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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 bfgs_method(f, x, line_search=1.0, maxiter=1000, tol=1e-15, num_store=None, hessinv_estimate=None, callback=None):
 r"""Quasi-Newton BFGS method to minimize a differentiable function. Can use either the regular BFGS method, or the limited memory BFGS method. Notes ----- This is a general and optimized implementation of a quasi-Newton method with BFGS update for solving a general unconstrained optimization problem .. math:: \min f(x) for a differentiable function :math:`f: \mathcal{X}\to \mathbb{R}` on a Hilbert space :math:`\mathcal{X}`. It does so by finding a zero of the gradient .. math:: \nabla f: \mathcal{X} \to \mathcal{X}. The QN method is an approximate Newton method, where the Hessian is approximated and gradually updated in each step. This implementation uses the rank-one BFGS update schema where the inverse of the Hessian is recalculated in each iteration. The algorithm is described in [GNS2009], Section 12.3 and in the `BFGS Wikipedia article <https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93\ Goldfarb%E2%80%93Shanno_algorithm>`_ Parameters f : `Functional` Functional with ``f.gradient``. x : ``f.domain`` element Starting point of the iteration line_search : float or `LineSearch`, optional Strategy to choose the step length. If a float is given, uses it as a fixed step length. maxiter : int, optional Maximum number of iterations. tol : float, optional Tolerance that should be used for terminating the iteration. num_store : int, optional Maximum number of correction factors to store. For ``None``, the method is the regular BFGS method. For an integer, the method becomes the Limited Memory BFGS method. hessinv_estimate : `Operator`, optional Initial estimate of the inverse of the Hessian operator. Needs to be an operator from ``f.domain`` to ``f.domain``. Default: Identity on ``f.domain`` callback : callable, optional Object executing code per iteration, e.g. plotting each iterate. References [GNS2009] Griva, I, Nash, S G, and Sofer, A. *Linear and nonlinear optimization*. Siam, 2009. """

    grad = f.gradient
    if x not in grad.domain:
        raise TypeError('`x` {!r} is not in the domain of `grad` {!r}'
                        ''.format(x, grad.domain))

    if not callable(line_search):
        line_search = ConstantLineSearch(line_search)

    ys = []
    ss = []

    grad_x = grad(x)
    for i in range(maxiter):
        # Determine a stepsize using line search
        search_dir = -_bfgs_direction(ss, ys, grad_x, hessinv_estimate)
        dir_deriv = search_dir.inner(grad_x)
        if np.abs(dir_deriv) == 0:
            return  # we found an optimum
        step = line_search(x, direction=search_dir, dir_derivative=dir_deriv)

        # Update x
        x_update = search_dir
        x_update *= step
        x += x_update

        grad_x, grad_diff = grad(x), grad_x
        # grad_diff = grad(x) - grad(x_old)
        grad_diff.lincomb(-1, grad_diff, 1, grad_x)

        y_inner_s = grad_diff.inner(x_update)

        # Test for convergence
        if np.abs(y_inner_s) < tol:
            if grad_x.norm() < tol:
                return
            else:
                # Reset if needed
                ys = []
                ss = []
                continue

        # Update Hessian
        ys.append(grad_diff)
        ss.append(x_update)
        if num_store is not None:
            # Throw away factors if they are too many.
            ss = ss[-num_store:]
            ys = ys[-num_store:]

        if callback is not None:
            callback(x)