From 817fba55a8ef4185eb047bdbae15e5fc78d382d7 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sun, 28 Dec 2025 10:24:09 +0100 Subject: [PATCH] [NumCS] Finish quadrature, a few fixes --- semester3/numcs/format.py | 37 ++------------ semester3/numcs/numcs-summary.pdf | Bin 1393162 -> 1402506 bytes semester3/numcs/numcs-summary.tex | 4 +- .../00_introduction/01_time-complexity.tex | 2 +- .../02_quadrature/01_equidistant-nodes.tex | 8 ---- .../01_equidistant-nodes/00_intro.tex | 31 ++++++++++++ .../01_equidistant-nodes/01_summed.tex | 43 +++++++++++++++++ .../02_romberg-scheme.tex | 45 ++++++++++++++++++ 8 files changed, 127 insertions(+), 43 deletions(-) delete mode 100644 semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex create mode 100644 semester3/numcs/parts/02_quadrature/01_equidistant-nodes/00_intro.tex create mode 100644 semester3/numcs/parts/02_quadrature/01_equidistant-nodes/01_summed.tex create mode 100644 semester3/numcs/parts/02_quadrature/01_equidistant-nodes/02_romberg-scheme.tex diff --git a/semester3/numcs/format.py b/semester3/numcs/format.py index 6512e81..b3ae450 100644 --- a/semester3/numcs/format.py +++ b/semester3/numcs/format.py @@ -1,33 +1,4 @@ -import numpy as np -import scipy as sp - - -def fastbroyd(x0, F, J, tol=1e-12, maxit=20): - x = x0.copy() # make sure we do not change the iput - lup = sp.linalg.lu_factor(J) # LU decomposition of J - s = sp.linalg.lu_solve(lup, F(x)) # start with a Newton corection - sn = np.dot(s, s) # squared norm of the correction - x -= s - f = F(x) # start with a full Newton step - dx = np.zeros((maxit, len(x))) # containers for storing corrections s and their sn: - dxn = np.zeros(maxit) - k = 0 - dx[k] = s - dxn[k] = sn - k += 1 # the number of the Broyden iteration - - # Broyden iteration - while sn > tol and k < maxit: - w = sp.linalg.lu_solve(lup, f) # f = F (actual Broyden iteration x) - # Using the Sherman-Morrison-Woodbury formel - for r in range(1, k): - w += dx[r] * (np.dot(dx[r - 1], w)) / dxn[r - 1] - z = np.dot(s, w) - s = (1 + z / (sn - z)) * w - sn = np.dot(s, s) - dx[k] = s - dxn[k] = sn - x -= s - f = F(x) - k += 1 # update x and iteration number k - return x, k # return the final value and the numbers of iterations needed +def simpson(f, a, b, N): + x, h = np.linspace(a, b, 2 * int(N) + 1, retstep=True) + I = h / 3.0 * (np.sum(f(x[::2])) + 4.0 * np.sum(f(x[1::2])) + f(x[0]) - f(x[-1])) + return I diff --git a/semester3/numcs/numcs-summary.pdf b/semester3/numcs/numcs-summary.pdf index f6f8e202d86ca839622816b63df4b628a5d6e93c..638a3a18771fed53cea250d6953942f0918bdfb5 100644 GIT binary patch delta 160314 zcmZs>V~j3bu&vv+ZQHhO+qUhuZQEY$)wXThw(VYhzkN<}lY6uO&5X=cCG*Fqnom`9 zsCx8_dc6;HhVv;LE#O8=S`k+ot@lfFnVB1?#Y0mXiW*mrO?nuNEdrVwYnxc3-=f>> zVC~EoKH#RY`Uc{NDu%r4(p|Ifr&==xRmJNd6+`B*{4ddBmK+shbF)?I{48$}aY-`< z4%%VcXT`m7e8<>1FQOtqK=xlas1$ZFKIG$0~j0rwf$A8OD#$(+1t 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Wie in Algorithmen und * ist auch hier wieder $\tco{\ldots}$ der Worst Case. diff --git a/semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex deleted file mode 100644 index 29bf562..0000000 --- a/semester3/numcs/parts/02_quadrature/01_equidistant-nodes.tex +++ /dev/null @@ -1,8 +0,0 @@ -% ┌ ┐ -% │ Author: Robin Bacher │ -% └ ┘ - -\newsection -\subsection{Äquidistante Punkte} -\label{sec:equidistant-nodes} -% TODO: If it is not in the lecture notes, include table from slide 41 of TA. Really handy diff --git a/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/00_intro.tex b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/00_intro.tex new file mode 100644 index 0000000..1968e1f --- /dev/null +++ b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/00_intro.tex @@ -0,0 +1,31 @@ +\newsection +\subsection{Äquidistante Punkte} +\label{sec:equidistant-nodes} +Untenstehend eine Liste verschiedener Quadraturverfahren (Reminder: Eine Funktion der Orgnung $2$ ist eine exakte Approximation einer konstanten oder linearen Funktion): + +\begin{tables}{cccc}{Eigenschaft & Mittelpunkt & Trapez & Simpson} + Knoten & 1 & 2 & 3 \\ + Ordnung & 2 & 2 & 4 \\ + Fehler & $\tco{h^2}$ & $\tco{h^2}$ & $\tco{h^4}$ \\ + Symmetrisch & Ja & Ja & Ja \\ +\end{tables} + +\shade{gray}{Mittelpunkt-Regel} $\displaystyle Q^M(f; a, b) = (b - a) f\left( \frac{a + b}{2} \right)$. +Gewicht: $\omega = b - a$ + +\shade{gray}{Trapez-Regel} $\displaystyle Q^T(f; a, b) = \frac{b - a}{2} (f(a) + f(b))$. +Fehler: $\displaystyle E(n) = \left| -\frac{1}{12} (b - a)^3 f^{(2)}(\xi) \right| \text{ mit } \xi \in [a, b]$\\ +Fehlerabschätzung: $\displaystyle |E_n| \leq \frac{(b - a)^3}{12} ||f''||_\infty$. +Gewichte: $\omega_1 = \omega_2 = \frac{b - a}{2}$ + +\shade{gray}{Simpson-Regel} $\displaystyle Q^S(f; a, b) = \frac{b - a}{6} \left( f(a) + 4f\left( \frac{a + b}{2} + f(b) \right) \right)$. +Fehler: $\displaystyle E(n) = \left| -\frac{1}{90} \left( \frac{b - a}{2} f^{(4)} \right) f^{(4)}(\xi) \right|$\\ +Fehlerabschätzung: $\displaystyle |E_n| \leq \left( \frac{b - a}{2} \right)^2 f^{(4)}(\xi)$. +Gewichte: $\frac{b - a}{6}, \frac{4(b - a)}{6}, \frac{b - a}{6}$ + + +\inlineremark Die Schranken für den Fehler erhält man aus den Lagrange-Polynomen vom Grad $n - 1$: +\rmvspace +\begin{align*} + f \in \C^n([a, b]) \Rightarrow \left| f(t) \dx t - Q_n(f) \right| \leq \frac{1}{n!} (b - a)^{n + 1} ||f^{(n)}||_{L^\infty([a, b])} +\end{align*} diff --git a/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/01_summed.tex b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/01_summed.tex new file mode 100644 index 0000000..45f2adb --- /dev/null +++ b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/01_summed.tex @@ -0,0 +1,43 @@ +\drmvspace +\subsubsection{Summierte Quadratur} +Mit Anwendung von Divide and Conquer kann die Präzision der Integration verbessert werden, man unterteilt dazu einfach das Integrationsintervall in viele kleine Intervalle: +\rmvspace +\begin{align*} + \int_{a}^{b} f(x) \dx x = \sum_{i = 0}^{N - 1} \int_{x_i}^{x_{i + 1}} f(x) \dx x = \sum_{i = 0}^{N - 1} Q_n (f; x_i, x_{i + 1}) +\end{align*} + +\drmvspace +Im Folgenden ist $h = \frac{b - a}{N}$, $x_0 = a$, $x_i = x_0 + ih$ und $x_N = b$ + +\shade{gray}{Summierte Mittelpunkt-Regel} +$\displaystyle I(f; a, b) \approx \sum_{i = 0}^{N - 1} h \cdot f \left( \frac{x_i + x_{i + 1}}{2} \right)$ + +\shade{gray}{Summierte Trapez-Regel} +$\displaystyle I(f; a, b) \approx \sum_{i = 0}^{N - 1} \frac{h}{2} (f(x_i) + f(x_{i + 