# week 11

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

Gradient descent methods. Step size choices. Backtracking rule.

Newton algorithm. Solving systems of nonlinear equations. Quadratic convergence,

#### Sample problems

• Compute first 5 iterations $$x_0=2,x_1,\ldots, x_5$$ of Newton method to solve
$x^2-5=0.$

Plot $$\log|x_k^2-5|$$.

• For $$x_0$$ close to $$0$$ the iterations of Newton method to solve
$x^3-5x=0$
quickly converge to $$0$$ (what else!?). Find the largest interval where this is true (i.e. find the smallest $$x_0>0$$ starting with which the iterations of the Newton method fail to converge).