def williamson(V, tol=1e-11):
    r"""Williamson decomposition of positive-definite (real) symmetric matrix.

    See :ref:`williamson`.

    Note that it is assumed that the symplectic form is

    .. math:: \Omega = \begin{bmatrix}0&I\\-I&0\end{bmatrix}

    where :math:`I` is the identity matrix and :math:`0` is the zero matrix.

    See https://math.stackexchange.com/questions/1171842/finding-the-symplectic-matrix-in-williamsons-theorem/2682630#2682630

    Args:
        V (array[float]): positive definite symmetric (real) matrix
        tol (float): the tolerance used when checking if the matrix is symmetric: :math:`|V-V^T| \leq` tol

    Returns:
        tuple[array,array]: ``(Db, S)`` where ``Db`` is a diagonal matrix
            and ``S`` is a symplectic matrix such that :math:`V = S^T Db S`
    """
    (n, m) = V.shape

    if n != m:
        raise ValueError("The input matrix is not square")

    diffn = np.linalg.norm(V - np.transpose(V))

    if diffn >= tol:
        raise ValueError("The input matrix is not symmetric")

    if n % 2 != 0:
        raise ValueError("The input matrix must have an even number of rows/columns")

    n = n // 2
    omega = sympmat(n)
    vals = np.linalg.eigvalsh(V)

    for val in vals:
        if val <= 0:
            raise ValueError("Input matrix is not positive definite")

    Mm12 = sqrtm(np.linalg.inv(V)).real
    r1 = Mm12 @ omega @ Mm12
    s1, K = schur(r1)
    X = np.array([[0, 1], [1, 0]])
    I = np.identity(2)
    seq = []

    # In what follows I construct a permutation matrix p  so that the Schur matrix has
    # only positive elements above the diagonal
    # Also the Schur matrix uses the x_1,p_1, ..., x_n,p_n  ordering thus I use rotmat to
    # go to the ordering x_1, ..., x_n, p_1, ... , p_n

    for i in range(n):
        if s1[2 * i, 2 * i + 1] > 0:
            seq.append(I)
        else:
            seq.append(X)

    p = block_diag(*seq)
    Kt = K @ p
    s1t = p @ s1 @ p
    dd = xpxp_to_xxpp(s1t)
    perm_indices = xpxp_to_xxpp(np.arange(2 * n))
    Ktt = Kt[:, perm_indices]
    Db = np.diag([1 / dd[i, i + n] for i in range(n)] + [1 / dd[i, i + n] for i in range(n)])
    S = Mm12 @ Ktt @ sqrtm(Db)
    return Db, np.linalg.inv(S).T