# Buggy code
def approx_jacobian(x, func, epsilon, *args):
    """
    Approximate the Jacobian matrix of a callable function.

    Parameters
    ----------
    x : array_like
        The state vector at which to compute the Jacobian matrix.
    func : callable f(x,*args)
        The vector-valued function.
    epsilon : float
        The perturbation used to determine the partial derivatives.
    args : sequence
        Additional arguments passed to func.

    Returns
    -------
    An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length
    of the outputs of `func`, and ``lenx`` is the number of elements in
    `x`.

    Notes
    -----
    The approximation is done using forward differences.

    """
    x0 = asfarray(x)
    f0 = atleast_1d(func(*((x0,) + args)))
    jac = zeros([len(x0), len(f0)])
    dx = zeros(len(x0))
    for i in range(len(x0)):
        dx[i] = epsilon
        jac[i] = (func(*((x0 + dx,) + args)) - f0) / epsilon
        dx[i] = 0.0

    return jac.transpose()


# Traceback (ValueError)
In [158]: minimize(func, [1.1, 1.2], method='Newton-CG', jac=False)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-158-94c7580e9508> in <module>()
----> 1 minimize(func, [1.1, 1.2], method='Newton-CG', jac=False)

~/miniconda3scipy/lib/python3.5/site-packages/scipy-1.2.0.dev0+6d455f3-py3.5-macosx-10.6-x86_64.egg/scipy/optimize/_minimize.py in minimize(fun, x0, args, method, jac, hess, hessp, bounds, constraints, tol, callback, options)
600     elif meth == 'newton-cg':
601         return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback,
--> 602                                   **options)
603     elif meth == 'l-bfgs-b':
604         return _minimize_lbfgsb(fun, x0, args, jac, bounds,

~/miniconda3scipy/lib/python3.5/site-packages/scipy-1.2.0.dev0+6d455f3-py3.5-macosx-10.6-x86_64.egg/scipy/optimize/optimize.py in _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback, xtol, eps, maxiter, disp, return_all, **unknown_options)
1494     _check_unknown_options(unknown_options)
1495     if jac is None:
-> 1496         raise ValueError('Jacobian is required for Newton-CG method')
1497     f = fun
1498     fprime = jac

ValueError: Jacobian is required for Newton-CG method

# Fixed code
def approx_jacobian(x, func, epsilon, *args):
    """
    Approximate the Jacobian matrix of a callable function.

    Parameters
    ----------
    x : array_like
        The state vector at which to compute the Jacobian matrix.
    func : callable f(x,*args)
        The vector-valued function.
    epsilon : float
        The perturbation used to determine the partial derivatives.
    args : sequence
        Additional arguments passed to func.

    Returns
    -------
    An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length
    of the outputs of `func`, and ``lenx`` is the number of elements in
    `x`.

    Notes
    -----
    The approximation is done using forward differences.

    """
    # approx_derivative returns (m, n) == (lenf, lenx)
    jac = approx_derivative(func, x, method="2-point", abs_step=epsilon, args=args)
    # if func returns a scalar jac.shape will be (lenx,). Make sure
    # it's at least a 2D array.
    return np.atleast_2d(jac)