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\\ + y[x_0,x_1,x_2] &= \frac{1}{2!h} \Delta^2 y_0 \\ + y[x_0,\ \ldots\ , x_n] &= \frac{1}{n! h^n} \Delta^n y_0 +\end{align*} + +\setcounter{all}{8} +\fancytheorem{Newton} Falls $\beta_j = y[x_0,\ \ldots\ , x_j]$ geht das resultierende Polynom durch alle $(x_i,y_i)$.\\ +\footnotesize +(D.h. die dividierten Differenzen sind korrekt.) +\normalsize + + +\newpage +\begin{multicols}{2} + + Matrixmultiplikation in $\mathcal{O}(n^3)$, Speicher $\mathcal{O}(n^2)$ + + \begin{code}{python} + # Slow matrix approach + def divdiff(x,y): + n = y.size + T = np.zeros((n,n)) + T[:,0] = y + + for l in range(1,n): + for i in range(n-l): + T[i, l] = (T[i+1,l-1] - T[i, l-1]) + T[i, l] /= (x[i+l] - x[i]) + \end{code} + + \newcolumn + + % Add the vectorized HW solution here + + Vektorisierter Ansatz in $\mathcal{O}(n^2)$, Speicher $\mathcal{O}(n)$ + + \begin{code}{python} + # NOT THE CORRECT CODE YET + def divdiff(x,y): + n = y.size + T = np.zeros((n,n)) + T[:,0] = y + + for l in range(1,n): + for i in range(n-l): + T[i, l] = (T[i+1,l-1]-T[i, l-1]) + T[i, l] /= (x[i+l] - x[i]) + \end{code} + +\end{multicols} + +\subsubsection{Auswertung} + +Auswertung eines Newton-Polynoms funktioniert in $\mathcal{O}(n)$ durch ein modifiziertes Horner-Schema: +\begin{align*} + p_0 &:= \beta_n \\ + p_1 &:= (x - x_{n-1})p_0 + \beta_{n-1} \\ + p_2 &:= (x - x_{n-2})p_1 + \beta_{n-2} \\ + \vdots \\ + p_n &= p(x) +\end{align*} + +\subsubsection{Fehler} + +\setcounter{all}{11} +\inlinetheorem $f$ $n$-mal diff.-bar, $y_i = f(x_i) \implies \exists \xi \in (\min_i x_i, \max_i x_i)$ s.d. $y[x_0,x_1,\ldots,x_n] = \frac{f^{(n)}(\xi)}{(n+1)!}$ + +\fancytheorem{Fehler} $f: [a,b] \to \R$ ist $(n+1)$-mal diff.-bar, $p$ ist das Polynom zu $f$ in $x_0,\ldots,x_n \in [a,b]$. +$$ + \forall x \in [a,b]\ \exists \xi \in (a,b):\quad\quad \underbrace{f(x)-p(x)}_{\text{Fehler}} = \prod_{i=0}^{n}(x-x_i)\cdot\frac{f^{(n+1)}(\xi)}{(n+1)!} +$$ + +Man bemerke: Die Wahl der Stützpunkte hat direkten Einfluss auf den Fehler. + +