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61 files changed, 15939 insertions, 667 deletions
diff --git a/buch/papers/laguerre/Makefile b/buch/papers/laguerre/Makefile index 0f0985a..85a1b83 100644 --- a/buch/papers/laguerre/Makefile +++ b/buch/papers/laguerre/Makefile @@ -3,9 +3,42 @@ # # (c) 2020 Prof Dr Andreas Mueller # +IMGFOLDER := images +PRESFOLDER := presentation -images: images/laguerre_polynomes.pdf +FIGURES := \ + images/targets.pdf \ + images/rel_error_complex.pdf \ + images/estimates.pdf \ + images/integrand.pdf \ + images/integrand_exp.pdf \ + images/laguerre_poly.pdf \ + images/rel_error_mirror.pdf \ + images/rel_error_range.pdf \ + images/rel_error_shifted.pdf \ + images/rel_error_simple.pdf \ + images/gammaplot.pdf -images/laguerre_polynomes.pdf: scripts/laguerre_plot.py - python3 scripts/laguerre_plot.py +.PHONY: all +all: images presentation +.PHONY: images +images: $(FIGURES) + +.PHONY: presentation +presentation: $(PRESFOLDER)/presentation.pdf + +images/%.pdf images/%.pgf: scripts/%.py scripts/gamma_approx.py + python3 $< + +images/gammaplot.pdf: images/gammaplot.tex images/gammapaths.tex + cd $(IMGFOLDER) && latexmk -quiet -pdf gammaplot.tex + +$(PRESFOLDER)/%.pdf: $(PRESFOLDER)/%.tex $(FIGURES) + cd $(PRESFOLDER) && latexmk -quiet -pdf $(<F) + +.PHONY: clean +clean: + rm $(FIGURES) + cd $(IMGFOLDER) && latexmk -C + cd $(PRESFOLDER) && latexmk -C
\ No newline at end of file diff --git a/buch/papers/laguerre/Makefile.inc b/buch/papers/laguerre/Makefile.inc index 12b0935..39b5d6f 100644 --- a/buch/papers/laguerre/Makefile.inc +++ b/buch/papers/laguerre/Makefile.inc @@ -10,6 +10,4 @@ dependencies-laguerre = \ papers/laguerre/definition.tex \ papers/laguerre/eigenschaften.tex \ papers/laguerre/quadratur.tex \ - papers/laguerre/gamma.tex - - + papers/laguerre/gamma.tex diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index d111f6f..4729a93 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -15,14 +15,17 @@ x y''(x) + (\nu + 1 - x) y'(x) + n y(x) n \in \mathbb{N}_0 , \quad x \in \mathbb{R} -. \label{laguerre:dgl} +. \end{align} +Spannenderweise wurde die verallgemeinerte Laguerre-Differentialgleichung +zuerst von Yacovlevich Sonine (1849 - 1915) beschrieben, +aber aufgrund ihrer Ähnlichkeit nach Laguerre benannt. Die klassische Laguerre-Diffentialgleichung erhält man, wenn $\nu = 0$. Hier wird die verallgemeinerte Laguerre-Differentialgleichung verwendet, -weil die Lösung mit der selben Methode berechnet werden kann, -aber man zusätzlich die Lösung für den allgmeinen Fall erhält. -Zur Lösung der Gleichung \eqref{laguerre:dgl} verwenden wir einen +weil die Lösung mit derselben Methode berechnet werden kann. +Zusätzlich erhält man aber die Lösung für den allgmeinen Fall. +Zur Lösung von \eqref{laguerre:dgl} verwenden wir einen Potenzreihenansatz. Da wir bereits wissen, dass die Lösung orthogonale Polynome sind, erscheint dieser Ansatz sinnvoll. @@ -44,7 +47,7 @@ y''(x) = \sum_{k=1}^\infty (k+1) k a_{k+1} x^{k-1} \end{align*} -in die Differentialgleichung ein, erhält man: +in die Differentialgleichung ein, erhält man \begin{align*} \sum_{k=1}^\infty (k+1) k a_{k+1} x^k + @@ -118,6 +121,15 @@ L_n^\nu(x) \sum_{k=0}^{n} \frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} x^k. \label{laguerre:allg_polynom} \end{align} +Die Laguerre-Polynome von Grad $0$ bis $7$ sind in +Abbildung~\ref{laguerre:fig:polyeval} dargestellt. +\begin{figure} +\centering +% \scalebox{0.8}{\input{papers/laguerre/images/laguerre_poly.pgf}} +\includegraphics[width=0.9\textwidth]{papers/laguerre/images/laguerre_poly.pdf} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\label{laguerre:fig:polyeval} +\end{figure} \subsection{Analytische Fortsetzung} Durch die analytische Fortsetzung erhalten wir zudem noch die zweite Lösung der @@ -126,8 +138,10 @@ Differentialgleichung mit der Form \Xi_n(x) = L_n(x) \ln(x) + \sum_{k=1}^\infty d_k x^k +. \end{align*} -Nach einigen mühsamen Rechnungen, +Nach einigen aufwändigen Rechnungen, +% die am besten ein Computeralgebrasystem übernimmt, die den Rahmen dieses Kapitel sprengen würden, erhalten wir \begin{align*} @@ -142,16 +156,5 @@ L_n(x) \ln(x) \end{align*} wobei $\alpha_0 = 0$ und $\alpha_k =\sum_{i=1}^k i^{-1}$, $\forall k \in \mathbb{N}$. -Die Laguerre-Polynome von Grad $0$ bis $7$ sind in -Abbildung~\ref{laguerre:fig:polyeval} dargestellt. -\begin{figure} -\centering -\includegraphics[width=0.7\textwidth]{% - papers/laguerre/images/laguerre_polynomes.pdf% -} -\caption{Laguerre-Polynome vom Grad $0$ bis $7$} -\label{laguerre:fig:polyeval} -\end{figure} - % https://www.math.kit.edu/iana1/lehre/hm3phys2012w/media/laguerre.pdf % http://www.physics.okayama-u.ac.jp/jeschke_homepage/E4/kapitel4.pdf diff --git a/buch/papers/laguerre/eigenschaften.tex b/buch/papers/laguerre/eigenschaften.tex index b0cc3a3..4adbe86 100644 --- a/buch/papers/laguerre/eigenschaften.tex +++ b/buch/papers/laguerre/eigenschaften.tex @@ -3,35 +3,31 @@ % % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % -\section{Eigenschaften - \label{laguerre:section:eigenschaften}} -{ -\large \color{red} -TODO: -Evtl. nur Orthogonalität hier behandeln, da nur diese für die Gauss-Quadratur -benötigt wird. -} - -Die Laguerre-Polynome besitzen einige interessante Eigenschaften -\rhead{Eigenschaften} - -\subsection{Orthogonalität - \label{laguerre:subsection:orthogonal}} -Im Abschnitt~\ref{laguerre:section:definition} haben wir behauptet, -dass die Laguerre-Polynome orthogonale Polynome sind. +\section{Orthogonalität + \label{laguerre:section:orthogonal}} +Im Abschnitt~\ref{laguerre:section:definition} +haben wir die Behauptung aufgestellt, +dass die Laguerre-Polynome orthogonal sind. Zu dieser Behauptung möchten wir nun einen Beweis liefern. -Wenn wir die Laguerre\--Differentialgleichung in ein -Sturm\--Liouville\--Problem umwandeln können, haben wir bewiesen, dass es sich -bei -den Laguerre\--Polynomen um orthogonale Polynome handelt (siehe +Wenn wir \eqref{laguerre:dgl} in ein +Sturm-Liouville-Problem umwandeln können, haben wir bewiesen, dass es sich +bei den Laguerre-Polynomen um orthogonale Polynome handelt (siehe Abschnitt~\ref{buch:integrale:subsection:sturm-liouville-problem}). -Der Sturm-Liouville-Operator hat die Form +Der Beweis kann äquivalent auch über den Sturm-Liouville-Operator \begin{align} S = \frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right). \label{laguerre:slop} \end{align} +und den Laguerre-Operator +\begin{align} +\Lambda += +x \frac{d}{dx^2} + (\nu + 1 -x) \frac{d}{dx} +\end{align} +erhalten werden, +indem wir diese Operatoren einander gleichsetzen. Aus der Beziehung \begin{align} S @@ -49,24 +45,27 @@ Ausserdem ist ersichtlich, dass $p(x)$ die Differentialgleichung \begin{align*} x \frac{dp}{dx} = --(\nu + 1 - x) p, +-(\nu + 1 - x) p \end{align*} erfüllen muss. Durch Separation erhalten wir dann \begin{align*} \int \frac{dp}{p} & = --\int \frac{\nu + 1 - x}{x}dx +-\int \frac{\nu + 1 - x}{x} \, dx += +-\int \frac{\nu + 1}{x} \, dx - \int 1\, dx \\ \log p & = --\log \nu + 1 - x + C +-(\nu + 1)\log x - x + c \\ p(x) & = -C x^{\nu + 1} e^{-x} +. \end{align*} -Eingefügt in