\(\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).

## No comments yet.