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Falls $N > 2m$, dann gilt $p_N(x) = f(x) \smallhspace \forall x$ +\numberingOff \inlineex Ein Beispiel aus der Musik: Wir haben ein analoges Signal und wollen es digitalisieren. Wir messen die Spannungswerte in äquidistanten Punkten. Falls wir jedoch die Frequenz der Messung zu niedrig wählen, so kann ein total falsches Interpolationspolynom entstehen, wie in der untenstehenden Abbildung \ref{fig:aliasing} zu sehen: +\numberingOn \begin{figure}[h!] \begin{center} \includegraphics[width=0.95\textwidth]{assets/01_interpolation/01_trigonometric/aliasing-in-music.png} diff --git a/semester3/numcs/parts/02_quadrature/02_non-equidistant/01_clenshaw-curtis.tex b/semester3/numcs/parts/02_quadrature/02_non-equidistant/01_clenshaw-curtis.tex index e50de18..71488f8 100644 --- a/semester3/numcs/parts/02_quadrature/02_non-equidistant/01_clenshaw-curtis.tex +++ b/semester3/numcs/parts/02_quadrature/02_non-equidistant/01_clenshaw-curtis.tex @@ -5,4 +5,33 @@ und Clenshaw und Curtis haben dann zusätzlich noch die Endknoten hinzugefügt ( Die Clenshaw-Curtis-Knoten sind die Chebyshev-Abszissen und die Formel verhält sich mit den entsprechenden Gewichten ähnlich gleich wie die Gauss-Quadratur. Da die Clenshaw-Curtis-Quadratur mithilfe der DFT berechnet werden kann ist sie sehr effizient. -Dazu müssen wir aber zuerst etwas umformen +Dazu müssen wir aber zuerst etwas umformen, mit $x = \cos(\theta)$, so dass das Integral eine periodische Funktion wird: +\rmvspace +\begin{align*} + \int_{-1}^{1} f(x) \dx x = \int_{0}^{\pi} f(\cos(\theta)) \sin(\theta) \dx \theta = f(\cos(\theta)) +\end{align*} + +\drmvspace +$F(\theta)$ ist $2\pi$-periodisch und gerade, kann sich also in eine Kosinius-Reihe entwickeln, also: $F(\theta) = \sum_{k = 0}^{\infty} a_k \cos(k \theta)$, woraus folgt, dass +\drmvspace +\begin{align*} + \int_{0}^{\pi} F(\theta) \sin(\theta) \dx \theta = \ldots = a_0 + \sum_{2 \leq k \text{ gerade}} \frac{2a_k}{1 - k^2} +\end{align*} + +\drmvspace +wobei sich die Koeffizienten $a_k$ mit FFT oder DCT berechnen lassen + +% TODO: Insert code from TA slides here + +Eine wichtige Erkenntnis ist, dass die Newton-Cotes bei grösserer Ordnung komplett unbrauchbar werden, wie das in Abbildung 5.5.24 im Skript zu sehen ist, +während die Clenshaw-Curtis-Quadratur ähnlich gut ist wie die Gauss-Quadratur (gleiche Konvergenzordnung). + +\begin{fullTable}{llllll}{Quadratur & Intervall & Gewichtsfunktion & Polynom & Notation & \texttt{scipy.special.}} + {Gewichtsfunktionen für Quadraturformeln} + Gauss & $(-1, 1)$ & $1$ & Legendre & $P_k$ & \texttt{roots\_legendre} \\ + Chebyshev I & $(-1, 1)$ & $\frac{1}{\sqrt{1 - x^2}}$ & Chebyshev I & $T_k$ & \texttt{roots\_chebyt} \\ + Chebyshev II & $(-1, 1)$ & $\sqrt{1 - x^2}$ & Chebyshev II & $U_k$ & \texttt{roots\_chebyu} \\ + Jacobi $\alpha, \beta > 1$ & $(-1, 1)$ & $(1 - x)^\alpha (1 + x)^\beta$ & Jacobi & $P_k^{(\alpha, \beta)}$ & \texttt{roots\_jacobi} \\ + Hermite & $\R$ & $e^{-x^2}$ & Hermite & $H_k$ & \texttt{roots\_hermite} \\ + Laguerre & $(0, \infty)$ & $x^\alpha e^{-x^2}$ & Laguerre & $L_k$ & \texttt{roots\_genlaguerre} \\ +\end{fullTable} diff --git a/semester3/numcs/parts/02_quadrature/03_adaptive.tex b/semester3/numcs/parts/02_quadrature/03_adaptive.tex index e41dac5..ffff2b5 100644 --- a/semester3/numcs/parts/02_quadrature/03_adaptive.tex +++ b/semester3/numcs/parts/02_quadrature/03_adaptive.tex @@ -1,2 +1,4 @@ \newsection \subsection{Adaptive Quadratur} +Der lokale Fehler einer zusammengesetzten Quadraturformel auf dem Gitter $\mathcal{M} := \{ a = x_0 < x_1 < \dots < x_m = b \}$ ist (für $f \in C^2([a, b])$). +Test