midterm: small correction

Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.md

@@ -366,11 +366,21 @@ respectively in the files `trust_region.m`, `dogleg.m`, and `cauchy.m`.
 
 ### Taylor expansion
 
+We first compute the gradient and the hessian of the energy function:
+
+$$\nabla f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix}
+\frac{d f(x)}{d x_1} \\ \frac{d f(x)}{d x_2} \end{bmatrix} =
+\begin{bmatrix}48x_1^3 - 16x_1x_2 + 2x_1 - 2\\2x_2 - 8x_1^2\end{bmatrix}$$
+
+$$ \nabla^2 f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix}
+\frac{d^2 f(x)}{d x_1^2} & \frac{d^2 f(x)}{dx_2 x_1} \\ \frac{d^2 f(x)}{dx_1
+x_2} & \frac{d^2 f(x)}{d x_2^2}\end{bmatrix} = \begin{bmatrix}144x_1^2 -16x_2 +
+2 - 16 & -16 \\ -16 & 2 \end{bmatrix}$$
+
 The Taylor expansion up the second order of the function is the following:
 
-$$f(x_0, w) = f(x_0) + \langle\begin{bmatrix}48x^3 - 16xy + 2x - 2\\2y - 8x^2
+$$f(x_0, w) = f(x_0) + \langle \nabla f(x_0), w\rangle + \frac12 \langle
-\end{bmatrix}, w\rangle + \frac12 \langle\begin{bmatrix}144x^2 -16y + 2 - 16 &
+\nabla^2 f(x_0) w, w\rangle$$
--16 \\ -16 & 2 \end{bmatrix}w, w\rangle$$
 
 ### Minimization