From 05338790545e2ac3582ef78a12df3087e49c14e2 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 30 Sep 2025 18:53:42 +0200 Subject: [PATCH] [NumCS] Catch up to current state --- .../analysis-ii/parts/diffeq/00_intro.tex | 2 +- semester3/numcs/numcs-summary.pdf | Bin 444345 -> 496601 bytes semester3/numcs/numcs-summary.tex | 13 +- .../00_introduction/00_rounding-errors.tex | 4 + .../00_introduction/01_time-complexity.tex | 4 + .../02_matrix-multiplication.tex | 4 + .../00_polynomial/00_intro.tex | 6 + .../03_lagrange-and-barzycentric-formula.tex | 6 +- .../04_chebychev-interpolation.tex | 9 -- .../04_chebyshev-interpolation.tex | 141 ++++++++++++++++++ .../01_trigonometric/00_fourier.tex | 97 ++++++++++++ .../01_trigonometric/00_intro.tex | 3 - 12 files changed, 270 insertions(+), 19 deletions(-) delete mode 100644 semester3/numcs/parts/01_interpolation/00_polynomial/04_chebychev-interpolation.tex create mode 100644 semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex create mode 100644 semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex delete mode 100644 semester3/numcs/parts/01_interpolation/01_trigonometric/00_intro.tex diff --git a/semester3/analysis-ii/parts/diffeq/00_intro.tex b/semester3/analysis-ii/parts/diffeq/00_intro.tex index fe40fc7..7dd78c4 100644 --- a/semester3/analysis-ii/parts/diffeq/00_intro.tex +++ b/semester3/analysis-ii/parts/diffeq/00_intro.tex @@ -1,6 +1,6 @@ \subsection{Introduction} \shortex $f'(x) = f(x)$ has only solution $f(x) = ae^x$ for any $a \in \R$; -$f' - a = 0$ has only solution $f(x) = \int_{x_0}^{x} a(t) \smallhspace \dx t$ +$f' - a = 0$ has only solution $f(x) = \int_{x_0}^{x} a(t) \dx t$ \setcounter{all}{6} \shorttheorem Let $F: \R^2 \rightarrow \R$ be a differential function of two variables. 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b/semester3/numcs/parts/00_introduction/01_time-complexity.tex @@ -1,3 +1,7 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + \newsection \subsection{Rechenaufwand} In NumCS wird die Anzahl elementarer Operationen wie Addition, Multiplikation, etc benutzt, um den Rechenaufwand zu beschreiben. diff --git a/semester3/numcs/parts/00_introduction/02_matrix-multiplication.tex b/semester3/numcs/parts/00_introduction/02_matrix-multiplication.tex index ce2c059..48eaa8d 100644 --- a/semester3/numcs/parts/00_introduction/02_matrix-multiplication.tex +++ b/semester3/numcs/parts/00_introduction/02_matrix-multiplication.tex @@ -1,3 +1,7 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + \newsection \subsection{Rechnen mit Matrizen} Wie in Lineare Algebra besprochen, ist das Resultat der Multiplikation einer Matrix $A \in \C^{m \times n}$ und einer Matrix $B \in \C^{n \times p}$ ist eine Matrix $AB = \in \C^{m \times p}$ diff --git a/semester3/numcs/parts/01_interpolation/00_polynomial/00_intro.tex b/semester3/numcs/parts/01_interpolation/00_polynomial/00_intro.tex index bad5dd7..a94f6bc 100644 --- a/semester3/numcs/parts/01_interpolation/00_polynomial/00_intro.tex +++ b/semester3/numcs/parts/01_interpolation/00_polynomial/00_intro.tex @@ -1,3 +1,9 @@ +% ┌ ┐ +% │ Author: Robin Bacher │ +% └ ┘ +% TODO: If you want your email to be in there, note it down here. +% I also did not touch the unedited files to avoid conflicts +% FIXME: Add subsection here and use \newsection on all further subsections to reset the counters and add a page break Bei der Interpolation versuchen wir eine Funktion $\tilde{f}$ durch eine Menge an Datenpunkten einer Funktion $f$ zu finden.