Gleichung~\eqref{laguerre:sl-lag} erhalten wir +Eingefügt in Gleichung~\eqref{laguerre:sl-lag} ergibt sich \begin{align*} \frac{C}{w(x)} \left( @@ -106,14 +105,9 @@ Für den rechten Rand ist die Bedingung (Gleichung~\eqref{laguerre:sllag_randb}) 0 \end{align*} für beliebige Polynomlösungen erfüllt für $k_\infty=0$ und $h_\infty=1$. -Damit können wir schlussfolgern, dass die Laguerre-Polynome orthogonal -bezüglich des Skalarproduktes auf dem Intervall $(0, \infty)$ mit der Laguerre\--Gewichtsfunktion -$w(x)=x^\nu e^{-x}$ sind. - - -\subsection{Rodrigues-Formel} - -\subsection{Drei-Terme Rekursion} - -\subsection{Beziehung mit der Hypergeometrischen Funktion} - +Damit können wir schlussfolgern: +Die verallgemeinerten Laguerre-Polynome sind orthogonal +bezüglich des Skalarproduktes auf dem Intervall $(0, \infty)$ +mit der verallgemeinerten Laguerre\--Gewichtsfunktion $w(x)=x^\nu e^{-x}$. +Die Laguerre-Polynome ($\nu=0$) sind somit orthognal im Intervall $(0, \infty)$ +mit der Gewichtsfunktion $w(x)=e^{-x}$. diff --git a/buch/papers/laguerre/gamma.tex b/buch/papers/laguerre/gamma.tex index e3838b0..2e5fc06 100644 --- a/buch/papers/laguerre/gamma.tex +++ b/buch/papers/laguerre/gamma.tex @@ -8,8 +8,8 @@ Die Gauss-Laguerre-Quadratur kann nun verwendet werden, um exponentiell abfallende Funktionen im Definitionsbereich $(0, \infty)$ zu berechnen. -Dabei bietet sich z.B. die Gamma-Funkion bestens an, wie wir in den folgenden -Abschnitten sehen werden. +Dabei bietet sich z.B. die Gamma-Funkion hervorragend an, +wie wir in den folgenden Abschnitten sehen werden. \subsection{Gamma-Funktion} Die Gamma-Funktion ist eine Erweiterung der Fakultät auf die reale und komplexe @@ -19,58 +19,514 @@ Integral der Form \begin{align} \Gamma(z) & = -\int_0^\infty t^{z-1} e^{-t} dt +\int_0^\infty x^{z-1} e^{-x} \, dx , \quad \text{wobei Realteil von $z$ grösser als $0$} -, \label{laguerre:gamma} +. \end{align} -welches alle Eigenschaften erfüllt, um mit der Gauss-Laguerre-Quadratur -berechnet zu werden. +Der Term $e^{-t}$ im Integranden und der Integrationsbereich erfüllen +genau die Bedingungen der Laguerre-Integration. +% Der Term $e^{-t}$ ist genau die Gewichtsfunktion der Laguerre-Integration und +% der Definitionsbereich passt ebenfalls genau für dieses Verfahren. +Weiter zu erwähnen ist, dass für die verallgemeinerte Laguerre-Integration die +Gewichtsfunktion $t^\nu e^{-t}$ exakt dem Integranden für $\nu=z-1$ entspricht. \subsubsection{Funktionalgleichung} -Die Funktionalgleichung besagt +Die Gamma-Funktion besitzt die gleiche Rekursionsbeziehung wie die Fakultät, +nämlich \begin{align} -z \Gamma(z) = \Gamma(z+1). +z \Gamma(z) += +\Gamma(z+1) +. \label{laguerre:gamma_funktional} \end{align} -Mittels dieser Gleichung kann der Wert an einer bestimmten, -geeigneten Stelle evaluiert werden und dann zurückverschoben werden, -um das gewünschte Resultat zu erhalten. + +\subsubsection{Reflektionsformel} +Die Reflektionsformel +\begin{align} +\Gamma(z) \Gamma(1 - z) += +\frac{\pi}{\sin \pi z} +,\quad +\text{für } +z \notin \mathbb{Z} +\label{laguerre:gamma_refform} +\end{align} +stellt eine Beziehung zwischen den zwei Punkten, +die aus der Spiegelung an der Geraden $\real z = 1/2$ hervorgehen, +her. +Dadurch lassen Werte der Gamma-Funktion sich für $z$ in der rechten Halbebene +leicht in die linke Halbebene übersetzen und umgekehrt. \subsection{Berechnung mittels Gauss-Laguerre-Quadratur} +In den vorherigen Abschnitten haben wir gesehen, +dass sich die Gamma-Funktion bestens für die Gauss-Laguerre-Quadratur eignet. +Nun bieten sich uns zwei Optionen, +diese zu berechnen: +\begin{enumerate} +\item Wir verwenden die verallgemeinerten Laguerre-Polynome, dann $f(x)=1$. +\item Wir verwenden die Laguerre-Polynome, dann $f(x)=x^{z-1}$. +\end{enumerate} +Die erste Variante wäre optimal auf das Problem angepasst, +allerdings müssten die Gewichte und Nullstellen für jedes $z$ +neu berechnet werden, +da sie per Definition von $z$ abhängen. +Dazu kommt, +dass die Berechnung der Gewichte $A_i$ nach \cite{laguerre:Cassity1965AbcissasCA} +\begin{align*} +A_i += +\frac{ +\Gamma(n) \Gamma(n+\nu) +} +{ +(n+\nu) +\left[L_{n-1}^{\nu}(x_i)\right]^2 +} +\end{align*} +Evaluationen der Gamma-Funktion benötigen. +Somit ist diese Methode eindeutig nicht geeignet für unser Vorhaben. + +Bei der zweiten Variante benötigen wir keine Neuberechung der Gewichte +und Nullstellen für unterschiedliche $z$. +In \eqref{laguerre:quadratur_gewichte} ist ersichtlich, +dass die Gewichte einfach zu berechnen sind. +Auch die Nullstellen können vorgängig, +mittels eines geeigneten Verfahrens, +aus den Polynomen bestimmt werden. +Als problematisch könnte sich höchstens +die zu integrierende Funktion $f(x)=x^{z-1}$ für $|z| \gg 0$ erweisen. +Somit entscheiden wir uns aufgrund der vorherigen Punkte, +die zweite Variante weiterzuverfolgen. + +\subsubsection{Direkter Ansatz} +Wenden wir also die Gauss-Laguerre-Quadratur aus +\eqref{laguerre:laguerrequadratur} auf die Gamma-Funktion +\eqref{laguerre:gamma} an, +ergibt sich +\begin{align} +\Gamma(z) +\approx +\sum_{i=1}^n x_i^{z-1} A_i. +\label{laguerre:naive_lag} +\end{align} + +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_simple.pgf} +\includegraphics{papers/laguerre/images/rel_error_simple.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des direkten Ansatzes +für verschiedene reele Werte von $z$ und Grade $n$ der Laguerre-Polynome} +\label{laguerre:fig:rel_error_simple} +\end{figure} -Fehlerterm: +Bevor wir die Gauss-Laguerre-Quadratur anwenden, +möchten wir als ersten Schritt eine Fehlerabschätzung durchführen. +Für den Fehlerterm \eqref{laguerre:lag_error} wird die $2n$-te Ableitung +der zu integrierenden Funktion $f(\xi)$ benötigt. +Für das Integral der Gamma-Funktion ergibt sich also \begin{align*} +\frac{d^{2n}}{d\xi^{2n}} f(\xi) + & = +\frac{d^{2n}}{d\xi^{2n}} \xi^{z-1} +\\ + & = +(z - 2n)_{2n} \xi^{z - 2n - 1} +. +\end{align*} +Eingesetzt im Fehlerterm \eqref{laguerre:lag_error} resultiert +\begin{align} R_n = (z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z-2n-1} -\end{align*} +, +\label{laguerre:gamma_err_simple} +\end{align} +wobei $\xi$ ein geeigneter Wert im Interval $(0, \infty)$ ist +und $n$ der Grad des verwendeten Laguerre-Polynoms. +Eine Fehlerabschätzung mit dem Fehlerterm stellt sich als unnütz heraus, +da $R_n$ für $z < 2n - 1$ bei $\xi \rightarrow 0$ eine Singularität aufweist +und für $z > 2n - 1$ bei $\xi \rightarrow \infty$ divergiert. +Nur für den unwahrscheinlichen Fall $ z = 2n - 1$ +wäre eine Fehlerabschätzung plausibel. -\subsubsection{Finden der optimalen Berechnungsstelle} +Wenden wir nun also direkt die Gauss-Laguerre-Quadratur auf die Gamma-Funktion +an. +Dazu benötigen wir die Gewichte nach +\eqref{laguerre:quadratur_gewichte} +und als Stützstellen die Nullstellen des Laguerre-Polynomes $L_n$. +Evaluieren wir den relativen Fehler unserer Approximation zeigt sich ein +Bild wie in Abbildung~\ref{laguerre:fig:rel_error_simple}. +Man kann sehen, +wie der relative Fehler Nullstellen aufweist für ganzzahlige $z \leq 2n$. +Laut der Theorie der Gauss-Quadratur auch ist das zu erwarten, +da die Approximation via Gauss-Quadratur +exakt ist für zu integrierende Polynome mit Grad $\leq 2n-1$ +und hinzukommt, +dass zudem von $z$ noch $1$ abgezogen wird im Exponenten. +Es ist ersichtlich, +dass sich für den Polynomgrad $n$ ein Interval gibt, +in dem der relative Fehler minimal ist. +Links steigt der relative Fehler besonders stark an, +während er auf der rechten Seite zu konvergieren scheint. +Um die linke Hälfte in den Griff zu bekommen, +könnten wir die Reflektionsformel der Gamma-Funktion ausnutzen. + +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_mirror.pgf} +\includegraphics{papers/laguerre/images/rel_error_mirror.