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zoYp3N8H%%9HyiGa?C-U0HT$X9#JZQ_oj?5%abGttAa;ZOu#jO8qmpjOHwk!lsYR?? z{2yp;SULpoHpvqaly&kKx#@8Bx9+@V=a{9mPAyBNW22CkkbMjWaC1Xms00276qV^p diff --git a/semester3/numcs/numcs-summary.tex b/semester3/numcs/numcs-summary.tex index 87de4f3..33050fb 100644 --- a/semester3/numcs/numcs-summary.tex +++ b/semester3/numcs/numcs-summary.tex @@ -132,6 +132,7 @@ Moral of the story: Use descriptive variable names and do NOT use $t$, $tt$, $tt \newsection \section{Numerische Quadratur} \input{parts/02_quadrature/00_introduction.tex} +\input{parts/02_quadrature/01_equidistant-nodes.tex} \end{document} diff --git a/semester3/numcs/parts/02_quadrature/00_introduction.tex b/semester3/numcs/parts/02_quadrature/00_introduction.tex index 09eac69..c7f4d6f 100644 --- a/semester3/numcs/parts/02_quadrature/00_introduction.tex +++ b/semester3/numcs/parts/02_quadrature/00_introduction.tex @@ -21,7 +21,7 @@ Wir wollen natürlich wieder $c_i^n \in [a, b]$ und $w_i^n$ so wählen, dass der \begin{align*} E(n) = \left| \int_{a}^{b} f(x) \dx - Q_n(f; a, b) \right| \end{align*} - Wir haben \bi{algebraische Konvergenz} wenn $E(n) = \tco{\frac{1}{n^p}}$ mit $p > 0$ und + Wir haben \bi{algebraische Konvergenz} wenn $E(n) = \tco{\frac{1}{n^p}}$ mit $p > 0$ und \bi{exponentielle Konvergenz} wenn $E(n) = \tco{q^n}$ mit $0 \leq q < 1$ \end{definition} @@ -58,4 +58,90 @@ Falls wir $c_k(x) = x^k$ haben (was oft der Fall ist, je nach Funktion aber kön Für ein Beispiel verweisen wir auf Beispiel \ref{all:2-3-2} \setLabelNumber{all}{6} -\fancyremark{Eigenschaften der Lagrange-Polynome} Zusätzlich zu den Eigenschaften in \ref{all:2-3-4} ist folgende Eigenschaft erwähnenswert +\fancyremark{Eigenschaften der Lagrange-Polynome} Zu den Eigenschaften aus \ref{all:2-3-4} fügen wir an (die Eigenschaften aus Bemerkung \ref{all:2-3-4} sind hier erneut aufgeführt) +\begin{multicols}{2} + \begin{enumerate} + \item $l_i(x_j) = 0 \smallhspace \forall j \neq i$ + \item $l_i(x_i) = 1 \smallhspace \forall i$ + \item $\deg(l_i) = n \smallhspace \forall i$ + \item $\sum_{k = 0}^{n} l_k(x) = 1 \smallhspace \forall x \in \R$ + \item $\sum_{k = 0}^{n} l_k^{(m)}(x) = 0 \text{ für } m > 0$ + \item $l_0, l_1, \ldots, l_n$ bilden Basis von $\mathcal{P}_{n + 1}$ + \end{enumerate} +\end{multicols} + +\vspace{-1.5pc} +wobei $\mathcal{P}_{n + 1}$ der Raum der Polynome von Grad maximal $n$ ist. + +\fancyremark{Quadraturgewichte aus den Lagrange-Polynomen} +Das Interpolationspolynom ist gegeben durch: +\drmvspace +\begin{align*} + p(x) = \sum_{j = 0}^{n} f(x_j) l_j(x) +\end{align*} + +\drmvspace +Durch die Eigenschaften der Lagrange-Polynome haben wir $p(x_j) = f(x_j)$ und die Konstruktion von $p(x)$ ist eindeutig in $\mathcal{P}_{n + 1}$. +Wir erhalten nun eine Quadraturformel, wenn wir $p$ als Approximation von $f$ verwenden: +\rmvspace +\begin{align*} + w_j = \int_{a}^{b} l_j(x), \smallhspace j = 0, 1, \ldots, n +\end{align*} + +\drmvspace +Durch die Konstruktion der Formel ist sie exakt für alle Polynome aus $\mathcal{P}_{n + 1}$ und der Fehler ist: +\rmvspace +\begin{align*} + \left| \int_{a}^{b} f(x) \dx x - \int_{a}^{b} p_n(x) \dx x \right| \leq \frac{1}{n!