\\ Die $x_i$ heissen Stützstellen/Knoten, für welche $\tilde{f}(x_i) = y_i$ gelten soll. (Interpolationsbedingung) \begin{align*} diff --git a/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex b/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex index 0ee43bf..648d393 100644 --- a/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex +++ b/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex @@ -1,6 +1,10 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + \newsection \subsection{Lagrange- und Baryzentrische Interpolationsformeln} -Session: Gemäss TA sehr gut beschrieben im alten Script +% Session: Gemäss TA sehr gut beschrieben im alten Script \begin{definition}[]{Lagrange Polynome} Für Knoten (auch gennannt Stützstellen) $x_0, x_1, \ldots, x_n \in \R$ definieren wir die Lagrange-Polynome: diff --git a/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebychev-interpolation.tex b/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebychev-interpolation.tex deleted file mode 100644 index 04c6e57..0000000 --- a/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebychev-interpolation.tex +++ /dev/null @@ -1,9 +0,0 @@ -\newsection -\subsection{Chebychev Interpolation} - -Session: Chebyshev Pol. : Abzisse = Extrema, Knoten = Nullstellen - -Lecture: Orthogonalität ist eine wichtige Eigenschaft: Siehe Lecture notes (handgeschr.) für Veranschaulichung. \\ -$\rightarrow$ Orth. liefert die Koeff. ohne Rechenaufwand. - -Lecture: Clenshaw-Alg. relativ zentral (Taschenrechner nutzen diesen intern) diff --git a/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex b/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex new file mode 100644 index 0000000..c6f7aa2 --- /dev/null +++ b/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex @@ -0,0 +1,141 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + +\newsection +\subsection{Chebyshev Interpolation} + +% Session: Chebyshev Pol. : Abszisse = Extrema, Knoten = Nullstellen +% +% Lecture: Orthogonalität ist eine wichtige Eigenschaft: Siehe Lecture notes (handgeschr.) für Veranschaulichung. \\ +% $\rightarrow$ Orth. liefert die Koeff. ohne Rechenaufwand. +% +% Lecture: Clenshaw-Alg. relativ zentral (Taschenrechner nutzen diesen intern) + + +\begin{definition}[]{Chebyshev-Polynome} + \begin{multicols}{2} + \fhl{Erster Art} + \rmvspace + \begin{align*} + T_n(x) = \cos(n \arccos(x)), \smallhspace x \in [-1, 1] + \end{align*} + + \fhl{Zweiter Art} + \rmvspace + \begin{align*} + U_n(x) = \frac{\sin((n + 1) \arccos(x))}{\sin(\arccos(x))}, \smallhspace x \in [-1, 1] + \end{align*} + \end{multicols} +\end{definition} +$T_n(x)$ scheint erst nicht ein Polynom zu sein, aber wir haben einen $\arccos$ in einem $\cos$. Zudem: + +\stepcounter{all} +\fancytheorem{Eigenschaften} +Das $n$-te Chebyshev-Polynom ist ein Polynom von Grad $n$ und für $x \in [-1, 1]$ gilt: +\begin{multicols}{2} + \begin{enumerate} + \item $T_0(x) = 1, T_1(x) = x$,\\ $T_{n + 1}(x) = 2x T_n(x) - T_{n - 1}(x)$ + \item $|T_n(x)| \leq 1$ + \item $T_n\left(\cos\left( \frac{k\pi}{n} \right)\right) = (-1)^k \text{ für } k = 0, \ldots, n$ + \item $T_n\left( \cos\left( \frac{(2k + 1) \pi}{2n} \right) \right) = 0 \text{ für } k = 0, \ldots, n - 1$ + \end{enumerate} +\end{multicols} + +\fancydef{Chebyshev-Knoten} Die $(n + 1)$ Chebyshev-Knoten $x_0, \ldots, x_n$ im Intervall $[-1, 1]$ sind die Nullstellen von $T_{n + 1}(x)$ + +\fancyremark{Chebyshev-Knoten für beliebiges Intervall} Für $I = [a, b]$ sind die Chebyshev-Knoten: +\rmvspace +\begin{align*} + x_k = a + \frac{1}{2} (b - a) \left( \cos \left( \frac{2k + 1}{2(n + 1)} \pi \right) + 1 \right) \smallhspace k = 0, \ldots, n +\end{align*} + +\fancydef{Chebyshev-Abszissen} Die $(n - 1)$ Chebyshev-Abszissen $x_0, \ldots, x_{n - 2}$ im Intervall $[-1, 1]$ sind die Extrema des Chebyshev-Polynoms $T_n(x)$ und zeitgleich die Nullstellen von $U_{n - 1}(x)$. +Je nach Kontext nimmt man noch die Grenzen des Intervalls ($1$ und $-1$) hinzu und hat dann $(n + 1)$ Abszissen. + +\fancyremark{Chebyshev-Abszissen für beliebiges Intervall} Für $I = [a, b]$ sind die Chebyshev-Abszissen: +\rmvspace +\begin{align*} + x_k = a + \frac{1}{2} (b - a) \left( \cos \left( \frac{k}{n} \pi \right) + 1 \right) \smallhspace k = 0, \ldots, n +\end{align*} +Oder $k = 1, \ldots, n - 1$ bei ausgeschlossenen Endpunkten $a$ und $b$ + +\inlineremark Gegen die Ränder des Intervalls werden die Chebyshev-Knoten dichter. + +\begin{theorem}[]{Orthogonalität} + Die Chebyshev-Polynome sind orthogonal bezüglich des Skalarprodukts + \rmvspace + \begin{align*} + \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \frac{1}{\sqrt{1 - x^2}} \dx x + \end{align*} + + Sie ($T_0, \ldots, T_n$) sind zudem orthogonal bezüglich des diskreten Skalarprodukts im Raum der Polynome von Grad $\leq n$ + \rmvspace + \begin{align*} + (f, g) = \sum_{l = 0}^{n} f(x_l)g(x_l) + \end{align*} + wobei $(x_0, \ldots, x_n)$ die Nullstellen von $T_{n + 1}$ sind. +\end{theorem} + + +% ──────────────────────────────────────────────────────────────────── +\stepcounter{all} +\newpage +\subsubsection{Fehler} +Was hat die neue Verteilung für einen Einfluss auf den Fehler? + +\begin{theorem}[]{Fehlerabschätzung} + Unter allen $(x_0, \ldots, x_n)$ mit $x_i \in \R$ wird (wobei $x_k$ die Nullstellen von $T_{n + 1}$ sind) + \rmvspace + \begin{align*} + \max_{x \in [-1, 1]} |(x - x_0) \cdot \ldots \cdot (x - x_n)| & & \text{minimal für } x_k = \cos \left( \frac{2k + 1}{2(n + 1)}\pi \right) + \end{align*} +\end{theorem} + +Folglich sind also die Nullstellen der Chebyshev-Polynome $T_n$ die bestmögliche Wahl für die Stützstellen. +Da die Abszissen mit FFT einfacher zu berechnen sind, werden diese oft bevorzugt berechnet. +Dies, da die Nullstellen von $T_n$ in den Extrema von $T_{2n}$ enthalten sind, während zudem zwischen zwei nebeneinanderliegenden Chebyshev-Abszissen jeweils eine Nullstelle von $T_{2n}$ liegt + +\stepcounter{all} +\fancytheorem{Lebesgue-Konstante} Für die Chebyshev-Interpolation: $\displaystyle \Lambda_n \approx \frac{2}{\pi} \log(n) \text{ für } n \rightarrow \infty$ + + +% ──────────────────────────────────────────────────────────────────── +\stepcounter{all} +\begin{theorem}[]{Interpolationspolynom} + Das Interpolationspolynom $p$ zu $f$ mit Chebyshev-Knoten gleich der Nullstellen von $T_{n + 1}$ ist gegeben durch + \begin{align*} + p(x) = c_0 + c_1 T_1(x) + \ldots + c_n T_n(x) + \end{align*} + wobei für die $c_k$ gilt: + \begin{align*} + c_k & = \frac{2}{n + 1} \sum_{l = 0}^{n} f\left( \underbrace{\cos \left( \frac{2l + 1}{n + 1} \frac{\pi}{2} \right)}_{= x_i (\text{Knoten})} \right) + \cos \left( k \frac{2l + 1}{n + 1} \frac{\pi}{2} \right) & & \text{für } k = 1, \ldots, n \\ + c_k & = \frac{1}{n + 1} \sum_{l = 0}^{n} f\left( \underbrace{\cos \left( \frac{2l + 1}{n + 1} \frac{\pi}{2} \right)}_{= x_i (\text{Knoten})} \right) + \cos \left( k \frac{2l + 1}{n + 1} \frac{\pi}{2} \right) & & \text{für } k = 0 + \end{align*} +\end{theorem} + +Für $n \geq 15$ berechnet man $c_k$ mit der Schnellen Fourier Transformation (FFT). + +\fancyremark{Laufzeit} Für die Interpolation ergibt sich folgender Aufwand: +\begin{center} + \begin{tabular}{ll} + Direkte Berechnung der $c_k$ & $\tco{(n + 1)^2}$ Operationen \\ + Dividierte Differenzen & $\tco{\frac{n (n + 1)}{2}}$ Operationen (zum Vergleich) \\ + $c_k$ mittels FFT & $\tco{n \log(n)}$ Operationen + \end{tabular} +\end{center} + + +\fancytheorem{Clenshaw-Algorithmus} Seien $d_{n + 2} = d_{n + 1} = 0$. Sei $d_k = c_k + (2x)d_{k + 1} - d_{k + 2} \text{ für } k = n, \ldots, 0$\\ +Dann gilt: $p(x) = \frac{1}{2}(d_0 - d_2)$ und man kann $p(x)$ mit Hilfe einer Rückwärtsrekursion berechnen + +Der Clenshaw-Algorithmus ist sehr stabil, auch wenn er mit (oft) unstabilen Rekursionen implementiert ist. + + + +% ──────────────────────────────────────────────────────────────────── +\subsection{DFT und Chebyshev-Interpolation} +Der folgende Abschnitt wurde ausgelassen, da dieser (noch) nicht während den Vorlesungen behandelt wurde. +Er behandelt primär die Implementation der Chebyshev-Interpolation. diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex new file mode 100644 index 0000000..174b58a --- /dev/null +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex @@ -0,0 +1,97 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + +% Lecture: Wir besitzen nicht das komplette Vorwissen in der Analysis für dieses Kapitel, d.h. wird totales Verständnis nicht +% +% Lecture: Intuitiv wird Fourier-Trans. zur Kompression genutzt, z.b. jpg format. +\subsection{Fourier-Reihen} +Eine Anwendung der Schnellen Fourier-Transformation (FFT) ist die Komprimierung eines Bildes und sie wird im JPEG-Format verwendet. + +\fancydef{Trigonometrisches Polynom von Grad $\leq m$} Die Funktion: +\rmvspace +\begin{align*} + p_m(t) := t \mapsto \sum_{j = -m}^{m} \gamma_j e^{2 \pi ijt} \text{ wobei } \gamma_j \in \C \text{ und } t \in \R +\end{align*} +% +% +\inlineremark $p_m : \R \rightarrow \C$ ist periodisch mit Periode $1$. +Falls $\gamma_{-j} = \overline{\gamma_j}$ für alle $j$, dann ist $p_m$ reellwertig und +% NOTE: Uhh... do we want to use the fancy symbols for real and imaginary part or just use $\text{Re}$? +$p_m$ kann folgendermassen dargestellt werden ($a_0 = 2\gamma_0, a_j = 2\Re(\gamma_j)$ und $b_j = -2\Im(\gamma_j)$): +\rmvspace +\begin{align*} + p_m(t) = \frac{a_0}{2} + \sum_{j = 1}^{m} (a_j \cos(2\pi jt) + b_j \sin(2\pi jt)) +\end{align*} + +\begin{definition}[]{$L^2$-Funktionen} + Wir definieren die $L^2$-Funktionen auf dem Intervall $(0, 1)$ als + \rmvspace + \begin{align*} + L^2(0, 1) := \{ f: (0, 1) \rightarrow \C \divides ||f||_{L^2(0, 1)} < \infty \} + \end{align*} + während die $L^2$-Norm auf $(0, 1)$ durch das Skalarprodukt + \rmvspace + \begin{align*} + \langle g, f \rangle_{L^2(0, 1)} := \int_{0}^{1} \overline{g(x)} f(x) \dx x + \end{align*} + über $||f||_{L^2(0, 1)} = \sqrt{\langle f, f \rangle_{L^2(0, 1)}}$ induziert wird +\end{definition} + +\inlineremark $L^2(a, b)$ lässt sich analog definieren mit +\rmvspace +\begin{align*} + \langle g, f \rangle_{L^2(a, b)} & := \int_{a}^{b} \overline{g(x)} f(x) \dx x \\ + & = (b - a) \int_{0}^{1} \overline{g(a + (b - a)t)} f(a + (b - a)t) \dx t +\end{align*} +In Anwendungen findet sich oft das Intervall $\left[ -\frac{T}{2}, \frac{T}{2} \right]$. +Dann verwandeln sich die Integrale in die Form $\frac{1}{T} \int_{\frac{T}{2}}^{-\frac{T}{2}} (\ldots) \dx t$ und $\exp(2\pi ijt)$ durch $\exp(i \frac{2\pi j}{T} t)$ ersetzt wird. + +\stepcounter{all} +\inlineremark Die Funktionen $\varphi_k(x) = \exp(2\pi ikx)$ sind orthogonal bezüglich des $L^2(0, 1)$-Skalarprodukts, bilden also eine Basis für den Unterraum der trigonometrischen polynome. + + +\inlinedef Eine Funktion $f$ ist der $L^2$-Grenzwert von Funktionenfolgen $f_n \in L^2(0, 1)$, wenn für $n \rightarrow \infty$ gilt, dass $||f - f_n||_{L^2(0, 1)} \rightarrow 0$ + + +\begin{theorem}[]{Fourier-Reihe} + Jede Funktion $f \in L^2(0, 1)$ ist der Grenzwert ihrer Fourier-Reihe: + \rmvspace + \begin{align*} + f(t) = \sum_{k = -\infty}^{\infty} \hat{f}(k) e^{2\pi ikt} + \end{align*} + wobei die Fourier-Koeffizienten + \rmvspace + \begin{align*} + \hat{f}(k) = \int_{0}^{1} f(t)e^{-2\pi ikt} \dx t \smallhspace k \in \Z + \end{align*} + definiert sind. Es gilt die Parseval'sche Gleichung: + \rmvspace + \begin{align*} + \sum_{k = -\infty}^{\infty} |\hat{f}(k)|^2 = ||f||_{L^2(0, 1)}^2 + \end{align*} +\end{theorem} +\inlineremark Oder viel einfacher und kürzer: Die Funktionen $\varphi_k(x)$ bilden eine vollständige Orthonormalbasis in $L^2(0, 1)$. + +\setcounter{all}{14} +\inlineremark Die Parseval'sche Gleichung beschreibt einfach gesagt einen ``schnellen'' Abfall der $\hat{f}(k)$. +Genauer gesagt, klingen die Koeffizienten schneller als $\frac{1}{\sqrt{k}}$ ab. +Sie sagt zudem aus, dass die $L^2$-Norm der Funktion aus einer Summe berechnet werden kann (nicht nur als Integral). +Wenn wir die Fourier-Reihe nach $t$ ableiten, erhalten wir +\rmvspace +\begin{align*} + f'(t) = \sum_{k = -\infty}^{\infty} 2\pi ik\hat{f}(k)e^{2\pi ikt} +\end{align*} + +\begin{theorem}[]{Fourier-Reihe} + Seien $f$ und $f'$ integrierbar auf $(0, 1)$, dann gilt $\hat{f'}(k) = 2\pi ik\hat{f}(k)$ für $k \in \Z$. + + Falls die Operationen erlaubt sind, dann gilt zudem: + \rmvspace + \begin{align*} + \hat{f^{(n)}} = (2\pi ik)^n \hat{f}(k) \text{ und } ||f^{(n)}||_{L^2}^2 = (2\pi)^{2n} \sum_{k = -\infty}^{\infty} k^{2n} |\hat{f}(k)|^2 + \end{align*} +\end{theorem} + + +\inlinetheorem Wenn $\displaystyle \int_{0}^{1} |f^{(n)}(t)|\dx t < \infty$, dann ist $\hat{f}(k) = \tco{k^{-n}}$ diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/00_intro.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_intro.tex deleted file mode 100644 index b0e4157..0000000 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/00_intro.tex +++ /dev/null @@ -1,3 +0,0 @@ -Lecture: Wir besitzen nicht das komplette Vorwissen in der Analysis für dieses Kapitel, d.h. wird totales Verständnis nicht - -Lecture: Intuitiv wird Fourier-Trans. zur Kompression genutzt, z.b. jpg format.