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fix error in polygonzugverfahren image, add tikz math library to lecture cls

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christian
2020-06-29 21:29:57 +02:00
parent df5ec7ac25
commit d93b24e356
3 changed files with 46 additions and 27 deletions
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ana16.tex
+45 -26
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@@ -1,5 +1,6 @@
\documentclass{lecture} \documentclass{lecture}
\usetikzlibrary{math}
\begin{document} \begin{document}
\newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} \newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}}
\newcommand{\graph}{\operatorname{Graph}} \newcommand{\graph}{\operatorname{Graph}}
@@ -119,7 +120,7 @@ Notationen: $x' = f(t,x), \dot x = f(t,x), \dv{x}{t} = f(t,x)$ (Dynamischer Proz
} }
] ]
\addplot3 (x,y,0); \addplot3 (x,y,0);
\addlegendentry{$f'(t,x)$} \addlegendentry{$f(t,x)$}
\addplot{(x-0.25)^2+0.15}; \addplot{(x-0.25)^2+0.15};
\addlegendentry{$y(t)$} \addlegendentry{$y(t)$}
\end{axis} \end{axis}
@@ -236,45 +237,63 @@ Reminder:
\[y^h(t)\coloneqq y_{n-1}^h + (t-t_{n-1})f(t_{n-1},y_{n-1}^h),\quad t\in [t_{n-1},t_n],\quad \forall n\ge 1\] \[y^h(t)\coloneqq y_{n-1}^h + (t-t_{n-1})f(t_{n-1},y_{n-1}^h),\quad t\in [t_{n-1},t_n],\quad \forall n\ge 1\]
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\begin{tikzpicture}[declare function={f1(\x) = 0.5*(2)^(\x-1) + 10/(\x+2); \begin{tikzpicture}[declare function={
f2(\x) = 0.5*(2)^(\x-1); g(\x) = 0.5*exp(\x-2); % base function for y(t)
f3(\x) = 0.5*(2)^(\x-1) - 10/(\x+2); f(\x) = 0.5*exp(\x-2); % derivative
f4(\x) = 0.5*(2)^(\x-1) - 20/(\x+2);}] %g(\x) = 0.5*(\x-2.7)^3 - 2*(\x-2.7)^2; % base function for y(t)
%f(\x) = 1.5*(\x-2.7)^2 - 4*(\x-2.7); % derivative
}]
\def\h{1} % step length (accuracy of approximation)
\def\torig{2} % y_0
\def\yorig{1} % t_0
\begin{axis}% \begin{axis}%
[grid=none, [grid=none,
%minor tick num=4,
grid style={line width=.1pt, draw=gray!10}, grid style={line width=.1pt, draw=gray!10},
major grid style={line width=.2pt,draw=gray!50}, major grid style={line width=.2pt,draw=gray!50},
axis lines=middle, axis lines=middle,
%enlargelimits={abs=0.2},
ymax=10, ymin=-1.5, ymax=10, ymin=-1.5,
restrict y to domain=-2:12,
xmin=-1, xmax=7, xmin=-1, xmax=7,
xtick={2,3,4,5}, xtick={2,3,4,5},
ytick=\empty, ytick=\empty,
xticklabels={$t_0$, $t_1$, $t_2$, $t_3$}, xticklabels={$t_0$, $t_1$, $t_2$, $t_3$},
%yticklabels={$y_0$, $y_1^{h}$, $y_2^{h}$},
xlabel=$t$, xlabel=$t$,
ylabel=$x$, ylabel=$y$,
legend pos=outer north east
] ]
\addplot[domain=0:10,samples=50,smooth,green] {f1(x)}; \def\d{0}
\addlegendentry{$y(t,t_0,y_0)$}; \def\t{0}
\addplot[domain=0:10,samples=50,smooth,blue] {f2(x)}; \foreach \i/\colour [remember=\d as \dlast (initially \yorig),
\addlegendentry{$y(t, t_1, y_1^{h})$}; remember=\t as \tlast (initially \torig)]
\addplot[domain=0:10,samples=50,smooth,orange] {f3(x)}; in {0/green,1/blue,2/orange,3/pink} {
\addlegendentry{$y(t, t_2, y_2^{h})$}; \tikzmath{\t=\tlast+\h;\d=g(\tlast)+\dlast+\h*f(\tlast)-g(\t);}
\addplot[domain=0:10,samples=50,smooth,pink] {f4(x)}; \colour
\addlegendentry{$y(t, t_3, y_3^{h})$}; \edef\temp{\noexpand
\draw (2,{f1(2)}) node[circle,fill,inner sep=0.5pt] {} \addplot[domain=0:10,samples=50,smooth,\colour] {g(x) + \dlast - g(\torig)};
-- (3,{f2(3)}) node[circle,fill,inner sep=0.5pt] {} }
-- (4, {f3(4)}) node[circle,fill,inner sep=0.5pt] {} \temp
-- (5, {f4(5)}) node[circle,fill,inner sep=0.5pt] {}; \if\i3
\draw[dashed,green] (2, {f1(2)}) -- (0, {f1(2)}) node[label=left:$y_0$](){}; \edef\temp{\noexpand
\draw[dashed,blue] (3, {f2(3)}) -- (0, {f2(3)}) node[label=left:$y_1$](){}; \draw[dashed,->] (\tlast,{g(\tlast) + \dlast - g(\torig)})
\draw[dashed,orange] (4, {f3(4)}) -- (0, {f3(4)}) node[label=left:$y_2$](){}; -- (\t,{g(\t) + \d - g(\torig)});
\draw[dashed,pink] (5, {f4(5)}) -- (0, {f4(5)}) node[label=left:$y_3$](){}; }
\else
\edef\temp{\noexpand
\draw (\tlast,{g(\tlast) + \dlast - g(\torig)}) node[circle,fill,inner sep=0.5pt] {}
-- (\t,{g(\t) + \d - g(\torig)}) node[circle,fill,inner sep=0.5pt] {};
}
\fi
\temp
\edef\temp{\noexpand
\draw[dashed,\colour] (\tlast, {g(\tlast) + \dlast - g(\torig)}) -- (0, {g(\tlast) + \dlast - g(\torig)}) node[label=left:$y_{\i}$](){};
}
\temp
\edef\temp{\noexpand\addlegendentry{$y(t,t_{\i},y_{\i})$};}
\temp
}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\caption{Eulersches Polygonzugverfahren} \caption{Eulersches Polygonzugverfahren, Steigung der Tangenten ist $f(t,y)$}
\end{figure} \end{figure}
\begin{enumerate}[1)] \begin{enumerate}[1)]
\item \textbf{z.Z.} dass dieses Verfahren durchführbar ist, d.h. $\graph(y^h)\subset D$. Sei $(t,y^h(t))\subset D$ für $t_0 \le t\le t_{k-1}$. Dann gilt \item \textbf{z.Z.} dass dieses Verfahren durchführbar ist, d.h. $\graph(y^h)\subset D$. Sei $(t,y^h(t))\subset D$ für $t_0 \le t\le t_{k-1}$. Dann gilt
analysisII.pdf
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lecture.cls
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@@ -27,7 +27,7 @@
\RequirePackage{environ} \RequirePackage{environ}
\RequirePackage{stackrel} \RequirePackage{stackrel}
\usetikzlibrary{quotes, angles} \usetikzlibrary{quotes, angles, math}
\pgfplotsset{compat=1.15} % or \pgfplotsset{compat=newest} \pgfplotsset{compat=1.15} % or \pgfplotsset{compat=newest}
\geometry{ \geometry{