From 7c05d155cd6b94ec25ea02bad65000f562eb6337 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 29 Dec 2025 08:55:47 +0100 Subject: [PATCH] [NumCS] Finish householder and givens --- semester3/numcs/numcs-summary.pdf | Bin 1436734 -> 1438932 bytes semester3/numcs/numcs-summary.tex | 12 ---- .../00_polynomial/00_intro.tex | 5 -- .../01_trigonometric/00_fourier.tex | 30 +-------- .../01_trigonometric/04_error-estimation.tex | 2 - .../02_non-equidistant/00_gauss.tex | 1 - .../numcs/parts/02_quadrature/03_adaptive.tex | 2 - .../parts/03_zeros/02_fix-point-iteration.tex | 3 - semester3/numcs/parts/04_linalg/00_intro.tex | 2 +- semester3/numcs/parts/04_linalg/02_qr.tex | 61 ++++++++++++++++++ .../01_linear/01_normal-equations.tex | 4 +- 11 files changed, 64 insertions(+), 58 deletions(-) diff --git a/semester3/numcs/numcs-summary.pdf b/semester3/numcs/numcs-summary.pdf index 72d9636da1340b8a8c941fad08b4faa2866ad75a..a5b87e48e5b70e9860631b5fa2a519aacc36a7d6 100644 GIT binary patch delta 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b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex index 271609a..abf76d9 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex @@ -29,9 +29,7 @@ Das Inverse davon nimmt eine Funktion der Frequenz und transformiert diese in ei % \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 $\text{Re}$? -% RESPONE: whatever he uses in the script, preferably \text{Re}() etc. -$p_m$ kann folgendermassen dargestellt werden ($a_0 = 2\gamma_0, a_j = 2\Re(\gamma_j)$ und $b_j = -2\Im(\gamma_j)$): +$p_m$ kann folgendermassen dargestellt werden ($a_0 = 2\gamma_0, a_j = 2\text{Re}(\gamma_j)$ und $b_j = -2\text{Im}(\gamma_j)$): \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)) @@ -123,8 +121,6 @@ Die dargestellte Funktion ist die Fourier-Reihe der charakteristischen Funktion \end{align*} Mit $c = \pi(a + b)$ und $d = \pi(b - a)$ -% TODO: Replace with rendered image from matplotlib (will be higher quality than screenshot from script and can tweak it to our liking) -% we will have it anyway after solving the exercises, so might as well \begin{figure}[h!] \begin{center} \includegraphics[width=0.95\textwidth]{assets/01_interpolation/01_trigonometric/overarcing.png} @@ -140,27 +136,3 @@ wobei $l = 0, 1 \ldots, N - 1$ und $N$ die Anzahl der Intervalle ist: \begin{align*} \hat{f}_N(k) := \frac{1}{N} \sum_{l = 0}^{N - 1} f(t_l) e^{-2\pi ikt_l} \approx \hat{f}(k) \end{align*} - -% TODO: Consider if we should use the below - -% \begin{tikzpicture} -% \begin{axis}[ -% legend pos=outer north east, -% title=Function plot of $f(x)$ (parts coloured), -% axis lines = box, -% xlabel = $x$, -% ylabel = $y$, -% variable = t, -% trig format plots = rad, -% ] -% \addplot [ -% domain=1:4, -% samples=70, -% color=blue, -% ] -% {log2(x)}; -% \addlegendentry{$ y=x^2 - x - 0.5$} -% \end{axis} -% \node (0) at (0, 0) {}; -% \end{tikzpicture} - diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/04_error-estimation.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/04_error-estimation.tex index a22e161..b984429 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/04_error-estimation.