}(b - a)^{n + 1} \max|f^{(n)}(z)| +\end{align*} + +\drmvspace +Wir wollen also ein kleines Intervall (oft $b - a < 1$ da wir so das Integral besser approximieren können) und wir setzen voraus, dass $f$ glatt ist. + +Da wir aber oft ein grösseres Intervall betrachten möchten, ist ein möglicher Ansatz, das grosse Intervall in kleinere Intervalle zu zerlegen. +Wir nehmen ein äquidistantes Gitter, mit $x_k = x_0 + k \cdot h$ für $h = \frac{b - a}{N}$ und $k = 0, \ldots, N$: +\rmvspace +\begin{align*} + \int_{a}^{b} f(x) \dx x = \sum_{k = 0}^{N - 1} \int_{x_k}^{x_{k + 1}} f(x) \dx x +\end{align*} + +\drmvspace +Die obige Formel wird auch die \textit{summierte} Quadraturformel genannt. Der Fehler ist dann also: +\rmvspace +\begin{align*} + \left| \int_{a}^{b} f(x) \dx x - \sum_{k = 0}^{N - 1} Q(f, x_k, x_{k + 1}) \right| \leq \ldots \leq C \frac{h^n}{n!}(b - a) & & + \text{ mit } C = \max_{z \in [a, b]} |f^{(n)}(z)| = ||f^n||_{\max} +\end{align*} + +\rmvspace +Der obige Ansatz ist gewissermassen ``divide and conquer'' (zu Deutsch: ``Teile und Herrsche'', wir werden aber DnC verwenden) +und wir der lokale Fehler liegt in $\tco{h^{n + 1}}$ und mit $N = (b - a) \div h$ Intervallen der Grösse $h$ haben wir einen globalen Fehler in $\tco{h^n}$. +Folglich ist also der Fehler kleiner, je kleiner $h$ ist. + +Wir benutzen erneut einen Variablenwechsel, um von einem Referenzintervall $[-1, 1]$ auf eines unserer Teilintervalle $[x_k, x_{k + 1}]$ zu wechseln. +Dies heisst also allgemein für Intervall $[a, b]$ nach $[-1, 1]$: +\begin{align*} + \int_{a}^{b} f(t) \dx t = \frac{1}{2} (b - a) \int_{-1}^{1} \hat{f}(\tau) \dx \tau & \text{ mit } \hat{f}(\tau) := f\left( \frac{1}{2}(1 - \tau) a + \frac{1}{2}(\tau + 1) b \right) +\end{align*} +Für dieses Referenzintervall können wir die Gewichte $\hat{w}_j$ und die Knoten $\hat{c}_j$ bestimmen. +% OMG, wtf, why can't he decide on using w, \omega or \hat{w} for the weights in the reference interval? That is so dumb. +% FIXME: Choose uniform naming (advocating for \hat{w}_j on the reference intervall and w_j on the [a, b] interval) +\rmvspace +\begin{align*} + \int_{a}^{b} f(t) \dx t \approx \frac{1}{2}(b - a) \sum_{j = 1}^{n} \hat{w}_j \hat{f}(\hat{c}_j) = \sum_{j = 1}^{n} w_j f(c_j) + & & \text{ mit } {c_j = \frac{1}{2} (1 - \hat{c}_j) a + \frac{1}{2}(1 + \hat{c}_j) b \atop{w_j = \frac{1}{2}(b - a)\hat{w}_j}} +\end{align*} + +\rmvspace\rmvspace +\inlinedef Die Ordnung einer Quadraturformel ist $n$ wenn sie Polynome vom Grad $(n - 1)$ exakt integriert. + +Dies folgt natürlich direkt davon, dass wir ein Polynom $n$-ten Grades mit $n + 1$ Koeffizienten darstellen können. + +\fancydef{Symmetrie} Eine Quadraturformel auf $[-1, 1]$ heisst symmetrisch, falls $\omega_i = \omega_{n + 1 - i}$ und \\ +$c_i = -c_{n + 1 -i}$ gilt für die Gewichte $\omega_i$ und Knoten $c_i$ + +\inlineremark Die Mittelpunkts-, Trapez- und Simpson-Regeln aus Abschnitt \ref{sec:equidistant-nodes} sind symmetrisch + +\inlinetheorem Die Ordnung einer symmetrischen Quadraturformel ist gerade + +\inlineproof Kann mittels Induktion bewiesen werden, siehe dazu Seite 123 im Skript diff --git a/semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex new file mode 100644 index 0000000..595128a --- /dev/null +++ b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex @@ -0,0 +1,3 @@ +\newsection +\subsection{Äquidistante Punkte} +\label{sec:equidistant-nodes} diff --git a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex index 97e8c7d..e841b11 100644 --- a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex +++ b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex @@ -1,4 +1,4 @@ -\newsection +\newpage \subsection{Nichtdeterminismus} Einfach gesagt werden hier Automaten behandelt, die zufällige (genannt \bi{nichtdeterministische}) Entscheidungen treffen. 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