How can I compute the first three iterates for the steepest descent sequence
$f(x_1,x_2) = \frac{(x_1^2+3x_2^2)}{2}$ beginning at $x_0 = (\frac{\sqrt{3}}{2}, \frac{1}{2 \sqrt{3}})^T$
$\nabla f(x_1,x_2) = \langle 2x_1, 3x_2\rangle$
and $\nabla f(\textbf{x}_0) = \langle .. , ..\rangle$
Assuming my calculations are correct, Where do I proceed from here for the next steps?
How do I draw the trajectory of the iterates towards the minimizer of $f$ ?
Your $\nabla f$ should be $<x_1,3 x_3>$. So the associated matrix $A=\begin{pmatrix} 1&0\\ 0&3\end{pmatrix}$, and $b=\begin{pmatrix} 0\\ 0\end{pmatrix}$
You can then use the formulas
$$r_n=b-Ax_n\\ \alpha_n=\frac{r_n^T r_n}{r_n^T A r^n}\\ x_{n+1}=x_n+\alpha r_n$$
to find the 3 iterations.
Your function is a paraboloid. You can either plot the points on a 3D paraboloid, or plot the points on a contour graph.