# Aufgabe 7

def graddesc( A, x0, step=0.1 ):
    xi = x0

    # create the result matrix and save the initial value
    results = np.empty([50, 2])
    results[0, 0] = xi[0]
    results[0, 1] = xi[1]

    # complete the matrix with the given formula
    for i in range(1, 50):
        xi = xi + step * (b - np.matmul(A, xi))
        results[i, 0] = xi[0]
        results[i, 1] = xi[1]

    return results

# generate the values with the same intial value as in the lecture
values = graddesc(A, np.array([-4, -4]))

plt.plot(values[:, 0], values[:, 1])
plot_f( plt.gca() )

# Aufgabe 8, keine Ahnung ob das richtig ist, sieht aber so aus

xvalues = np.arange(1, len(history) + 1)  # ascending array from 1 to 307
yvalues = np.empty(len(history))
for i in range(len(history)):
    yvalues[i] = np.linalg.norm(A@history[i] - f)  # add the residual norm in this iteration
plt.semilogy(xvalues, yvalues)