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des Ansatzes mit Spiegelung negativer Realwerte +für verschiedene reele Werte von $z$ und Grade $n$ der Laguerre-Polynome} +\label{laguerre:fig:rel_error_mirror} +\end{figure} + +Spiegelt man nun $z$ mit negativem Realteil mittels der Reflektionsformel, +ergibt sich ein stabilerer Fehler in der linken Hälfte, +wie in Abbildung~\ref{laguerre:fig:rel_error_mirror}. +Die Spiegelung bringt nur für wenige Werte einen, +für praktische Anwendungen geeigneten, +relativen Fehler. +Wie wir aber in Abbildung~\ref{laguerre:fig:rel_error_simple} sehen konnten, +gibt es für jeden Polynomgrad $n$ ein Intervall $[a(n), a(n) + 1]$, +$a(n) \in \mathbb{Z}$, +in welchem der relative Fehler minimal ist. +Die Funktionalgleichung der Gamma-Funktion \eqref{laguerre:gamma_funktional} +könnte uns hier helfen, +das Problem in den Griff zu bekommen. + +\subsubsection{Analyse des Integranden} +Wie wir im vorherigen Abschnitt gesehen haben, +scheint der Integrand problematisch. +Darum möchten wir jetzt den Integranden analysieren, +um ihn besser verstehen zu können und +dadurch geeignete Gegenmassnahmen zu entwickeln. + +% Dieser Abschnitt soll eine grafisches Verständnis dafür schaffen, +% wieso der Integrand so problematisch ist. +% Was das heisst sollte in Abbildung~\ref{laguerre:fig:integrand} +% und Abbildung~\ref{laguerre:fig:integrand_exp} grafisch dargestellt werden. +\begin{figure} +\centering +% \input{papers/laguerre/images/integrand.pgf} +\includegraphics{papers/laguerre/images/integrand.pdf} +%\vspace{-12pt} +\caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\label{laguerre:fig:integrand} +\end{figure} + +In Abbildung~\ref{laguerre:fig:integrand} ist der Integrand $x^z$ für +unterschiedliche Werte von $z$ dargestellt. +Dies entspricht der zu integrierenden Funktion $f(x)$ +der Gauss-Laguerre-Quadratur für die Gamma-Funktion. +Man erkennt, +dass für kleine $z$ sich ein singulärer Integrand ergibt +und auch für grosse $z$ wächst der Integrand sehr schnell an. +Das heisst, +die Ableitungen im Fehlerterm divergieren noch schneller +und das wirkt sich negativ auf die Genauigkeit der Approximation aus. +Somit lässt sich hier sagen, +dass kleine Exponenten um $0$ genauere Resultate liefern sollten. + +\begin{figure} +\centering +% \input{papers/laguerre/images/integrand_exp.pgf} +\includegraphics{papers/laguerre/images/integrand_exp.pdf} +%\vspace{-12pt} +\caption{Integrand $x^z e^{-x}$ mit unterschiedlichen Werten für $z$} +\label{laguerre:fig:integrand_exp} +\end{figure} + +In Abbildung~\ref{laguerre:fig:integrand_exp} fügen wir +die Dämpfung der Gewichtsfunktion $w(x)$ +der Gauss-Laguerre-Quadratur wieder hinzu +und erhalten so wieder den kompletten Integranden $x^{z-1} e^{-x}$ +der Gamma-Funktion. +Für negative $z$ ergeben sich immer noch Singularitäten, +wenn $x \rightarrow 0$. +Um $1$ wächst der Term $x^z$ schneller als die Dämpfung $e^{-x}$, +aber für $x \rightarrow \infty$ geht der Integrand gegen $0$. +Das führt zu glockenförmigen Kurven, +die für grosse Exponenten $z$ nach der Stelle $x=1$ schnell anwachsen. +Zu grosse Exponenten $z$ sind also immer noch problematisch. +Kleine positive $z$ scheinen nun also auch zulässig zu sein. +Damit formulieren wir die Vermutung, +dass $a(n)$, +welches das Intervall $[a(n), a(n) + 1]$ definiert, +in dem der relative Fehler minimal ist, +grösser als $0$ und kleiner als $2n-1$ ist. + +\subsubsection{Ansatz mit Verschiebungsterm} +% Mittels der Funktionalgleichung \eqref{laguerre:gamma_funktional} +% kann der Wert von $\Gamma(z)$ im Interval $z \in [a,a+1]$, +% in dem der relative Fehler minimal ist, +% evaluiert werden und dann mit der Funktionalgleichung zurückverschoben werden. Nun stellt sich die Frage, ob die Approximation mittels Gauss-Laguerre-Quadratur verbessert werden kann, -wenn man das Problem an einer geeigneten Stelle evaluiert und -dann zurückverschiebt mit der Funktionalgleichung. -Dazu wollen wir den Fehlerterm in -Gleichung~\eqref{laguerre:lagurre:lag_error} anpassen und dann minimieren. -Zunächst wollen wir dies nur für $z\in \mathbb{R}$ und $0<z<1$ definieren. -Zudem nehmen wir an, dass die optimale Stelle $x^* \in \mathbb{R}$, $z < x^*$ -ist. -Dann fügen wir einen Verschiebungsterm um $m$ Stellen ein, daraus folgt +wenn man das Problem in einem geeigneten Intervall $[a(n), a(n)+1]$, +$a(n) \in \mathbb{Z}$, +evaluiert und dann mit der +Funktionalgleichung \eqref{laguerre:gamma_funktional} zurückverschiebt. +Für dieses Vorhaben führen wir einen Verschiebungsterm $m \in \mathbb{Z}$ ein. +Passen wir \eqref{laguerre:naive_lag} +mit dem Verschiebungsterm $m$ +%,der $z$ and die Stelle $z_m = z + m$ verschiebt, +an, +ergibt sich +\begin{align} +\Gamma(z) +\approx +s(z, m) \sum_{i=1}^n x_i^{z + m - 1} A_i +% && +% \text{mit } +% s(z, m) +% = +% \begin{cases} +% \displaystyle +% \frac{1}{(z - m)_m} & \text{wenn } m \geq 0\\ +% (z + m)_{-m} & \text{wenn } m < 0 +% \end{cases} +% . +\label{laguerre:shifted_lag} +\end{align} +mit \begin{align*} -R_n +s(z, m) = -\frac{(z - 2n)_{2n}}{(z - m)_m} \frac{(n!)^2}{(2n)!} \xi^{z + m - 2n - 1} +\begin{cases} +\displaystyle +\frac{1}{(z)_m} & \text{wenn } m \geq 0 \\ +(z + m)_{-m} & \text{wenn } m < 0 +\end{cases} . \end{align*} -{ -\large \color{red} -TODO: -Geeignete Minimierung für Fehler finden, so dass sie mit den emprisich -bestimmen optimalen Punkten übereinstimmen. -} +\subsubsection{Finden der optimalen Berechnungsstelle} +Um die optimale Stelle $z^*(n) \in \left[a(n), a(n) + 1\right]$, +$z^*(n) \in \mathbb{R}$, +zu finden, +erweitern wir denn Fehlerterm \eqref{laguerre:gamma_err_simple} +und erhalten +\begin{align} +R_{n,m}(\xi) += +s(z, m) \cdot (z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z + m - 2n - 1} +,\quad +\text{für } +\xi \in (0, \infty) +\label{laguerre:gamma_err_shifted} +. +\end{align} + +\begin{figure} +\centering +\includegraphics{papers/laguerre/images/targets.pdf} +% %\vspace{-12pt} +\caption{$a$ in Abhängigkeit von $z$ und $n$} +\label{laguerre:fig:targets} +\end{figure} +% wobei ist +% mit $z^*(n) \in \mathbb{R}$ wollen wir finden, +% in dem wir den Fehlerterm \eqref{laguerre:lag_error} anpassen +% und in einem nächsten Schritt minimieren. +% Zudem nehmen wir an, +% dass $z < z^*(n)$ ist. +% Wir fügen einen Verschiebungsterm um $m \in \mathbb{N}$ Stellen ein, +% daraus folgt +% +% Damit wir den idealen Verschiebungsterm $m^*$ finden können, +% müssen wir mittels des Fehlerterms \eqref{laguerre:gamma_err_shifted} +% ein Optimierungsproblem +% +% Das Optimierungsproblem