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/04_error-estimation.tex @@ -18,13 +18,11 @@ \numberingOff \inlineex Zur Fehlerbetrachtung verwenden wir drei Funktionen $f : [0, 1] \rightarrow \R$, welche wir mit trigonometrischer Interpolation an den Punkten $\frac{k}{N}$ approximieren: \begin{enumerate}[label=(\Roman*)] - % FIXME: Possibly wrong function definition in script \item Stufenfunktion (periodische Fortsetzung von $f$) $f : [0, 1] \rightarrow \R$ mit $f(t) = \begin{cases} 0 & \text{für } \left| t - \frac{1}{2} \right| > \frac{1}{4} \\ 1 & \text{für } \left| t - \frac{1}{2} \right| \leq \frac{1}{4} \end{cases}$ \item Periodische, glatte Funktion $h: \R \rightarrow \R$ mit $h(t) = \displaystyle \frac{1}{\sqrt{1 + \frac{1}{2} \sin(2\pi t)}}$ - % TODO: Is it $h$ or $g$ here? $g$ makes little sense imho \item Hutfunktion (periodische Fortsetzung von $h$) $g : [0, 1] \rightarrow \R$ mit $g(t) = \left| t - \frac{1}{2} \right|$ \end{enumerate} Die untenstehende Abbildung \ref{fig:interpolation-error-examples} beinhaltet einen Plot, auf dem die Konvergenz in Abhängigkeit des Grades des Interpolationspolynoms aufgetragen ist. diff --git a/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex b/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex index d8f8204..035d0ee 100644 --- a/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex +++ b/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex @@ -97,7 +97,6 @@ Die zugehörige Quadraturformel ist exakt für Polynome $2s - 1$-ten Grades. Die % Yeah, there is *no way* we would have figured that out... and he even provides a proof for it... yikes... \inlinetheorem Die Gewichte der Gauss-Legendre-Quadraturformel sind positiv. -% TODO: Consider adding code \begin{tables}{cccc}{Algorithmus & Laufzeit & Genauigkeit Knoten & Genauigkeit Gewichte} GW (1969) & $\tco{s^3}$ / $\tco{s^2}$ & $\tco{1}$ & $\tco{s^2}$ \\ Bogaert-Townsend & $\tco{s}$ & $\tco{1}$ & $\tco{1}$ \\ diff --git a/semester3/numcs/parts/02_quadrature/03_adaptive.tex b/semester3/numcs/parts/02_quadrature/03_adaptive.tex index 77acd2b..b183cef 100644 --- a/semester3/numcs/parts/02_quadrature/03_adaptive.tex +++ b/semester3/numcs/parts/02_quadrature/03_adaptive.tex @@ -17,5 +17,3 @@ Auf Seiten 150 - 151 im Skript findet sich Code, um eine adaptive Quadratur durc Mit \texttt{scipy.integrate.quadrature} können wir die Gauss-Quadratur verwenden. Für $x \in \R^d$, also eine mehrdimensionale Funktion der Dimension $d$ können wir \texttt{scipy.integrate.nquad} verwenden. Mehr dazu im nächsten Kapitel - -% TODO: Possibly explain the graphs and / or add code for computation using scipy 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 42bf3da..0d15430 100644 --- a/semester3/numcs/parts/03_zeros/02_fix-point-iteration.tex +++ b/semester3/numcs/parts/03_zeros/02_fix-point-iteration.tex @@ -6,7 +6,6 @@ \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, @@ -28,8 +27,6 @@ $\phi_2(x) = \frac{1 + x}{1 + e^x}$ quadratische Konvergenz und $\phi_3(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. diff --git a/semester3/numcs/parts/04_linalg/00_intro.tex b/semester3/numcs/parts/04_linalg/00_intro.tex index a450613..3e9d1ee 100644 --- a/semester3/numcs/parts/04_linalg/00_intro.tex +++ b/semester3/numcs/parts/04_linalg/00_intro.tex @@ -43,6 +43,7 @@ Wenn sie zudem normiert sind (also $||q_i||_2 = 1 \smallhspace \forall i \leq n$ \drmvspace \shade{gray}{Perturbierte LGS} +\setLabelNumber{all}{18} Statt $Ax = b$ ist das LGS ungenau gegeben: $(A + \Delta A)(\tilde{x} - x) = \Delta b - \Delta Ax$. @@ -79,7 +80,6 @@ Passen oft nicht (direkt) in den Speicher: effizientere Speicherung nötig, mög % TODO: Update with notes from TA and script \textbf{Dünnbesetzte Matrizen} - $\text{nnz}(A) := |\{ (i,j) \ |\ a_{ij} \in A, a_{ij} \neq 0 \}| \ll m\cdot n$ $\limit{l}{\infty} \frac{\text{nnz}(A^{(l)})}{n_l m_l} = 0$ diff --git a/semester3/numcs/parts/04_linalg/02_qr.tex b/semester3/numcs/parts/04_linalg/02_qr.tex index 148ad83..7d644b9 100644 --- a/semester3/numcs/parts/04_linalg/02_qr.tex +++ b/semester3/numcs/parts/04_linalg/02_qr.tex @@ -80,6 +80,24 @@ Wir haben jedoch das Problem, dass die Berechnung von $r$ überlaufen kann. Dies \end{multicols} Es ist wichtig, dass wir das $r = \text{sign}(a) \sqrt{a^2 + b^2}$ mit Vorzeichen berechnen, um Auslöschung zu vermeiden +Man kann nun mit der Givens-Rotation die QR-Zerlegung durchführen: +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{GivensQRDecomposition}(A)} + \begin{algorithmic}[1] + \State $m \gets \texttt{A.shape[0]}$ + \State $n \gets \texttt{A.shape[1]}$ + \State $q \gets$ Initialisiere ein $n \times n$ array + \For{$j = 1, 2, \ldots, n$} + \For{$i = m, \ldots, 2$} + \State Nullsetze $a_{m, j}, a_{m - 1, j}, \ldots, a_{2, j}$ durch Givens-Rotationen $G_{m, j}, G_{m - 1, j}, \ldots, G_{2, j}$ + \EndFor + \EndFor + \State \Return $l$ + \end{algorithmic} + \end{spacing} +\end{algorithm} + \newpage \bg{purple}{Gram-Schmidt} @@ -131,4 +149,47 @@ Es wurden zwei Algorithmen behandelt, beide unten in Pseudocode: Falls $R$ nicht benötigt wird, kann viel Speicher gespart werden, indem man das $r_{ik}$ als eine scoped variable verwendet. +\newpage \bg{purple}{Householder-Reflektor} + +Wir konstruieren eine Matrix $H = I - 2 \displaystyle\frac{vv^\top}{v^\top v} = I - 2uu^\top \text{ mit } u = \frac{v}{||v||}$. + +Dabei ist die Matrix $H$ orthogonal ($H^\top H = I$) und symmetrisch ($H^\top = H$). + +Um nun die QR-Zerlegung durchzuführen mit der Householder-Reflektion fehlen uns die Householder-Reflektoren. +Um diese zu erstellen wollen wir das $v$ so wählen, dass $Hx = ||x|| e_1$ gilt. So werden also $m - 1$ Elemente auf einmal auf Null gesetzt. + +Der Ansatz dazu ist entsprechend $v = x - \alpha e_1$ mit $\alpha = -\text{sign}(x_1)||x||$ (minus, um numerische Stabilität zu erhalten) und wir haben dann: +\drmvspace +\begin{align*} + Hx = \alpha e_1 \Longleftrightarrow Hx = x - 2 \frac{v^\top x}{v^\top v} v +\end{align*} + +\drmvspace +Dann müssen wir nur noch $v^\top x$ und $v^\top v$ berechnen und auflösen. +Der vollständige QR-Algorithmus lautet: +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{HouseholderQR}(A)} + \begin{algorithmic}[1] + \State $n \gets \texttt{A.shape[0]}$ + \State $H \gets$ Initialisiere ein $n \times n$ array + \For{$k = 1, 2, \ldots, n$} + \State $x \gets A(k : m, k$ \Comment{Wähle subvektor der $k$-ten Spalte} + \State $H_k \gets$ Konstruiere Householder-Reflektor für $x$ + \State $A(k : m, k : n) \gets H_k A(k : m, k : n)$ \Comment{Update} + \EndFor + \State $Q = H_1 H_2 \cdots H_n$ + \State $R = H_n H_{n - 1} \cdots H_1 A$ + \State \Return $Q$, $R$ + \end{algorithmic} + \end{spacing} +\end{algorithm} + +Die Laufzeiten der verschiedenen Methoden im Vergleich: +\begin{itemize} + \item Householder-QR: $\approx 2mn^2$ Flops + \item Gram-Schmidt: $\approx 2mn^2$ Flops + \item Givens: $\approx 3mn^2$ Flops +\end{itemize} +Jedoch ist die Householder-Methode bedeutend stabiler als die anderen beiden. diff --git a/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex b/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex index 7360883..eaa466d 100644 --- a/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex +++ b/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex @@ -37,6 +37,4 @@ $B_a = A^H & 0 \end{bmatrix} $ -% TODO: What the f does the kappa mean??? -ersetzen, wobei wir $a$ so wählen, dass $\kappa(B_a)$ minimal wird. -% TODO: Consider adding the code and the example, but likely skip +ersetzen, wobei wir $a$ so wählen, dass $\kappa(B_a)$ minimal wird (Zur Erinnerung, $\kappa$ ist die Konditionszahl der Matrix).