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).
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