daraus lässt sich als +Daraus formulieren wir das Optimierungproblem +\begin{align*} +m^* += +\operatorname*{argmin}_m \max_\xi R_{n,m}(\xi) +. +\end{align*} +Allerdings ist die Funktion $R_{n,m}(\xi)$ unbeschränkt und +hat die gleichen Probleme wie die Fehlerabschätzung des direkten Ansatzes. +Dazu müssten wir $\xi$ versuchen, +unter Kontrolle zu bringen, +was ein äussersts schwieriges Unterfangen zu sein scheint. +Da die Gauss-Quadratur aber sowieso +nur wirklich praktisch sinnvoll für kleine $n$ ist, +können die Intervalle $[a(n), a(n)+1]$ empirisch gesucht werden. + +Wir bestimmen nun die optimalen Verschiebungsterme empirisch +für $n = 2,\ldots, 12$ im Intervall $z \in (0, 1)$, +da $z$ sowieso um den Term $m$ verschoben wird, +reicht die $m^*$ nur in diesem Intervall zu analysieren. +In Abbildung~\ref{laguerre:fig:targets} sind die empirisch bestimmten $m^*$ +abhängig von $z$ und $n$ dargestellt. +In $n$-Richtung lässt sich eine klare lineare Abhängigkeit erkennen und +die Beziehung zu $z$ ist negativ, +d.h. wenn $z$ grösser ist, wird $m^*$ kleiner. +Allerdings ist die genaue Beziehung zu $z$ +aus dieser Grafik nicht offensichtlich, +aber sie scheint regelmässig zu sein. +Es lässt die Vermutung aufkommen, +dass die Restriktion von $m^* \in \mathbb{Z}$ Rundungsprobleme verursacht. +Wir versuchen, +dieses Problem via lineare Regression und geeignete Rundung zu beheben. +Den linearen Regressor +\begin{align*} +\hat{m} += +\alpha n + \beta +\end{align*} +machen wir nur abhängig von $n$ +in dem wir den Mittelwert $\overline{m}$ von $m^*$ über $z$ berechnen. + +\begin{figure} +\centering +% \input{papers/laguerre/images/estimates.pgf} +\includegraphics{papers/laguerre/images/estimates.pdf} +%\vspace{-12pt} +\caption{Schätzung Mittelwert von $m$ und Fehler} +\label{laguerre:fig:schaetzung} +\end{figure} -\subsection{Resultate} +In Abbildung~\ref{laguerre:fig:schaetzung} sind die Resultate +der linearen Regression aufgezeigt mit $\alpha = 1.34094$ und $\beta = +0.854093$. +Die lineare Beziehung ist ganz klar ersichtlich und der Fit scheint zu genügen. +Der optimale Verschiebungsterm kann nun mit +\begin{align*} +m^* +\approx +\lceil \hat{m} - z \rceil += +\lceil \alpha n + \beta - z \rceil +\end{align*} +% kann nun mit dem linearen Regressor und $z$ +gefunden werden. + +\subsubsection{Evaluation des Schätzers} +In einem ersten Schritt möchten wir analysieren, +wie gut die Abschätzung des optimalen Verschiebungsterms ist. +Dazu bestimmen wir den relativen Fehler für verschiedene Verschiebungsterme $m$ +rund um $m^*$ bei gegebenem Polynomgrad $n = 8$ für $z \in (0, 1)$. +Abbildung~\ref{laguerre:fig:rel_error_shifted} sind die relativen Fehler +der Approximation dargestellt. +Man kann deutlich sehen, +dass der relative Fehler anwächst, +je weiter der Verschiebungsterm vom idealen Wert abweicht. +Zudem scheint der Schätzer den optimalen Verschiebungsterm gut zu bestimmen, +da der Schätzer zuerst der grünen Linie folgt und +dann beim Übergang auf die orange Linie wechselt. +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_shifted.pgf} +\includegraphics{papers/laguerre/images/rel_error_shifted.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des Ansatzes mit Verschiebungsterm +für verschiedene reele Werte von $z$ und Verschiebungsterme $m$. +Das verwendete Laguerre-Polynom besitzt den Grad $n = 8$. +$m^*$ bezeichnet hier den optimalen Verschiebungsterm.} +\label{laguerre:fig:rel_error_shifted} +\end{figure} + +\subsubsection{Resultate} +Das Verfahren scheint für den Grad $n=8$ und $z \in (0,1)$ gut zu funktioneren. +Es stellt sich nun die Frage, +wie der relative Fehler sich für verschiedene $z$ und $n$ verhält. +In Abbildung~\ref{laguerre:fig:rel_error_range} sind die relativen Fehler für +unterschiedliche $n$ dargestellt. +Der relative Fehler scheint immer noch Nullstellen aufzuweisen +für ganzzahlige $z$. +Durch das Verschieben ergibt sich jetzt aber, +wie zu erwarten war, +ein periodischer relativer Fehler mit einer Periodendauer von $1$. +Zudem lässt sich erkennen, +dass der Fehler abhängig von der Ordnung $n$ +des verwendeten Laguerre-Polynoms ist. +Wenn der Grad $n$ um $1$ erhöht wird, +verbessert sich die Genauigkeit des Resultats um etwa eine signifikante Stelle. + +In Abbildung~\ref{laguerre:fig:rel_error_complex} +ist der Betrag des relativen Fehlers in der komplexen Ebene dargestellt. +Je stärker der Imaginäranteil von $z$ von $0$ abweicht, +umso schlechter wird die Genauigkeit der Approximation. +Das erstaunt nicht weiter, +da die Gauss-Quadratur eigentlich nur für reelle Zahlen definiert ist. +Wenn der Imaginäranteil von $z$ ungefähr $0$ ist, +lässt sich das gleiche Bild beobachten wie in +Abbildung~\ref{laguerre:fig:rel_error_range}. + +\begin{figure} +\centering +% \input{papers/laguerre/images/rel_error_range.pgf} +\includegraphics{papers/laguerre/images/rel_error_range.pdf} +%\vspace{-12pt} +\caption{Relativer Fehler des Ansatzes mit optimalen Verschiebungsterm +für verschiedene reele Werte von $z$ und Laguerre-Polynome vom Grad $n$} +\label{laguerre:fig:rel_error_range} +\end{figure} + +\begin{figure} +\centering +\includegraphics{papers/laguerre/images/rel_error_complex.pdf} +%\vspace{-12pt} +\caption{Absolutwert des relativen Fehlers in der komplexen Ebene} +\label{laguerre:fig:rel_error_complex} +\end{figure} + +\subsubsection{Vergleich mit Lanczos-Methode} +Nun stellt sich die Frage, +wie das in diesem Abschnitt beschriebene Approximationsverfahren +der Gamma-Funktion sich gegenüber den üblichen Approximationsverfahren schlägt. +Eine häufig verwendete Methode ist die Lanczos-Approximation, +welche gegeben ist durch +\begin{align} +\Gamma(z + 1) +\approx +\sqrt{2\pi} \left( z + \sigma + \frac{1}{2} \right)^{z + 1/2} +e^{-(z + \sigma + 1/2)} \sum_{k=0}^n g_k H_k(z) +, +\end{align} +wobei +\begin{align*} +g_k = \frac{e^\sigma \varepsilon_k (-1)^k}{\sqrt{2\pi}} +\sum_{r=0}^k (-1)^r \, \binom{k}{r} \, (k)_r +\left( \frac{e}{r + \sigma + \frac{1}{2}}\right)^{r + 1/2} +, +\end{align*} +\begin{align*} +\varepsilon_k += +\begin{cases} +1 & \text{für } k = 0 \\ +2 & \text{sonst} +\end{cases} +\quad \text{und}\quad +H_k(z) += +\frac{(-1)^k (-z)_k}{(z+1)_k} +\end{align*} +mit $H_0 = 1$ und $\sum_0^n g_k = 1$ (siehe \cite{laguerre:lanczos}). +Diese Methode wurde zum Beispiel in +{\em GNU Scientific Library}, {\em Boost}, {\em CPython} und +{\em musl} implementiert. +Diese Methode erreicht für $n = 7$ typischerweise Genauigkeit von $13$ +korrekten, signifikanten Stellen für reele Argumente. +Zum Vergleich: die vorgestellte Methode erreicht für $n = 7$ +eine minimale Genauigkeit von $6$ korrekten, signifikanten Stellen +für reele Argumente. +Das Resultat ist etwas enttäuschend, +aber nicht unerwartet, +da die Lanczos-Methode spezifisch auf dieses Problem zugeschnitten ist und +unsere Methode eine erweiterte allgemeine Methode ist. +Was die Komplexität der Berechnungen im Betrieb angeht, +ist die Gauss-Laguerre-Quadratur wesentlich ressourcensparender, +weil sie nur aus $n$ Funktionsevaluationen, +wenigen Multiplikationen und Additionen besteht. +Demzufolge könnte diese Methode Anwendung in Systemen mit wenig Rechenleistung +und/oder knappen Energieressourcen finden.
