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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def solve_mbar(u_kn_nonzero, N_k_nonzero, f_k_nonzero, solver_protocol=None):
 """Solve MBAR self-consistent equations using some sequence of equation solvers. Parameters u_kn_nonzero : np.ndarray, shape=(n_states, n_samples), dtype='float' The reduced potential energies, i.e. -log unnormalized probabilities for the nonempty states N_k_nonzero : np.ndarray, shape=(n_states), dtype='int' The number of samples in each state for the nonempty states f_k_nonzero : np.ndarray, shape=(n_states), dtype='float' The reduced free energies for the nonempty states solver_protocol: tuple(dict()), optional, default=None Optional list of dictionaries of steps in solver protocol. If None, a default protocol will be used. Returns ------- f_k : np.ndarray The converged reduced free energies. all_results : list(dict()) List of results from each step of solver_protocol. Each element in list contains the results dictionary from solve_mbar_once() for the corresponding step. Notes ----- This function requires that N_k_nonzero > 0--that is, you should have already dropped all the states for which you have no samples. Internally, this function works in a reduced coordinate system defined by subtracting off the first component of f_k and fixing that component to be zero. This function calls `solve_mbar_once()` multiple times to achieve converged results. Generally, a single call to solve_mbar_once() will not give fully converged answers because of limited numerical precision. Each call to `solve_mbar_once()` re-conditions the nonlinear equations using the current guess. """

    if solver_protocol is None:
        solver_protocol = DEFAULT_SOLVER_PROTOCOL
    for protocol in solver_protocol:
        if protocol['method'] is None:
            protocol['method'] = DEFAULT_SOLVER_METHOD

    all_results = []
    for k, options in enumerate(solver_protocol):
        f_k_nonzero, results = solve_mbar_once(u_kn_nonzero, N_k_nonzero, f_k_nonzero, **options)
        all_results.append(results)
        all_results.append(("Final gradient norm: %.3g" % np.linalg.norm(mbar_gradient(u_kn_nonzero, N_k_nonzero, f_k_nonzero))))
    return f_k_nonzero, all_results