1})) = \frac{h}{2} \left( f(a) + 2 \sum_{i = 1}^{N - 1} f(x_i) + f(b) \right)$\\ +Fehler: $\displaystyle E(n) \leq \frac{h^2}{12} (b - a) \max_{x \in [a, b]} |f''(x)|$. Ist exakt bei periodischen, unendlich oft differenzierbaren Funktionen. +Untenstehend eine Implementation der Trapez-Regel in Numpy + +\rmvspace +\begin{code}{python} +def trapezoidal(f, a, b, N): + x, h = np.linspace(a, b, int(N) + 1, retstep=True) + I = h / 2.0 * (f(x[0]) + 2.0 * np.sum(f(x[1:-1])) + f(x[-1])) + return I +\end{code} + +\newpage +\shade{gray}{Summierte Simpson-Regel} +\rmvspace +\begin{align*} + I(f; a, b) & \approx \frac{h}{6} \left( f(a) + 2 \sum_{i = 1}^{N - 1} f(x_i) + 4 \sum_{i = 1}^{N} f\left( \frac{x_{i - 1} + x_i}{2} \right) + f(b) \right) \\ + & = \frac{\tilde{h}}{3} \left( f(\tilde{x_0}) + 2 \sum_{i = 1}^{N - 1} f(\tilde{x}_{2i}) + 4 \sum_{i = 1}^{N} f(\tilde{x}_{2i - 1}) + f(\tilde{x}_{2N}) \right) + \mediumhspace \text{ mit } \tilde{h} = \frac{h}{2} \text{, } \tilde{x}_{2i} = x_i \text{ und } \tilde{x}_{2i - 1} = \frac{x_{i - 1} + x_i}{2} +\end{align*} + +\drmvspace Untenstehend eine Implementation der Simpson-Regel +\begin{code}{python} +def simpson(f, a, b, N): + x, h = np.linspace(a, b, 2 * int(N) + 1, retstep=True) + I = h / 3.0 * (np.sum(f(x[::2])) + 4.0 * np.sum(f(x[1::2])) + f(x[0]) - f(x[-1])) + return I +\end{code} diff --git a/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/02_romberg-scheme.tex b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/02_romberg-scheme.tex new file mode 100644 index 0000000..5afce86 --- /dev/null +++ b/semester3/numcs/parts/02_quadrature/01_equidistant-nodes/02_romberg-scheme.tex @@ -0,0 +1,45 @@ +\subsubsection{Romberg Schema} +Für glatte Funktionen haben wir: $T (h) = I [f] + c_1 h^2 + c_2 h^4 + \ldots + c_p h^{2p} + \tco{h^{2p+2}}$ + +Die Idee des Romberg-Schemas ist es, die führenden Fehlerterme durch Linearkombinationen zu eliminieren + +\bg{orange}{Schritt 1} Berechnung von $T(h)$ und $T(\frac{h}{2})$: +\rmvspace +\begin{align*} + T(h) & = I + c_1 h^2 + c_2 h^4 + \ldots \\ + T\left( \frac{h}{2} \right) & = I + c_1 \frac{h^2}{4} + c_2 \frac{h^4}{16} + \ldots \\ +\end{align*} + +\drmvspace +\bg{orange}{Schritt 2} Linearkombination zur Elimination des $h^2$-Terms (Ordnung dann $4$): +\rmvspace +\begin{align*} + R_{1, 1} = \frac{4 T(h / 2) - T(h)}{3} = I + c_2' h^4 + \ldots +\end{align*} + +\drmvspace +\bg{orange}{Schritt 3} Wiederholen bis zur gewünschten Präzision mit der allgemeinen Rekursionsformel: +\rmvspace +\begin{align*} + R_{l, k} = \frac{4^k R_{l, k - 1} - R_{l - 1, k - 1}}{4^k - 1} +\end{align*} + +\drmvspace +Der Einfachheit halber können die Terme auch in das sogenannte ``Romberg-Tableau'' eingefüllt werden: +\begin{table}[h!] + \begin{center} + \begin{tabular}[c]{c|cccc} + k & 0 & 1 & 2 & 3\\ + \hline + 0 & $T(h)$ & $R_{0, 1}$ \\ + 1 & $T(h / 2)$ & $R_{1, 1}$ & $R_{1, 2}$ \\ + 1 & $T(h / 4)$ & $R_{2, 1}$ & $R_{2, 2}$ & $R_{2, 3}$ \\ + 1 & $T(h / 8)$ & $R_{3, 1}$ & $R_{3, 2}$ & $R_{3, 3}$ \\ + \end{tabular} + \end{center} +\end{table} +Das Romberg-Schema konvergiert sehr schnell für glatte Funktionen. + + +\subsubsection{Anwendung} +In der Praxis keine Newton-Cotes höherer Ordnung mit äquidistanten Stützpunkten