\ No newline at end of file diff --git a/buch/papers/laguerre/images/estimates.pdf b/buch/papers/laguerre/images/estimates.pdf Binary files differnew file mode 100644 index 0000000..bd995de --- /dev/null +++ b/buch/papers/laguerre/images/estimates.pdf diff --git a/buch/papers/laguerre/images/gammapaths.tex b/buch/papers/laguerre/images/gammapaths.tex new file mode 100644 index 0000000..efa0863 --- /dev/null +++ b/buch/papers/laguerre/images/gammapaths.tex @@ -0,0 +1,1024 @@ +\def\gammaplus{({\dx*0.0190},{\dy*52.0728}) + -- ({\dx*0.0200},{\dy*49.4422}) + -- ({\dx*0.0400},{\dy*24.4610}) + -- ({\dx*0.0600},{\dy*16.1457}) + -- ({\dx*0.0800},{\dy*11.9966}) + -- ({\dx*0.1000},{\dy*9.5135}) + -- ({\dx*0.1200},{\dy*7.8633}) + -- ({\dx*0.1400},{\dy*6.6887}) + -- ({\dx*0.1600},{\dy*5.8113}) + -- ({\dx*0.1800},{\dy*5.1318}) + -- ({\dx*0.2000},{\dy*4.5908}) + -- ({\dx*0.2200},{\dy*4.1505}) + -- ({\dx*0.2400},{\dy*3.7855}) + -- ({\dx*0.2600},{\dy*3.4785}) + -- ({\dx*0.2800},{\dy*3.2169}) + -- ({\dx*0.3000},{\dy*2.9916}) + -- ({\dx*0.3200},{\dy*2.7958}) + -- ({\dx*0.3400},{\dy*2.6242}) + -- ({\dx*0.3600},{\dy*2.4727}) + -- ({\dx*0.3800},{\dy*2.3383}) + -- ({\dx*0.4000},{\dy*2.2182}) + -- ({\dx*0.4200},{\dy*2.1104}) + -- ({\dx*0.4400},{\dy*2.0132}) + -- ({\dx*0.4600},{\dy*1.9252}) + -- ({\dx*0.4800},{\dy*1.8453}) + -- ({\dx*0.5000},{\dy*1.7725}) + -- ({\dx*0.5200},{\dy*1.7058}) + -- ({\dx*0.5400},{\dy*1.6448}) + -- ({\dx*0.5600},{\dy*1.5886}) + -- ({\dx*0.5800},{\dy*1.5369}) + -- ({\dx*0.6000},{\dy*1.4892}) + -- ({\dx*0.6200},{\dy*1.4450}) + -- ({\dx*0.6400},{\dy*1.4041}) + -- ({\dx*0.6600},{\dy*1.3662}) + -- ({\dx*0.6800},{\dy*1.3309}) + -- ({\dx*0.7000},{\dy*1.2981}) + -- ({\dx*0.7200},{\dy*1.2675}) + -- ({\dx*0.7400},{\dy*1.2390}) + -- ({\dx*0.7600},{\dy*1.2123}) + -- ({\dx*0.7800},{\dy*1.1875}) + -- ({\dx*0.8000},{\dy*1.1642}) + -- ({\dx*0.8200},{\dy*1.1425}) + -- ({\dx*0.8400},{\dy*1.1222}) + -- ({\dx*0.8600},{\dy*1.1031}) + -- ({\dx*0.8800},{\dy*1.0853}) + -- ({\dx*0.9000},{\dy*1.0686}) + -- ({\dx*0.9200},{\dy*1.0530}) + -- ({\dx*0.9400},{\dy*1.0384}) + -- ({\dx*0.9600},{\dy*1.0247}) + -- ({\dx*0.9800},{\dy*1.0119}) + -- ({\dx*1.0000},{\dy*1.0000}) + -- ({\dx*1.0200},{\dy*0.9888}) + -- ({\dx*1.0400},{\dy*0.9784}) + -- ({\dx*1.0600},{\dy*0.9687}) + -- ({\dx*1.0800},{\dy*0.9597}) + -- ({\dx*1.1000},{\dy*0.9514}) + -- ({\dx*1.1200},{\dy*0.9436}) + -- ({\dx*1.1400},{\dy*0.9364}) + -- ({\dx*1.1600},{\dy*0.9298}) + -- ({\dx*1.1800},{\dy*0.9237}) + -- ({\dx*1.2000},{\dy*0.9182}) + -- ({\dx*1.2200},{\dy*0.9131}) + -- ({\dx*1.2400},{\dy*0.9085}) + -- ({\dx*1.2600},{\dy*0.9044}) + -- ({\dx*1.2800},{\dy*0.9007}) + -- ({\dx*1.3000},{\dy*0.8975}) + -- ({\dx*1.3200},{\dy*0.8946}) + -- ({\dx*1.3400},{\dy*0.8922}) + -- ({\dx*1.3600},{\dy*0.8902}) + -- ({\dx*1.3800},{\dy*0.8885}) + -- ({\dx*1.4000},{\dy*0.8873}) + -- ({\dx*1.4200},{\dy*0.8864}) + -- ({\dx*1.4400},{\dy*0.8858}) + -- ({\dx*1.4600},{\dy*0.8856}) + -- ({\dx*1.4800},{\dy*0.8857}) + -- ({\dx*1.5000},{\dy*0.8862}) + -- ({\dx*1.5200},{\dy*0.8870}) + -- ({\dx*1.5400},{\dy*0.8882}) + -- ({\dx*1.5600},{\dy*0.8896}) + -- ({\dx*1.5800},{\dy*0.8914}) + -- ({\dx*1.6000},{\dy*0.8935}) + -- ({\dx*1.6200},{\dy*0.8959}) + -- ({\dx*1.6400},{\dy*0.8986}) + -- ({\dx*1.6600},{\dy*0.9017}) + -- ({\dx*1.6800},{\dy*0.9050}) + -- ({\dx*1.7000},{\dy*0.9086}) + -- ({\dx*1.7200},{\dy*0.9126}) + -- ({\dx*1.7400},{\dy*0.9168}) + -- ({\dx*1.7600},{\dy*0.9214}) + -- ({\dx*1.7800},{\dy*0.9262}) + -- ({\dx*1.8000},{\dy*0.9314}) + -- ({\dx*1.8200},{\dy*0.9368}) + -- ({\dx*1.8400},{\dy*0.9426}) + -- ({\dx*1.8600},{\dy*0.9487}) + -- ({\dx*1.8800},{\dy*0.9551}) + -- ({\dx*1.9000},{\dy*0.9618}) + -- ({\dx*1.9200},{\dy*0.9688}) + -- ({\dx*1.9400},{\dy*0.9761}) + -- ({\dx*1.9600},{\dy*0.9837}) + -- ({\dx*1.9800},{\dy*0.9917}) + -- ({\dx*2.0000},{\dy*1.0000}) + -- ({\dx*2.0200},{\dy*1.0086}) + -- ({\dx*2.0400},{\dy*1.0176}) + -- ({\dx*2.0600},{\dy*1.0269}) + -- ({\dx*2.0800},{\dy*1.0365}) + -- ({\dx*2.1000},{\dy*1.0465}) + -- ({\dx*2.1200},{\dy*1.0568}) + -- ({\dx*2.1400},{\dy*1.0675}) + -- ({\dx*2.1600},{\dy*1.0786}) + -- ({\dx*2.1800},{\dy*1.0900}) + -- ({\dx*2.2000},{\dy*1.1018}) + -- ({\dx*2.2200},{\dy*1.1140}) + -- ({\dx*2.2400},{\dy*1.1266}) + -- ({\dx*2.2600},{\dy*1.1395}) + -- ({\dx*2.2800},{\dy*1.1529}) + -- ({\dx*2.3000},{\dy*1.1667}) + -- ({\dx*2.3200},{\dy*1.1809}) + -- ({\dx*2.3400},{\dy*1.1956}) + -- ({\dx*2.3600},{\dy*1.2107}) + -- ({\dx*2.3800},{\dy*1.2262}) + -- ({\dx*2.4000},{\dy*1.2422}) + -- ({\dx*2.4200},{\dy*1.2586}) + -- ({\dx*2.4400},{\dy*1.2756}) + -- ({\dx*2.4600},{\dy*1.2930}) + -- ({\dx*2.4800},{\dy*1.3109}) + -- ({\dx*2.5000},{\dy*1.3293}) + -- ({\dx*2.5200},{\dy*1.3483}) + -- ({\dx*2.5400},{\dy*1.3678}) + -- ({\dx*2.5600},{\dy*1.3878}) + -- ({\dx*2.5800},{\dy*1.4084}) + -- ({\dx*2.6000},{\dy*1.4296}) + -- ({\dx*2.6200},{\dy*1.4514}) + -- ({\dx*2.6400},{\dy*1.4738}) + -- ({\dx*2.6600},{\dy*1.4968}) + -- ({\dx*2.6800},{\dy*1.5204}) + -- ({\dx*2.7000},{\dy*1.5447}) + -- ({\dx*2.7200},{\dy*1.5696}) + -- ({\dx*2.7400},{\dy*1.5953}) + -- ({\dx*2.7600},{\dy*1.6216}) + -- ({\dx*2.7800},{\dy*1.6487}) + -- ({\dx*2.8000},{\dy*1.6765}) + -- ({\dx*2.8200},{\dy*1.7051}) + -- ({\dx*2.8400},{\dy*1.7344}) + -- ({\dx*2.8600},{\dy*1.7646}) + -- ({\dx*2.8800},{\dy*1.7955}) + -- ({\dx*2.9000},{\dy*1.8274}) + -- ({\dx*2.9200},{\dy*1.8600}) + -- ({\dx*2.9400},{\dy*1.8936}) + -- ({\dx*2.9600},{\dy*1.9281}) + -- ({\dx*2.9800},{\dy*1.9636}) + -- ({\dx*3.0000},{\dy*2.0000}) + -- ({\dx*3.0200},{\dy*2.0374}) + -- ({\dx*3.0400},{\dy*2.0759}) + -- ({\dx*3.0600},{\dy*2.1153}) + -- ({\dx*3.0800},{\dy*2.1559}) + -- ({\dx*3.1000},{\dy*2.1976}) + -- ({\dx*3.1200},{\dy*2.2405}) + -- ({\dx*3.1400},{\dy*2.2845}) + -- ({\dx*3.1600},{\dy*2.3297}) + -- ({\dx*3.1800},{\dy*2.3762}) + -- ({\dx*3.2000},{\dy*2.4240}) + -- ({\dx*3.2200},{\dy*2.4731}) + -- ({\dx*3.2400},{\dy*2.5235}) + -- ({\dx*3.2600},{\dy*2.5754}) + -- ({\dx*3.2800},{\dy*2.6287}) + -- ({\dx*3.3000},{\dy*2.6834}) + -- ({\dx*3.3200},{\dy*2.7397}) + -- ({\dx*3.3400},{\dy*2.7976}) + -- ({\dx*3.3600},{\dy*2.8571}) + -- ({\dx*3.3800},{\dy*2.9183}) + -- ({\dx*3.4000},{\dy*2.9812}) + -- ({\dx*3.4200},{\dy*3.0459}) + -- ({\dx*3.4400},{\dy*3.1124}) + -- ({\dx*3.4600},{\dy*3.1807}) + -- ({\dx*3.4800},{\dy*3.2510}) + -- ({\dx*3.5000},{\dy*3.3234}) + -- ({\dx*3.5200},{\dy*3.3977}) + -- ({\dx*3.5400},{\dy*3.4742}) + -- ({\dx*3.5600},{\dy*3.5529}) + -- ({\dx*3.5800},{\dy*3.6338}) + -- ({\dx*3.6000},{\dy*3.7170}) + -- ({\dx*3.6200},{\dy*3.8027}) + -- ({\dx*3.6400},{\dy*3.8908}) + -- ({\dx*3.6600},{\dy*3.9814}) + -- ({\dx*3.6800},{\dy*4.0747}) + -- ({\dx*3.7000},{\dy*4.1707}) + -- ({\dx*3.7200},{\dy*4.2694}) + -- ({\dx*3.7400},{\dy*4.3711}) + -- ({\dx*3.7600},{\dy*4.4757}) + -- ({\dx*3.7800},{\dy*4.5833}) + -- ({\dx*3.8000},{\dy*4.6942}) + -- ({\dx*3.8200},{\dy*4.8083}) + -- ({\dx*3.8400},{\dy*4.9257}) + -- ({\dx*3.8600},{\dy*5.0466}) + -- ({\dx*3.8800},{\dy*5.1711}) + -- ({\dx*3.9000},{\dy*5.2993}) + -- ({\dx*3.9200},{\dy*5.4313}) + -- ({\dx*3.9400},{\dy*5.5673}) + -- ({\dx*3.9600},{\dy*5.7073}) + -- ({\dx*3.9800},{\dy*5.8515}) + -- ({\dx*4.0000},{\dy*6.0000}) + -- ({\dx*4.0200},{\dy*6.1530}) + -- ({\dx*4.0400},{\dy*6.3106}) + -- ({\dx*4.0600},{\dy*6.4730}) + -- ({\dx*4.0800},{\dy*6.6403}) + -- ({\dx*4.0810},{\dy*6.6488})} +\def\gammaone{({\dx*-0.9810},{\dy*-53.0814}) + -- ({\dx*-0.9800},{\dy*-50.4512}) + -- ({\dx*-0.9600},{\dy*-25.4802}) + -- ({\dx*-0.9400},{\dy*-17.1763}) + -- ({\dx*-0.9200},{\dy*-13.0397}) + -- ({\dx*-0.9000},{\dy*-10.5706}) + -- ({\dx*-0.8800},{\dy*-8.9355}) + -- ({\dx*-0.8600},{\dy*-7.7775}) + -- ({\dx*-0.8400},{\dy*-6.9182}) + -- ({\dx*-0.8200},{\dy*-6.2583}) + -- ({\dx*-0.8000},{\dy*-5.7386}) + -- ({\dx*-0.7800},{\dy*-5.3211}) + -- ({\dx*-0.7600},{\dy*-4.9809}) + -- ({\dx*-0.7400},{\dy*-4.7006}) + -- ({\dx*-0.7200},{\dy*-4.4678}) + -- ({\dx*-0.7000},{\dy*-4.2737}) + -- ({\dx*-0.6800},{\dy*-4.1114}) + -- ({\dx*-0.6600},{\dy*-3.9760}) + -- ({\dx*-0.6400},{\dy*-3.8636}) + -- ({\dx*-0.6200},{\dy*-3.7714}) + -- ({\dx*-0.6000},{\dy*-3.6969}) + -- ({\dx*-0.5800},{\dy*-3.6386}) + -- ({\dx*-0.5600},{\dy*-3.5950}) + -- ({\dx*-0.5400},{\dy*-3.5652}) + -- ({\dx*-0.5200},{\dy*-3.5487}) + -- ({\dx*-0.5000},{\dy*-3.5449}) + -- ({\dx*-0.4800},{\dy*-3.5538}) + -- ({\dx*-0.4600},{\dy*-3.5756}) + -- ({\dx*-0.4400},{\dy*-3.6105}) + -- ({\dx*-0.4200},{\dy*-3.6594}) + -- ({\dx*-0.4000},{\dy*-3.7230}) + -- ({\dx*-0.3800},{\dy*-3.8027}) + -- ({\dx*-0.3600},{\dy*-3.9004}) + -- ({\dx*-0.3400},{\dy*-4.0181}) + -- ({\dx*-0.3200},{\dy*-4.1590}) + -- ({\dx*-0.3000},{\dy*-4.3269}) + -- ({\dx*-0.2800},{\dy*-4.5267}) + -- ({\dx*-0.2600},{\dy*-4.7652}) + -- ({\dx*-0.2400},{\dy*-5.0514}) + -- ({\dx*-0.2200},{\dy*-5.3976}) + -- ({\dx*-0.2000},{\dy*-5.8211}) + -- ({\dx*-0.1800},{\dy*-6.3472}) + -- ({\dx*-0.1600},{\dy*-7.0135}) + -- ({\dx*-0.1400},{\dy*-7.8795}) + -- ({\dx*-0.1200},{\dy*-9.0442}) + -- ({\dx*-0.1000},{\dy*-10.6863}) + -- ({\dx*-0.0800},{\dy*-13.1627}) + -- ({\dx*-0.0600},{\dy*-17.3067}) + -- ({\dx*-0.0400},{\dy*-25.6183}) + -- ({\dx*-0.0200},{\dy*-50.5974}) + -- ({\dx*-0.0190},{\dy*-53.2279})} +\def\gammatwo{({\dx*-1.9810},{\dy*26.7952}) + -- ({\dx*-1.9800},{\dy*25.4804}) + -- ({\dx*-1.9600},{\dy*13.0001}) + -- ({\dx*-1.9400},{\dy*8.8538}) + -- ({\dx*-1.9200},{\dy*6.7915}) + -- ({\dx*-1.9000},{\dy*5.5635}) + -- ({\dx*-1.8800},{\dy*4.7529}) + -- ({\dx*-1.8600},{\dy*4.1815}) + -- ({\dx*-1.8400},{\dy*3.7599}) + -- ({\dx*-1.8200},{\dy*3.4386}) + -- ({\dx*-1.8000},{\dy*3.1881}) + -- ({\dx*-1.7800},{\dy*2.9894}) + -- ({\dx*-1.7600},{\dy*2.8301}) + -- ({\dx*-1.7400},{\dy*2.7015}) + -- ({\dx*-1.7200},{\dy*2.5976}) + -- ({\dx*-1.7000},{\dy*2.5139}) + -- ({\dx*-1.6800},{\dy*2.4473}) + -- ({\dx*-1.6600},{\dy*2.3952}) + -- ({\dx*-1.6400},{\dy*2.3559}) + -- ({\dx*-1.6200},{\dy*2.3280}) + -- ({\dx*-1.6000},{\dy*2.3106}) + -- ({\dx*-1.5800},{\dy*2.3029}) + -- ({\dx*-1.5600},{\dy*2.3045}) + -- ({\dx*-1.5400},{\dy*2.3151}) + -- ({\dx*-1.5200},{\dy*2.3346}) + -- ({\dx*-1.5000},{\dy*2.3633}) + -- ({\dx*-1.4800},{\dy*2.4012}) + -- ({\dx*-1.4600},{\dy*2.4490}) + -- ({\dx*-1.4400},{\dy*2.5073}) + -- ({\dx*-1.4200},{\dy*2.5770}) + -- ({\dx*-1.4000},{\dy*2.6593}) + -- ({\dx*-1.3800},{\dy*2.7556}) + -- ({\dx*-1.3600},{\dy*2.8679}) + -- ({\dx*-1.3400},{\dy*2.9986}) + -- ({\dx*-1.3200},{\dy*3.1508}) + -- ({\dx*-1.3000},{\dy*3.3283}) + -- ({\dx*-1.2800},{\dy*3.5365}) + -- ({\dx*-1.2600},{\dy*3.7819}) + -- ({\dx*-1.2400},{\dy*4.0737}) + -- ({\dx*-1.2200},{\dy*4.4243}) + -- ({\dx*-1.2000},{\dy*4.8510}) + -- ({\dx*-1.1800},{\dy*5.3790}) + -- ({\dx*-1.1600},{\dy*6.0461}) + -- ({\dx*-1.1400},{\dy*6.9118}) + -- ({\dx*-1.1200},{\dy*8.0752}) + -- ({\dx*-1.1000},{\dy*9.7148}) + -- ({\dx*-1.0800},{\dy*12.1877}) + -- ({\dx*-1.0600},{\dy*16.3271}) + -- ({\dx*-1.0400},{\dy*24.6330}) + -- ({\dx*-1.0200},{\dy*49.6053}) + -- ({\dx*-1.0190},{\dy*52.2354})} +\def\gammathree{({\dx*-2.9810},{\dy*-8.9887}) + -- ({\dx*-2.9800},{\dy*-8.5505}) + -- ({\dx*-2.9600},{\dy*-4.3919}) + -- ({\dx*-2.9400},{\dy*-3.0115}) + -- ({\dx*-2.9200},{\dy*-2.3259}) + -- ({\dx*-2.9000},{\dy*-1.9184}) + -- ({\dx*-2.8800},{\dy*-1.6503}) + -- ({\dx*-2.8600},{\dy*-1.4621}) + -- ({\dx*-2.8400},{\dy*-1.3239}) + -- ({\dx*-2.8200},{\dy*-1.2194}) + -- ({\dx*-2.8000},{\dy*-1.1386}) + -- ({\dx*-2.7800},{\dy*-1.0753}) + -- ({\dx*-2.7600},{\dy*-1.0254}) + -- ({\dx*-2.7400},{\dy*-0.9859}) + -- ({\dx*-2.7200},{\dy*-0.9550}) + -- ({\dx*-2.7000},{\dy*-0.9311}) + -- ({\dx*-2.6800},{\dy*-0.9132}) + -- ({\dx*-2.6600},{\dy*-0.9004}) + -- ({\dx*-2.6400},{\dy*-0.8924}) + -- ({\dx*-2.6200},{\dy*-0.8886}) + -- ({\dx*-2.6000},{\dy*-0.8887}) + -- ({\dx*-2.5800},{\dy*-0.8926}) + -- ({\dx*-2.5600},{\dy*-0.9002}) + -- ({\dx*-2.5400},{\dy*-0.9115}) + -- ({\dx*-2.5200},{\dy*-0.9264}) + -- ({\dx*-2.5000},{\dy*-0.9453}) + -- ({\dx*-2.4800},{\dy*-0.9682}) + -- ({\dx*-2.4600},{\dy*-0.9955}) + -- ({\dx*-2.4400},{\dy*-1.0276}) + -- ({\dx*-2.4200},{\dy*-1.0649}) + -- ({\dx*-2.4000},{\dy*-1.1080}) + -- ({\dx*-2.3800},{\dy*-1.1578}) + -- ({\dx*-2.3600},{\dy*-1.2152}) + -- ({\dx*-2.3400},{\dy*-1.2815}) + -- ({\dx*-2.3200},{\dy*-1.3581}) + -- ({\dx*-2.3000},{\dy*-1.4471}) + -- ({\dx*-2.2800},{\dy*-1.5511}) + -- ({\dx*-2.2600},{\dy*-1.6734}) + -- ({\dx*-2.2400},{\dy*-1.8186}) + -- ({\dx*-2.2200},{\dy*-1.9929}) + -- ({\dx*-2.2000},{\dy*-2.2050}) + -- ({\dx*-2.1800},{\dy*-2.4674}) + -- ({\dx*-2.1600},{\dy*-2.7991}) + -- ({\dx*-2.1400},{\dy*-3.2298}) + -- ({\dx*-2.1200},{\dy*-3.8091}) + -- ({\dx*-2.1000},{\dy*-4.6261}) + -- ({\dx*-2.0800},{\dy*-5.8595}) + -- ({\dx*-2.0600},{\dy*-7.9258}) + -- ({\dx*-2.0400},{\dy*-12.0750}) + -- ({\dx*-2.0200},{\dy*-24.5571}) + -- ({\dx*-2.0190},{\dy*-25.8719})} +\def\gammafour{({\dx*-3.9950},{\dy*8.3966}) + -- ({\dx*-3.9800},{\dy*2.1484}) + -- ({\dx*-3.9600},{\dy*1.1091}) + -- ({\dx*-3.9400},{\dy*0.7643}) + -- ({\dx*-3.9200},{\dy*0.5933}) + -- ({\dx*-3.9000},{\dy*0.4919}) + -- ({\dx*-3.8800},{\dy*0.4253}) + -- ({\dx*-3.8600},{\dy*0.3788}) + -- ({\dx*-3.8400},{\dy*0.3448}) + -- ({\dx*-3.8200},{\dy*0.3192}) + -- ({\dx*-3.8000},{\dy*0.2996}) + -- ({\dx*-3.7800},{\dy*0.2845}) + -- ({\dx*-3.7600},{\dy*0.2727}) + -- ({\dx*-3.7400},{\dy*0.2636}) + -- ({\dx*-3.7200},{\dy*0.2567}) + -- ({\dx*-3.7000},{\dy*0.2516}) + -- ({\dx*-3.6800},{\dy*0.2481}) + -- ({\dx*-3.6600},{\dy*0.2460}) + -- ({\dx*-3.6400},{\dy*0.2452}) + -- ({\dx*-3.6200},{\dy*0.2455}) + -- ({\dx*-3.6000},{\dy*0.2469}) + -- ({\dx*-3.5800},{\dy*0.2493}) + -- ({\dx*-3.5600},{\dy*0.2529}) + -- ({\dx*-3.5400},{\dy*0.2575}) + -- ({\dx*-3.5200},{\dy*0.2632}) + -- ({\dx*-3.5000},{\dy*0.2701}) + -- ({\dx*-3.4800},{\dy*0.2782}) + -- ({\dx*-3.4600},{\dy*0.2877}) + -- ({\dx*-3.4400},{\dy*0.2987}) + -- ({\dx*-3.4200},{\dy*0.3114}) + -- ({\dx*-3.4000},{\dy*0.3259}) + -- ({\dx*-3.3800},{\dy*0.3425}) + -- ({\dx*-3.3600},{\dy*0.3617}) + -- ({\dx*-3.3400},{\dy*0.3837}) + -- ({\dx*-3.3200},{\dy*0.4091}) + -- ({\dx*-3.3000},{\dy*0.4385}) + -- ({\dx*-3.2800},{\dy*0.4729}) + -- ({\dx*-3.2600},{\dy*0.5133}) + -- ({\dx*-3.2400},{\dy*0.5613}) + -- ({\dx*-3.2200},{\dy*0.6189}) + -- ({\dx*-3.2000},{\dy*0.6891}) + -- ({\dx*-3.1800},{\dy*0.7759}) + -- ({\dx*-3.1600},{\dy*0.8858}) + -- ({\dx*-3.1400},{\dy*1.0286}) + -- ({\dx*-3.1200},{\dy*1.2209}) + -- ({\dx*-3.1000},{\dy*1.4923}) + -- ({\dx*-3.0800},{\dy*1.9024