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\begin{code}{python} # Slow matrix approach - def divdiff(x,y): + def divdiff_slow(x,y): n = y.size T = np.zeros((n,n)) T[:,0] = y @@ -104,6 +104,8 @@ Falls $x_j = x_0 + \underbrace{j \cdot h}_{:= \Delta^j}$ gilt vereinfacht sich e 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]) + + return T[0,:] \end{code} \newcolumn @@ -113,16 +115,15 @@ Falls $x_j = x_0 + \underbrace{j \cdot h}_{:= \Delta^j}$ gilt vereinfacht sich e 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 + # Fast vectorized approach + def divdiff_fast(x,y): + n = y.shape[0] - 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]) + for k in range(1, n): + y[k:] = (y[k:] - y[(k-1):n-1]) + y[k:] /= (x[k:] - x[0:n-k]) + + return y \end{code} \end{multicols} @@ -130,6 +131,9 @@ Falls $x_j = x_0 + \underbrace{j \cdot h}_{:= \Delta^j}$ gilt vereinfacht sich e \subsubsection{Auswertung} Auswertung eines Newton-Polynoms funktioniert in $\mathcal{O}(n)$ durch ein modifiziertes Horner-Schema: + +\begin{multicols}{2} + \begin{align*} p_0 &:= \beta_n \\ p_1 &:= (x - x_{n-1})p_0 + \beta_{n-1} \\ @@ -138,6 +142,22 @@ Auswertung eines Newton-Polynoms funktioniert in $\mathcal{O}(n)$ durch ein modi p_n &= p(x) \end{align*} +\newcolumn + +\begin{code}{python} + def evalNewton(x_data, beta, x): + p = np.zeros(x.shape[0]) + p += beta[beta.shape[0]-1] + + for i in range(1, n+1): + p = (x - x_data[n-i])*p + beta[n-i] + + return p +\end{code} + +\end{multicols} + + \subsubsection{Fehler} \setcounter{all}{11}