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\end{fullTable} \drmvspace -\inlineremark Für das \textit{a posteriori} Abbruchkriterium mit linearer Konvergenz und bekanntem $L$ gilt folgende Abschätzung: -\drmvspace -\begin{align*} - ||x^{(k + 1)} - x^*|| \leq \frac{L}{1 - L} ||x^{(k + 1)} - x^{(k)}|| -\end{align*} +\inlineremark Für das \textit{a posteriori} Abbruchkriterium mit linearer Konvergenz und bekanntem $L$ gilt die Abschätzung aus Lemma \ref{all:6-3-6} mit Korollar \ref{all:6-3-17} diff --git a/semester3/numcs/parts/03_zeros/02_fix-point-iteration.tex b/semester3/numcs/parts/03_zeros/02_fix-point-iteration.tex index 29b60c4..42bf3da 100644 --- a/semester3/numcs/parts/03_zeros/02_fix-point-iteration.tex +++ b/semester3/numcs/parts/03_zeros/02_fix-point-iteration.tex @@ -1,7 +1,66 @@ -\drmvspace\drmvspace -\newsectionNoPB +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + +\rmvspace\newsectionNoPB \subsection{Fixpunktiteration} Ein $1$-Punkt-Verfahren benötigt nur den vorigen Wert: $x^{(k + 1)} = \phi(x^{(k)})$ % FIXME: Below konsistent is probably wrong, but is what is in the script \inlinedef Eine Fixpunktiteration heisst konsistent mit $F(x) = 0$ falls $F(x) = 0 \Leftrightarrow \phi(x) = x$ + +\inlineex Für $F(x) = xe^x - 1$ mit $x \in [0, 1]$ liefert $\phi_1(x) = e^{-x}$ lineare Konvergenz, +$\phi_2(x) = \frac{1 + x}{1 + e^x}$ quadratische Konvergenz und $\phi_3(x) = x + 1 - xe^x$ eine divergente Folge. + +\setLabelNumber{all}{5} +\fancydef{Kontraktion} $\phi$ falls es ein $L < 1$ gibt, so dass $||\phi(x) - \phi(y)|| \leq L||x - y|| \ \forall x, y$ + +\inlineremark Falls $x^*$ ein Fixpunkt der Kontraktion $\phi$ ist, dann ist +\drmvspace +\begin{align*} + ||x^{(k + 1)} - x^*|| = ||\phi(x^{(k)}) - \phi(x^*)|| \leq L||x^{(k)} - x^*|| +\end{align*} + +\drmvspace +\begin{theorem}[]{Banach'scher Fixpunktsatz} + Sei $D \subseteq \K^n$ ($\K = \R, \C$) mit $D$ abgeschlossen und $\phi: D \rightarrow D$ eine Kontraktion. + Dann existiert ein eindeutiger Fixpunkt $x^*$, für welchen also gilt, dass $\phi(x^*) = x^*$. + Dieser ist der Grenzwert der Folge $x^{(k + 1)} = \phi(x^{(k)})$. +\end{theorem} + +% NOTE: If need be, we can switch to theorem here, or I can add a new environment for "support theorem" or the like, +% I however feel like a Lemma suits the idea of "Hilfstheorem" quite well +\inlinelemma Für $U \subseteq \R^n$ konvex und $\phi : U \rightarrow \R^n$ stetig differenzierbar mit $L := \sup_{x \in U} ||D_\phi(x)|| < 1$ +($D_\phi(x)$ ist die Jacobi-Matrix von $\phi(x)$). +Wenn $\phi(x^*) = x^*$ für $x^* \in U$, dann konvergiert $x^{(k + 1)} = \phi(x^{(k)})$ gegen $x^*$ lokal mindestens linear. +Dies ist eine hinreichende (= sufficient) Bedingung. + +\setLabelNumber{all}{11} +\inlinelemma Für $\phi : \R^n \rightarrow \R^n$ mit $\phi(x^*) = x^*$ und $\phi$ stetig differenzierbar in $x^*$. +Ist $||D_\phi(x^*)|| < 1$, dann konvergiert $x^{(k + 1)} = \phi(x^{(k)})$ lokal und mindestens linear mit $L = ||D_\phi(x^*)||$ + +\stepLabelNumber{all} +\fancytheorem{Satz von Taylor} Sei $I \subseteq \R$ ein Intervall, $\phi : I \rightarrow \R$ $(m + 1)$-mal differenzierbar und $x \in I$. +Dann gilt für jedes $y \in I$ +\drmvspace +\begin{align*} + \phi(y) - \phi(x) = \sum_{k = 1}^{m} \frac{1}{k!} \left( \phi^{(k)}(x) (y - x)^k \right) + \tco{|y - x|^{m + 1}} +\end{align*} + +\drmvspace +\inlinelemma Sei $I$ und $\phi$ wie in Satz \ref{all:6-3-13}. Sei zudem $\phi^{(l)}(x^*) = 0$ für $l \in \{ 1, \ldots, m \}$ mit $m \geq 1$. +Dann konvergiert $x^{(k + 1)} = \phi(x^{(k)})$ lokal gegen $x^*$ mit Ordnung $p \geq m + 1$ + +\stepLabelNumber{all} +\inlinelemma Konvergiert $\phi$ linear mit $L < 1$, dann gilt: +\drmvspace +\begin{align*} + ||x^{*} - x^{(k)}|| \leq \frac{L^{k - l}}{1 - L} ||x^{(l + 1)} - x^{(l)}|| +\end{align*} + +\drmvspace +\inlinecorollary für $l = 0$ haben wir ein \textit{a priori} und für $l = k - 1$ ein \textit{a posteriori} Abbruchkriterium: +\drmvspace +\begin{align*} + ||x^* - x^{(k)}|| \leq \frac{L^k}{1 - L} ||x^{(1)} - x^{(0)}|| \leq \tau & & ||x^* - x^{(k)}|| \leq \frac{L}{1 - L} ||x^{(k)} - x^{(k - 1)}|| \leq \tau +\end{align*} diff --git a/semester3/numcs/parts/03_zeros/03_interval-splitting.tex b/semester3/numcs/parts/03_zeros/03_interval-splitting.tex new file mode 100644 index 0000000..551a905 --- /dev/null +++ b/semester3/numcs/parts/03_zeros/03_interval-splitting.tex @@ -0,0 +1,2 @@ +\newsection +\subsection{Intervallhalbierungsverfahren}