Gradient Descent Example

Consider the quadratic function in two dimensions:

$f(\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right])=\frac{1}{2}{\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^T\left[\begin{array}{cc}2& 1\\ 1& 20\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]-{\left[\begin{array}{c}5\\ 3\end{array}\right]}^T\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

The gradient is:

gradf$(\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right])={\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^T\left[\begin{array}{cc}2& 1\\ 1& 20\end{array}\right]-{\left[\begin{array}{c}5\\ 3\end{array}\right]}^T$

Suppose $x_0=[-3,-1{]}^T,$ gradient descent algorithm with $=0\mathrm{.}085$ gives $x_1=[-1\mathrm{.}98,1\mathrm{.}21{]}^T$ and $x_2=[-1\mathrm{.}32,-0\mathrm{.}42{]}^T$