human
Complete this python program:
def _root_df_sane(func, x0, args=(), ftol=1e-08, fatol=1e-300, maxfev=1000, fnorm=None, callback=None, disp=False, M=10, eta_strategy=None, sigma_eps=1e-10, sigma_0=1.0, line_search='cruz', **unknown_options):
assistant
#Here is the Python program you requested.

    """Solve nonlinear equation with the DF-SANE method
    Options
    ftol : float, optional
    Relative norm tolerance.
    fatol : float, optional
    Absolute norm tolerance.
    Algorithm terminates when ``||func(x)|| < fatol + ftol ||func(x_0)||``.
    fnorm : callable, optional
    Norm to use in the convergence check. If None, 2-norm is used.
    maxfev : int, optional
    Maximum number of function evaluations.
    disp : bool, optional
    Whether to print convergence process to stdout.
    eta_strategy : callable, optional
    Choice of the ``eta_k`` parameter, which gives slack for growth
    of ``||F||**2``.  Called as ``eta_k = eta_strategy(k, x, F)`` with
    `k` the iteration number, `x` the current iterate and `F` the current
    residual. Should satisfy ``eta_k > 0`` and ``sum(eta, k=0..inf) < inf``.
    Default: ``||F||**2 / (1 + k)**2``.
    sigma_eps : float, optional
    The spectral coefficient is constrained to ``sigma_eps < sigma < 1/sigma_eps``.
    Default: 1e-10
    sigma_0 : float, optional
    Initial spectral coefficient.
    Default: 1.0
    M : int, optional
    Number of iterates to include in the nonmonotonic line search.
    Default: 10
    line_search : {\cruz\, \cheng\}
    Type of line search to employ. \cruz\ is the original one defined in
    [Martinez & Raydan. Math. Comp. 75, 1429 (2006)], \cheng\ is
    a modified search defined in [Cheng & Li. IMA J. Numer. Anal. 29, 814 (2009)].
    Default: \cruz\
    References
    .. [1] "Spectral residual method without gradient information for solving
    large-scale nonlinear systems of equations." W. La Cruz,
    J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006).
    .. [2] W. La Cruz, Opt. Meth. Software, 29, 24 (2014).
    .. [3] W. Cheng, D.-H. Li. IMA J. Numer. Anal. **29**, 814 (2009).
    """
    _check_unknown_options(unknown_options)
    if (line_search not in ('cheng', 'cruz')):
        raise ValueError(("Invalid    value    %r    for    'line_search'" % (line_search,)))
    nexp = 2
    if (eta_strategy is None):
        def eta_strategy(k, x, F):
            return (f_0 / ((1 + k) ** 2))
    if (fnorm is None):
        def fnorm(F):
            return (f_k ** (1.0 / nexp))
    def fmerit(F):
        return (np.linalg.norm(F) ** nexp)
    nfev = [0]
    (f, x_k, x_shape, f_k, F_k, is_complex) = _wrap_func(func, x0, fmerit, nfev, maxfev, args)
    k = 0
    f_0 = f_k
    sigma_k = sigma_0
    F_0_norm = fnorm(F_k)
    prev_fs = collections.deque([f_k], M)
    Q = 1.0
    C = f_0
    converged = False
    message = 'too    many    function    evaluations    required'
    while True:
        F_k_norm = fnorm(F_k)
        if disp:
            print(('iter    %d:    ||F||    =    %g,    sigma    =    %g' % (k, F_k_norm, sigma_k)))
        if (callback is not None):
            callback(x_k, F_k)
        if (F_k_norm < ((ftol * F_0_norm) + fatol)):
            message = 'successful    convergence'
            converged = True
            break
        if (abs(sigma_k) > (1 / sigma_eps)):
            sigma_k = ((1 / sigma_eps) * np.sign(sigma_k))
        elif (abs(sigma_k) < sigma_eps):
            sigma_k = sigma_eps
        d = ((- sigma_k) * F_k)
        eta = eta_strategy(k, x_k, F_k)
        try:
            if (line_search == 'cruz'):
                (alpha, xp, fp, Fp) = _nonmonotone_line_search_cruz(f, x_k, d, prev_fs, eta=eta)
            elif (line_search == 'cheng'):
                (alpha, xp, fp, Fp, C, Q) = _nonmonotone_line_search_cheng(f, x_k, d, f_k, C, Q, eta=eta)
        except _NoConvergence:
            break
        s_k = (xp - x_k)
        y_k = (Fp - F_k)
        sigma_k = (np.vdot(s_k, s_k) / np.vdot(s_k, y_k))
        x_k = xp
        F_k = Fp
        f_k = fp
        if (line_search == 'cruz'):
            prev_fs.append(fp)
        k += 1
    x = _wrap_result(x_k,...