\newsection
\subsection{Gedämpftes Newton-Verfahren}
Wir wenden einen einen Dämpfungsfaktor $\lambda^{(k)}$ an, welcher heuristisch gewählt wird:
\rmvspace
\begin{align*}
    x^{(k + 1)} := x^{(k)} - \lambda^{(k)}DF(x^{(k)})^{-1} F(x^{(k)})
\end{align*}

\drmvspace
Wir wählen $\lambda^{(k)}$ so, dass für $\Delta x^{(k)} = DF(x^{(k)})^{-1} F(x^{(k)})$ und $\Delta(\lambda^{(k)}) = DF(x^{(k)})^{-1} F(x^{(k)} - \lambda^{(k)} \Delta x^{(k)})$
\rmvspace
\begin{align*}
    ||\Delta x(\lambda^{(k)})||_2 \leq \left( 1 - \frac{\lambda^{(k)}}{2} \right) ||\Delta x^{(k)}||_2
\end{align*}

\innumpy Das gedämpfte Newton-Verfahren lässt sich mit Funktionen aus \verb|scipy.linalg| implementieren:

\begin{code}{python}
def dampened_newton(x: np.ndarray, F, DF, q=0.5, rtol=1e-10, atol=1e-12):
    """ Dampened Newton with dampening factor q """

    lup    = lu_factor(DF(x))         # LU factorization for efficiency, direct works too
    s      = lu_solve(lup, F(x))      # 1st proper Newton correction
    damp   = 1                        # Start with no dampening
    x_damp = x - damp*s    
    s_damp = lu_solve(lup, F(x_damp)) # 1st simplified Newton correction (Reuse Jacobian)
    
    while norm(s_damp) > rtol * norm(x_damp) and norm(s_damp) > atol:
        while norm(s_damp) > (1-damp*q) * norm(s):  # Reduce dampening if step aggresive
            damp *= q
            if damp < 1e-4: return x            # Conclude dampening doesn't work anymore
            x_damp = x - damp*s                 # Try weaker dampening instead
            s_damp = lu_solve(lup, F(x_damp)) 
        
        x = x_damp     # Accept this dampened iteration, continue with next proper step
        
        lup    = lu_factor(DF(x))
        s      = lu_solve(lup, F(x))         # Next proper Newton correction
        damp   = np.min( damp/q, 1 )
        x_damp = x - damp*s
        s_damp = lu_solve(lup, F(x_damp))    # Next simplified Newton correction
    
    return x_damp
\end{code}

\newpage