}) + -- ({\dx*-3.0600},{\dy*2.5901}) + -- ({\dx*-3.0400},{\dy*3.9720}) + -- ({\dx*-3.0200},{\dy*8.1315}) + -- ({\dx*-3.0050},{\dy*33.1259})} +\def\gammafive{({\dx*-4.9990},{\dy*-8.3476}) + -- ({\dx*-4.9800},{\dy*-0.4314}) + -- ({\dx*-4.9600},{\dy*-0.2236}) + -- ({\dx*-4.9400},{\dy*-0.1547}) + -- ({\dx*-4.9200},{\dy*-0.1206}) + -- ({\dx*-4.9000},{\dy*-0.1004}) + -- ({\dx*-4.8800},{\dy*-0.0872}) + -- ({\dx*-4.8600},{\dy*-0.0779}) + -- ({\dx*-4.8400},{\dy*-0.0712}) + -- ({\dx*-4.8200},{\dy*-0.0662}) + -- ({\dx*-4.8000},{\dy*-0.0624}) + -- ({\dx*-4.7800},{\dy*-0.0595}) + -- ({\dx*-4.7600},{\dy*-0.0573}) + -- ({\dx*-4.7400},{\dy*-0.0556}) + -- ({\dx*-4.7200},{\dy*-0.0544}) + -- ({\dx*-4.7000},{\dy*-0.0535}) + -- ({\dx*-4.6800},{\dy*-0.0530}) + -- ({\dx*-4.6600},{\dy*-0.0528}) + -- ({\dx*-4.6400},{\dy*-0.0528}) + -- ({\dx*-4.6200},{\dy*-0.0531}) + -- ({\dx*-4.6000},{\dy*-0.0537}) + -- ({\dx*-4.5800},{\dy*-0.0544}) + -- ({\dx*-4.5600},{\dy*-0.0555}) + -- ({\dx*-4.5400},{\dy*-0.0567}) + -- ({\dx*-4.5200},{\dy*-0.0582}) + -- ({\dx*-4.5000},{\dy*-0.0600}) + -- ({\dx*-4.4800},{\dy*-0.0621}) + -- ({\dx*-4.4600},{\dy*-0.0645}) + -- ({\dx*-4.4400},{\dy*-0.0673}) + -- ({\dx*-4.4200},{\dy*-0.0704}) + -- ({\dx*-4.4000},{\dy*-0.0741}) + -- ({\dx*-4.3800},{\dy*-0.0782}) + -- ({\dx*-4.3600},{\dy*-0.0830}) + -- ({\dx*-4.3400},{\dy*-0.0884}) + -- ({\dx*-4.3200},{\dy*-0.0947}) + -- ({\dx*-4.3000},{\dy*-0.1020}) + -- ({\dx*-4.2800},{\dy*-0.1105}) + -- ({\dx*-4.2600},{\dy*-0.1205}) + -- ({\dx*-4.2400},{\dy*-0.1324}) + -- ({\dx*-4.2200},{\dy*-0.1467}) + -- ({\dx*-4.2000},{\dy*-0.1641}) + -- ({\dx*-4.1800},{\dy*-0.1856}) + -- ({\dx*-4.1600},{\dy*-0.2129}) + -- ({\dx*-4.1400},{\dy*-0.2485}) + -- ({\dx*-4.1200},{\dy*-0.2963}) + -- ({\dx*-4.1000},{\dy*-0.3640}) + -- ({\dx*-4.0800},{\dy*-0.4663}) + -- ({\dx*-4.0600},{\dy*-0.6380}) + -- ({\dx*-4.0400},{\dy*-0.9832}) + -- ({\dx*-4.0200},{\dy*-2.0228}) + -- ({\dx*-4.0010},{\dy*-41.6040})} +\def\gammasix{({\dx*-5.9998},{\dy*6.9470}) + -- ({\dx*-5.9800},{\dy*0.0721}) + -- ({\dx*-5.9600},{\dy*0.0375}) + -- ({\dx*-5.9400},{\dy*0.0260}) + -- ({\dx*-5.9200},{\dy*0.0204}) + -- ({\dx*-5.9000},{\dy*0.0170}) + -- ({\dx*-5.8800},{\dy*0.0148}) + -- ({\dx*-5.8600},{\dy*0.0133}) + -- ({\dx*-5.8400},{\dy*0.0122}) + -- ({\dx*-5.8200},{\dy*0.0114}) + -- ({\dx*-5.8000},{\dy*0.0108}) + -- ({\dx*-5.7800},{\dy*0.0103}) + -- ({\dx*-5.7600},{\dy*0.0099}) + -- ({\dx*-5.7400},{\dy*0.0097}) + -- ({\dx*-5.7200},{\dy*0.0095}) + -- ({\dx*-5.7000},{\dy*0.0094}) + -- ({\dx*-5.6800},{\dy*0.0093}) + -- ({\dx*-5.6600},{\dy*0.0093}) + -- ({\dx*-5.6400},{\dy*0.0094}) + -- ({\dx*-5.6200},{\dy*0.0095}) + -- ({\dx*-5.6000},{\dy*0.0096}) + -- ({\dx*-5.5800},{\dy*0.0098}) + -- ({\dx*-5.5600},{\dy*0.0100}) + -- ({\dx*-5.5400},{\dy*0.0102}) + -- ({\dx*-5.5200},{\dy*0.0105}) + -- ({\dx*-5.5000},{\dy*0.0109}) + -- ({\dx*-5.4800},{\dy*0.0113}) + -- ({\dx*-5.4600},{\dy*0.0118}) + -- ({\dx*-5.4400},{\dy*0.0124}) + -- ({\dx*-5.4200},{\dy*0.0130}) + -- ({\dx*-5.4000},{\dy*0.0137}) + -- ({\dx*-5.3800},{\dy*0.0145}) + -- ({\dx*-5.3600},{\dy*0.0155}) + -- ({\dx*-5.3400},{\dy*0.0166}) + -- ({\dx*-5.3200},{\dy*0.0178}) + -- ({\dx*-5.3000},{\dy*0.0192}) + -- ({\dx*-5.2800},{\dy*0.0209}) + -- ({\dx*-5.2600},{\dy*0.0229}) + -- ({\dx*-5.2400},{\dy*0.0253}) + -- ({\dx*-5.2200},{\dy*0.0281}) + -- ({\dx*-5.2000},{\dy*0.0316}) + -- ({\dx*-5.1800},{\dy*0.0358}) + -- ({\dx*-5.1600},{\dy*0.0413}) + -- ({\dx*-5.1400},{\dy*0.0483}) + -- ({\dx*-5.1200},{\dy*0.0579}) + -- ({\dx*-5.1000},{\dy*0.0714}) + -- ({\dx*-5.0800},{\dy*0.0918}) + -- ({\dx*-5.0600},{\dy*0.1261}) + -- ({\dx*-5.0400},{\dy*0.1951}) + -- ({\dx*-5.0200},{\dy*0.4029}) + -- ({\dx*-5.0002},{\dy*41.6525})} +\def\gammasinplus{({\dx*0.0190},{\dy*52.1325}) + -- ({\dx*0.0200},{\dy*49.5050}) + -- ({\dx*0.0400},{\dy*24.5863}) + -- ({\dx*0.0600},{\dy*16.3331}) + -- ({\dx*0.0800},{\dy*12.2453}) + -- ({\dx*0.1000},{\dy*9.8225}) + -- ({\dx*0.1200},{\dy*8.2314}) + -- ({\dx*0.1400},{\dy*7.1145}) + -- ({\dx*0.1600},{\dy*6.2930}) + -- ({\dx*0.1800},{\dy*5.6676}) + -- ({\dx*0.2000},{\dy*5.1786}) + -- ({\dx*0.2200},{\dy*4.7879}) + -- ({\dx*0.2400},{\dy*4.4701}) + -- ({\dx*0.2600},{\dy*4.2074}) + -- ({\dx*0.2800},{\dy*3.9874}) + -- ({\dx*0.3000},{\dy*3.8006}) + -- ({\dx*0.3200},{\dy*3.6401}) + -- ({\dx*0.3400},{\dy*3.5005}) + -- ({\dx*0.3600},{\dy*3.3776}) + -- ({\dx*0.3800},{\dy*3.2680}) + -- ({\dx*0.4000},{\dy*3.1692}) + -- ({\dx*0.4200},{\dy*3.0790}) + -- ({\dx*0.4400},{\dy*2.9955}) + -- ({\dx*0.4600},{\dy*2.9173}) + -- ({\dx*0.4800},{\dy*2.8433}) + -- ({\dx*0.5000},{\dy*2.7725}) + -- ({\dx*0.5200},{\dy*2.7039}) + -- ({\dx*0.5400},{\dy*2.6369}) + -- ({\dx*0.5600},{\dy*2.5709}) + -- ({\dx*0.5800},{\dy*2.5055}) + -- ({\dx*0.6000},{\dy*2.4402}) + -- ({\dx*0.6200},{\dy*2.3748}) + -- ({\dx*0.6400},{\dy*2.3090}) + -- ({\dx*0.6600},{\dy*2.2425}) + -- ({\dx*0.6800},{\dy*2.1752}) + -- ({\dx*0.7000},{\dy*2.1071}) + -- ({\dx*0.7200},{\dy*2.0380}) + -- ({\dx*0.7400},{\dy*1.9679}) + -- ({\dx*0.7600},{\dy*1.8969}) + -- ({\dx*0.7800},{\dy*1.8249}) + -- ({\dx*0.8000},{\dy*1.7520}) + -- ({\dx*0.8200},{\dy*1.6783}) + -- ({\dx*0.8400},{\dy*1.6039}) + -- ({\dx*0.8600},{\dy*1.5289}) + -- ({\dx*0.8800},{\dy*1.4534}) + -- ({\dx*0.9000},{\dy*1.3776}) + -- ({\dx*0.9200},{\dy*1.3017}) + -- ({\dx*0.9400},{\dy*1.2258}) + -- ({\dx*0.9600},{\dy*1.1501}) + -- ({\dx*0.9800},{\dy*1.0747}) + -- ({\dx*1.0000},{\dy*1.0000}) + -- ({\dx*1.0200},{\dy*0.9261}) + -- ({\dx*1.0400},{\dy*0.8531}) + -- ({\dx*1.0600},{\dy*0.7814}) + -- ({\dx*1.0800},{\dy*0.7110}) + -- ({\dx*1.1000},{\dy*0.6423}) + -- ({\dx*1.1200},{\dy*0.5755}) + -- ({\dx*1.1400},{\dy*0.5106}) + -- ({\dx*1.1600},{\dy*0.4480}) + -- ({\dx*1.1800},{\dy*0.3879}) + -- ({\dx*1.2000},{\dy*0.3304}) + -- ({\dx*1.2200},{\dy*0.2757}) + -- ({\dx*1.2400},{\dy*0.2240}) + -- ({\dx*1.2600},{\dy*0.1754}) + -- ({\dx*1.2800},{\dy*0.1302}) + -- ({\dx*1.3000},{\dy*0.0885}) + -- ({\dx*1.3200},{\dy*0.0503}) + -- ({\dx*1.3400},{\dy*0.0159}) + -- ({\dx*1.3600},{\dy*-0.0146}) + -- ({\dx*1.3800},{\dy*-0.0412}) + -- ({\dx*1.4000},{\dy*-0.0638}) + -- ({\dx*1.4200},{\dy*-0.0822}) + -- ({\dx*1.4400},{\dy*-0.0965}) + -- ({\dx*1.4600},{\dy*-0.1065}) + -- ({\dx*1.4800},{\dy*-0.1123}) + -- ({\dx*1.5000},{\dy*-0.1138}) + -- ({\dx*1.5200},{\dy*-0.1110}) + -- ({\dx*1.5400},{\dy*-0.1039}) + -- ({\dx*1.5600},{\dy*-0.0926}) + -- ({\dx*1.5800},{\dy*-0.0772}) + -- ({\dx*1.6000},{\dy*-0.0575}) + -- ({\dx*1.6200},{\dy*-0.0339}) + -- ({\dx*1.6400},{\dy*-0.0062}) + -- ({\dx*1.6600},{\dy*0.0254}) + -- ({\dx*1.6800},{\dy*0.0607}) + -- ({\dx*1.7000},{\dy*0.0996}) + -- ({\dx*1.7200},{\dy*0.1421}) + -- ({\dx*1.7400},{\dy*0.1879}) + -- ({\dx*1.7600},{\dy*0.2368}) + -- ({\dx*1.7800},{\dy*0.2888}) + -- ({\dx*1.8000},{\dy*0.3436}) + -- ({\dx*1.8200},{\dy*0.4010}) + -- ({\dx*1.8400},{\dy*0.4609}) + -- ({\dx*1.8600},{\dy*0.5229}) + -- ({\dx*1.8800},{\dy*0.5869}) + -- ({\dx*1.9000},{\dy*0.6527}) + -- ({\dx*1.9200},{\dy*0.7201}) + -- ({\dx*1.9400},{\dy*0.7887}) + -- ({\dx*1.9600},{\dy*0.8584}) + -- ({\dx*1.9800},{\dy*0.9289}) + -- ({\dx*2.0000},{\dy*1.0000}) + -- ({\dx*2.0200},{\dy*1.0714}) + -- ({\dx*2.0400},{\dy*1.1429}) + -- ({\dx*2.0600},{\dy*1.2142}) + -- ({