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zcM>t#lJA}(f}FO3ECj+RYQigSpr(KQpfq{N37l1dAM403r8Kqt_?Nm2zM(`gPSSn) zr{jN2A#3?5CjToXgqpz&36OdT(B7&L(v<-X6o$`rb{C@fY_83q%NGDC3OgG&8xIl< KjkJmk(*FU&CCd*0 diff --git a/semester3/numcs/numcs-summary.tex b/semester3/numcs/numcs-summary.tex index 832b69d..bd3c31f 100644 --- a/semester3/numcs/numcs-summary.tex +++ b/semester3/numcs/numcs-summary.tex @@ -119,5 +119,12 @@ Moral of the story: Use descriptive variable names and do NOT use $t$, $tt$, $tt \input{parts/01_interpolation/01_trigonometric/03_interpolation/01_zero-padding.tex} \input{parts/01_interpolation/01_trigonometric/04_error-estimation.tex} +% ── piece-wise interpolation ──────────────────────────────────────── +\newsection +\section{Stückweise Polynomiale Interpolation} +\input{parts/01_interpolation/02_piece-wise/00_intro.tex} +\input{parts/01_interpolation/02_piece-wise/01_hermite-interpolation.tex} +\input{parts/01_interpolation/02_piece-wise/02_splines.tex} + \end{document} diff --git a/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex b/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex index 6a26663..a6f81ba 100644 --- a/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex +++ b/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex @@ -1 +1,36 @@ % Lecture: \chi here are used as RELU function! +Globale Interpolation (also Interpolation auf dem ganzen Intervall $]-\infty, \infty[$) funktioniert nur dann gut, wenn: +\rmvspace +\begin{enumerate}[label=(\alph*), noitemsep] + \item die gegebenen Interpolationspunkte als Chebyshev-Knoten oder -Abszissen verwendet werden können + \item die Funktion glatt ist +\end{enumerate} +Es müssen beide obige Eigenschaften zutreffen. +Eine Idee um die Einschränkungen zu reduzieren oder komplett zu entfernen ist es, das Intervall zu unterteilen, oder formaler, +das Intervall $I = [a, b]$ in viele kleinere Intervalle zu zerlegen. + +Wir haben dann ein Polynom vom Grad $n$ auf jedem Teilintervall mit $n + 1$ Punkten, was den Fehler verringert: +\begin{align*} + |f(x) - s(x)| < \frac{h^{n + 1}}{(n + 1)!} ||f^{(n + 1)}||_{\infty} +\end{align*} + +Seien $N + 1$ Messpunkte gegeben. Wir verwenden sie als Knoten +(im Englischen \textit{breakpoints} gennant. Die Knoten sind also nicht dasselbe wie in den vorigen Kapiteln, es gibt aber keinen wirklich sinnvollen Namen im Deutschen) +diese $N + 1$ Messpunkte. Die Knoten dienen Paarweise als Abgrenzung der neuen, kleinen Intervalle, die wir erstellt haben. +Die linearen Interpolanten für jedes Intervall sind (mit $h_j = x_j - x_{j - 1}$): +\begin{align*} + s_j(x) = y_{j - 1}\frac{x_j - x}{h_j} + y_j \frac{x - x_{j - 1}}{h_j} \mediumhspace \text{für } x \in [x_{j - 1}, x_j] +\end{align*} +% NOTE: This might just be little more than a page in the script, but he talked about it for like 20 min in the lecture... +% and he even went as far as telling people to really pay attention... +% so I guess that means it is *not* important (could well not be important actually) +Wie man nun zu dieser Formel kommt: +Sei $\chi(t) = t \smallhspace \forall t \in [0, 1]$. +Die Funktion $f(t) = y_0 \chi(1 - t) + y_1 \chi(t)$ hat also die Interpolationseigenschaften $f(0) = y_0$ und $f(1) = y_1$ und ist linear in $t$. +Die Interpolation $s_j(x)$ auf $[x_{j - 1}, x_j]$ entsteht dann also aus $f$ mit Variablenwechsel $t = \frac{x - x_{j - 1}}{h_j} \in [0, 1] \leftrightarrow x = x_{j - 1} + h_j t$, +also gilt: +\begin{align*} + s_j(x) = y_{j - 1} \chi \left( \frac{x_j - x}{h_j} \right) + y_j \chi \left( \frac{x - x_{j - 1}}{h_j} \right) \mediumhspace \text{für } x \in [x_{j - 1}, x_j] +\end{align*} +Dies ist eine lokale Interpolation und $s_j$ ist $0$ ausser im definierten Intervall. +Die Idee des Variablenwechsel ist es, das Intervall, auf welchem die Funktion definiert ist von $[0, 1]$ nach $[x_{j - 1}, x_j]$ zu