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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 broydens_method(f, x, line_search=1.0, impl='first', maxiter=1000, tol=1e-15, hessinv_estimate=None, callback=None):
 r"""Broyden's first method, a quasi-Newton scheme. Notes ----- This is a general and optimized implementation of Broyden's method, a quasi-Newton method 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} using a Newton-type update scheme with approximate Hessian. The algorithm is described in [Bro1965] and [Kva1991], and in a `Wikipedia article <https://en.wikipedia.org/wiki/Broyden's_method>`_. 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. impl : {'first', 'second'}, optional What version of Broydens method to use. First is also known as Broydens 'good' method, while the second is known as Broydens 'bad' method. maxiter : int, optional Maximum number of iterations. ``tol``. tol : float, optional Tolerance that should be used for terminating the iteration. 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 [Bro1965] Broyden, C G. *A class of methods for solving nonlinear simultaneous equations*. Mathematics of computation, 33 (1965), pp 577--593. [Kva1991] Kvaalen, E. *A faster Broyden method*. BIT Numerical Mathematics 31 (1991), pp 369--372. """

    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)

    impl, impl_in = str(impl).lower(), impl
    if impl not in ('first', 'second'):
        raise ValueError('`impl` {!r} not understood'
                         ''.format(impl_in))

    ss = []
    ys = []

    grad_x = grad(x)
    for i in range(maxiter):
        # find step size
        search_dir = -_broydens_direction(ss, ys, grad_x,
                                          hessinv_estimate, impl)
        dir_deriv = search_dir.inner(grad_x)
        if np.abs(dir_deriv) == 0:
            return  # we found an optimum

        step = line_search(x, search_dir, dir_deriv)

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

        # compute new gradient
        grad_x, grad_x_old = grad(x), grad_x
        delta_grad = grad_x - grad_x_old

        # update hessian.
        # TODO: reuse from above
        v = _broydens_direction(ss, ys, delta_grad, hessinv_estimate,
                                impl)
        if impl == 'first':
            divisor = x_update.inner(v)

            # Test for convergence
            if np.abs(divisor) < tol:
                if grad_x.norm() < tol:
                    return
                else:
                    # Reset if needed
                    ys = []
                    ss = []
                    continue
            u = (x_update - v) / divisor
            ss.append(u)
            ys.append(x_update)
        elif impl == 'second':
            divisor = delta_grad.inner(delta_grad)

            # Test for convergence
            if np.abs(divisor) < tol:
                if grad_x.norm() < tol:
                    return
                else:
                    # Reset if needed
                    ys = []
                    ss = []
                    continue
            u = (x_update - v) / divisor
            ss.append(u)
            ys.append(delta_grad)

        if callback is not None:
            callback(x)