mat[i])])
            vector_list.append(new_vec)

        ## Make a matrix from the short vectors
        if len(vector_list) > 0:
            M1 = Matrix(vector_list)

    ## Organize these vectors by length (and also introduce their negatives)
    max_len = vec_len // 2
    vector_list_by_length = [[] for _ in range(max_len + 1)]
    for v in vector_list:
        l = self(v)
        vector_list_by_length[l].append(v)
        vector_list_by_length[l].append(vector([-x  for x in v]))

    ## Make a matrix from the column vectors (in order of ascending length).
    sorted_list = []
    for i in range(len(vector_list_by_length)):
        for v in vector_list_by_length[i]:
            sorted_list.append(v)
    sorted_matrix = Matrix(sorted_list).transpose()

    ## Determine a basis of vectors of minimal length
    pivots = sorted_matrix.pivots()
    basis = tuple(sorted_matrix.column(i) for i in pivots)
    for v in basis:
        v.set_immutable()

    ## Return the appropriate result
    if show_lengths:
        pivot_lengths = tuple(self(v) for v in basis)
        return basis, pivot_lengths
    else:
        return basis