verschieben. diff --git a/semester3/numcs/parts/01_interpolation/02_piece-wise/01_hermite-interpolation.tex b/semester3/numcs/parts/01_interpolation/02_piece-wise/01_hermite-interpolation.tex index 2acc7a3..fa619ec 100644 --- a/semester3/numcs/parts/01_interpolation/02_piece-wise/01_hermite-interpolation.tex +++ b/semester3/numcs/parts/01_interpolation/02_piece-wise/01_hermite-interpolation.tex @@ -1,2 +1,43 @@ -% 3rd order polynomial has 4 degrees of freedom (a, b, c and d), we have 4 coefficients for cubic hermite interpolation -% CHIP algorithm ist formerhaltend +\subsection{Kubische Hermite-Interpolation} +Die Kubische Hermite-Interpolation (CHIP) produziert eine auf $[a, b]$ stetig differenzierbare Funktion, welche auf den Teilintervallen $[x_{j - 1}, x_j]$ jeweils ein Polynom von Grad 3 ist. +Wichtige Eigenschaft von Polynomen $n$-ten Grades ist, dass sie $n + 1$ Freiheitsgrade haben (da sie $n + 1$ freie Variabeln enthalten). + +Nutzen wir wieder das Konzept von oben, und wählen eine Funktion $\varphi(t) = t^2(3 - 2t)$ für $t \in [0, 1]$, +so erfüllt $f(t) = y_0\varphi(1 - t) + y_1 \varphi(t)$ wieder unsere Interpolationseigenschaften $f(0) = y_0$ und $f(1) = y_1$ und wir vollziehen denselben Variablenwechsel wie oben. +So erhalten wir: +\begin{align*} + p_j(x) = y_{j - 1} \varphi \left( \frac{x_j - x}{h_j} \right) + y_j \varphi \left( \frac{x - x_{j - 1}}{h_j} \right) \mediumhspace \text{für } x \in [x_{j - 1}, x_j] +\end{align*} +Wir haben folgende Ableitungen: $\varphi'(t) = 6t(1 - t)$, also sind die Nullstellen dieser Funktion bei $t \in \{ 0, 1 \}$, +weshalb auch die Ableitungen von $p_j$ an den Stellen $x_{j - 1}$ und $x_j$ verschwinden. + +Für die Ableitungen definieren wir eine zweite Funktion $\psi(t) = t^2(t - 1)$, welche offensichtlich die Nullstellen an $t \in \{ 0, 1 \}$ hat +und deren Ableitung $\psi'(t) = t(3t - 2)$. Mit demselben Variablenwechsel müssen wir die Kettenregel beachten: +\begin{align*} + q_j(x) = c_{j - 1} h_j \psi \left( \frac{x - x_{j - 1}}{h_j} \right) - c_j h_j \psi \left( \frac{x_j - x}{h_j} \right) \mediumhspace \text{für } x \in [x_{j - 1}, x_j] +\end{align*} +Die Interpolationsfunktion ist dann einfach die Summe $s_j(x) = p_j(x) + q_j(x) \mediumhspace \text{für } x \in [x_{j - 1}, x_j]$ + +\fhlc{Cyan}{In Numpy} verwendet man \texttt{scipy.interpolate.Akima1DInterpolator} oder \texttt{PchipInterpolator}, welcher ``formerhaltender'' ist, +also wenn eine Funktion lokal monoton ist, so ist der Interpolant dort auch monoton. +Bei anderen Interpolationsmethoden ist dies nicht garantiert (so auch nicht beim \texttt{Akima1DInterpolator}) + +\begin{footnotesize} + \textit{Wenn man den Parameter \texttt{method="makima"} bei \texttt{Akima1DInterpolator} mitgibt, wird eine neuere modifizierte Variante davon ausgeführt} +\end{footnotesize} + +\stepcounter{all} +\begin{theorem}[]{Fehler der CHIP} + % FIXME: Explain what the C is or if it is \C, then replace + Sei $f \in C^4[a, b]$ und $s$ der stückweise CHIP mit exakten Werten der Ableitungen $s'(x_j) = f'(x_j), s(x_j) = f(x_j)$ für $j = 0, \ldots, N$ + % und sei $s_{|[x_{j - 1, x_j}]|}$ ein Polynom vom Grad $3$, für $j = 1, \ldots, N$. + und sei $s_j$ ein Polynom vom Grad $3$, für $j = 1, \ldots, N$. Dann gilt: + \rmvspace + \begin{align*} + ||f^{(k)} - s^{(k)}||_{L^\infty} \leq \frac{1}{384} h^{4 - k} ||f^{(4)}||_{L^\infty} + \end{align*} + + \rmvspace + mit $h = \max_{j = 1, \ldots, N} (x_j - x_{j - 1})$ und $k = 0, 1$ +\end{theorem} +% FIXME: There is an important *looking* remark below this theorem... only issue... half of it is missing... Gemmini gobble gobble diff --git a/semester3/numcs/parts/01_interpolation/02_piece-wise/02_splines.tex b/semester3/numcs/parts/01_interpolation/02_piece-wise/02_splines.tex new file mode 100644 index 0000000..e69de29

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