\dx*2.0800},{\dy*1.2852}) + -- ({\dx*2.1000},{\dy*1.3555}) + -- ({\dx*2.1200},{\dy*1.4249}) + -- ({\dx*2.1400},{\dy*1.4933}) + -- ({\dx*2.1600},{\dy*1.5603}) + -- ({\dx*2.1800},{\dy*1.6258}) + -- ({\dx*2.2000},{\dy*1.6896}) + -- ({\dx*2.2200},{\dy*1.7514}) + -- ({\dx*2.2400},{\dy*1.8111}) + -- ({\dx*2.2600},{\dy*1.8685}) + -- ({\dx*2.2800},{\dy*1.9234}) + -- ({\dx*2.3000},{\dy*1.9757}) + -- ({\dx*2.3200},{\dy*2.0253}) + -- ({\dx*2.3400},{\dy*2.0719}) + -- ({\dx*2.3600},{\dy*2.1155}) + -- ({\dx*2.3800},{\dy*2.1560}) + -- ({\dx*2.4000},{\dy*2.1932}) + -- ({\dx*2.4200},{\dy*2.2272}) + -- ({\dx*2.4400},{\dy*2.2578}) + -- ({\dx*2.4600},{\dy*2.2851}) + -- ({\dx*2.4800},{\dy*2.3089}) + -- ({\dx*2.5000},{\dy*2.3293}) + -- ({\dx*2.5200},{\dy*2.3463}) + -- ({\dx*2.5400},{\dy*2.3599}) + -- ({\dx*2.5600},{\dy*2.3701}) + -- ({\dx*2.5800},{\dy*2.3770}) + -- ({\dx*2.6000},{\dy*2.3807}) + -- ({\dx*2.6200},{\dy*2.3812}) + -- ({\dx*2.6400},{\dy*2.3786}) + -- ({\dx*2.6600},{\dy*2.3731}) + -- ({\dx*2.6800},{\dy*2.3647}) + -- ({\dx*2.7000},{\dy*2.3537}) + -- ({\dx*2.7200},{\dy*2.3402}) + -- ({\dx*2.7400},{\dy*2.3242}) + -- ({\dx*2.7600},{\dy*2.3062}) + -- ({\dx*2.7800},{\dy*2.2861}) + -- ({\dx*2.8000},{\dy*2.2643}) + -- ({\dx*2.8200},{\dy*2.2409}) + -- ({\dx*2.8400},{\dy*2.2162}) + -- ({\dx*2.8600},{\dy*2.1903}) + -- ({\dx*2.8800},{\dy*2.1637}) + -- ({\dx*2.9000},{\dy*2.1364}) + -- ({\dx*2.9200},{\dy*2.1087}) + -- ({\dx*2.9400},{\dy*2.0810}) + -- ({\dx*2.9600},{\dy*2.0535}) + -- ({\dx*2.9800},{\dy*2.0264}) + -- ({\dx*3.0000},{\dy*2.0000}) + -- ({\dx*3.0200},{\dy*1.9746}) + -- ({\dx*3.0400},{\dy*1.9505}) + -- ({\dx*3.0600},{\dy*1.9280}) + -- ({\dx*3.0800},{\dy*1.9072}) + -- ({\dx*3.1000},{\dy*1.8886}) + -- ({\dx*3.1200},{\dy*1.8723}) + -- ({\dx*3.1400},{\dy*1.8587}) + -- ({\dx*3.1600},{\dy*1.8480}) + -- ({\dx*3.1800},{\dy*1.8404}) + -- ({\dx*3.2000},{\dy*1.8362}) + -- ({\dx*3.2200},{\dy*1.8356}) + -- ({\dx*3.2400},{\dy*1.8390}) + -- ({\dx*3.2600},{\dy*1.8464}) + -- ({\dx*3.2800},{\dy*1.8581}) + -- ({\dx*3.3000},{\dy*1.8744}) + -- ({\dx*3.3200},{\dy*1.8954}) + -- ({\dx*3.3400},{\dy*1.9213}) + -- ({\dx*3.3600},{\dy*1.9523}) + -- ({\dx*3.3800},{\dy*1.9885}) + -- ({\dx*3.4000},{\dy*2.0301}) + -- ({\dx*3.4200},{\dy*2.0773}) + -- ({\dx*3.4400},{\dy*2.1301}) + -- ({\dx*3.4600},{\dy*2.1886}) + -- ({\dx*3.4800},{\dy*2.2530}) + -- ({\dx*3.5000},{\dy*2.3234}) + -- ({\dx*3.5200},{\dy*2.3997}) + -- ({\dx*3.5400},{\dy*2.4821}) + -- ({\dx*3.5600},{\dy*2.5706}) + -- ({\dx*3.5800},{\dy*2.6652}) + -- ({\dx*3.6000},{\dy*2.7660}) + -- ({\dx*3.6200},{\dy*2.8729}) + -- ({\dx*3.6400},{\dy*2.9859}) + -- ({\dx*3.6600},{\dy*3.1051}) + -- ({\dx*3.6800},{\dy*3.2303}) + -- ({\dx*3.7000},{\dy*3.3616}) + -- ({\dx*3.7200},{\dy*3.4989}) + -- ({\dx*3.7400},{\dy*3.6421}) + -- ({\dx*3.7600},{\dy*3.7911}) + -- ({\dx*3.7800},{\dy*3.9459}) + -- ({\dx*3.8000},{\dy*4.1064}) + -- ({\dx*3.8200},{\dy*4.2724}) + -- ({\dx*3.8400},{\dy*4.4440}) + -- ({\dx*3.8600},{\dy*4.6209}) + -- ({\dx*3.8800},{\dy*4.8030}) + -- ({\dx*3.9000},{\dy*4.9903}) + -- ({\dx*3.9200},{\dy*5.1826}) + -- ({\dx*3.9400},{\dy*5.3799}) + -- ({\dx*3.9600},{\dy*5.5819}) + -- ({\dx*3.9800},{\dy*5.7887}) + -- ({\dx*4.0000},{\dy*6.0000}) + -- ({\dx*4.0200},{\dy*6.2158}) + -- ({\dx*4.0400},{\dy*6.4359}) + -- ({\dx*4.0600},{\dy*6.6603}) + -- ({\dx*4.0800},{\dy*6.8889}) + -- ({\dx*4.0810},{\dy*6.9005})} +\def\gammasinone{({\dx*-0.9810},{\dy*-53.1410}) + -- ({\dx*-0.9800},{\dy*-50.5140}) + -- ({\dx*-0.9600},{\dy*-25.6055}) + -- ({\dx*-0.9400},{\dy*-17.3637}) + -- ({\dx*-0.9200},{\dy*-13.2884}) + -- ({\dx*-0.9000},{\dy*-10.8796}) + -- ({\dx*-0.8800},{\dy*-9.3036}) + -- ({\dx*-0.8600},{\dy*-8.2033}) + -- ({\dx*-0.8400},{\dy*-7.3999}) + -- ({\dx*-0.8200},{\dy*-6.7941}) + -- ({\dx*-0.8000},{\dy*-6.3263}) + -- ({\dx*-0.7800},{\dy*-5.9586}) + -- ({\dx*-0.7600},{\dy*-5.6655}) + -- ({\dx*-0.7400},{\dy*-5.4296}) + -- ({\dx*-0.7200},{\dy*-5.2384}) + -- ({\dx*-0.7000},{\dy*-5.0827}) + -- ({\dx*-0.6800},{\dy*-4.9557}) + -- ({\dx*-0.6600},{\dy*-4.8523}) + -- ({\dx*-0.6400},{\dy*-4.7685}) + -- ({\dx*-0.6200},{\dy*-4.7012}) + -- ({\dx*-0.6000},{\dy*-4.6480}) + -- ({\dx*-0.5800},{\dy*-4.6072}) + -- ({\dx*-0.5600},{\dy*-4.5773}) + -- ({\dx*-0.5400},{\dy*-4.5573}) + -- ({\dx*-0.5200},{\dy*-4.5467}) + -- ({\dx*-0.5000},{\dy*-4.5449}) + -- ({\dx*-0.4800},{\dy*-4.5519}) + -- ({\dx*-0.4600},{\dy*-4.5677}) + -- ({\dx*-0.4400},{\dy*-4.5928}) + -- ({\dx*-0.4200},{\dy*-4.6279}) + -- ({\dx*-0.4000},{\dy*-4.6740}) + -- ({\dx*-0.3800},{\dy*-4.7325}) + -- ({\dx*-0.3600},{\dy*-4.8052}) + -- ({\dx*-0.3400},{\dy*-4.8944}) + -- ({\dx*-0.3200},{\dy*-5.0033}) + -- ({\dx*-0.3000},{\dy*-5.1359}) + -- ({\dx*-0.2800},{\dy*-5.2972}) + -- ({\dx*-0.2600},{\dy*-5.4942}) + -- ({\dx*-0.2400},{\dy*-5.7359}) + -- ({\dx*-0.2200},{\dy*-6.0350}) + -- ({\dx*-0.2000},{\dy*-6.4089}) + -- ({\dx*-0.1800},{\dy*-6.8830}) + -- ({\dx*-0.1600},{\dy*-7.4952}) + -- ({\dx*-0.1400},{\dy*-8.3052}) + -- ({\dx*-0.1200},{\dy*-9.4124}) + -- ({\dx*-0.1000},{\dy*-10.9953}) + -- ({\dx*-0.0800},{\dy*-13.4114}) + -- ({\dx*-0.0600},{\dy*-17.4941}) + -- ({\dx*-0.0400},{\dy*-25.7436}) + -- ({\dx*-0.0200},{\dy*-50.6602}) + -- ({\dx*-0.0190},{\dy*-53.2876})} +\def\gammasintwo{({\dx*-1.9810},{\dy*26.8549}) + -- ({\dx*-1.9800},{\dy*25.5432}) + -- ({\dx*-1.9600},{\dy*13.1254}) + -- ({\dx*-1.9400},{\dy*9.0411}) + -- ({\dx*-1.9200},{\dy*7.0402}) + -- ({\dx*-1.9000},{\dy*5.8725}) + -- ({\dx*-1.8800},{\dy*5.1211}) + -- ({\dx*-1.8600},{\dy*4.6073}) + -- ({\dx*-1.8400},{\dy*4.2416}) + -- ({\dx*-1.8200},{\dy*3.9745}) + -- ({\dx*-1.8000},{\dy*3.7759}) + -- ({\dx*-1.7800},{\dy*3.6268}) + -- ({\dx*-1.7600},{\dy*3.5146}) + -- ({\dx*-1.7400},{\dy*3.4305}) + -- ({\dx*-1.7200},{\dy*3.3681}) + -- ({\dx*-1.7000},{\dy*3.3229}) + -- ({\dx*-1.6800},{\dy*3.2916}) + -- ({\dx*-1.6600},{\dy*3.2715}) + -- ({\dx*-1.6400},{\dy*3.2607}) + -- ({\dx*-1.6200},{\dy*3.2578}) + -- ({\dx*-1.6000},{\dy*3.2616}) + -- ({\dx*-1.5800},{\dy*3.2715}) + -- ({\dx*-1.5600},{\dy*3.2868}) + -- ({\dx*-1.5400},{\dy*3.3072}) + -- ({\dx*-1.5200},{\dy*3.3327}) + -- ({\dx*-1.5000},{\dy*3.3633}) + -- ({\dx*-1.4800},{\dy*3.3993}) + -- ({\dx*-1.4600},{\dy*3.4412}) + -- ({\dx*-1.4400},{\dy*3.4896}) + -- ({\dx*-1.4200},{\dy*3.5456}) + -- ({\dx*-1.4000},{\dy*3.6103}) + -- ({\dx*-1.3800},{\dy*3.6854}) + -- ({\dx*-1.3600},{\dy*3.7727}) + -- ({\dx*-1.3400},{\dy*3.8749}) + -- ({\dx*-1.3200},{\dy*3.9951}) + -- ({\dx*-1.3000},{\dy*4.1374}) + -- ({\dx*-1.2800},{\dy*4.3070}) + -- ({\dx*-1.2600},{\dy*4.5109}) + -- ({\dx*-1.2400},{\dy*4.7583}) + -- ({\dx*-1.2200},{\dy*5.0617}) + -- ({\dx*-1.2000},{\dy*5.4387}) + -- ({\dx*-1.1800},{\dy*5.9148}) + -- ({\dx*-1.1600},{\dy*6.5279}) + -- ({\dx*-1.1400},{\dy*7.3376}) + -- ({\dx*-1.1200},{\dy*8.4433}) + -- ({\dx*-1.1000},{\dy*10.0238}) + -- ({\dx*-1.0800},{\dy*12.4364}) + -- ({\dx*-1.0600},{\dy*16.5145}) + -- ({\dx*-1.0400},{\dy*24.7583}) + -- ({\dx*-1.0200},{\dy*49.6681}) + -- ({\dx*-1.0190},{\dy*52.2951})} +\def\gammasinthree{({\dx*-2.9810},{\dy*-9.0483}) + -- ({\dx*-2.9800},{\dy*-8.6133}) + -- ({\dx*-2.9600},{\dy*-4.5173}) + -- ({\dx*-2.9400},{\dy*-3.1989}) + -- ({\dx*-2.9200},{\dy*-2.5746}) + -- ({\dx*-2.9000},{\dy*-2.2274}) + -- ({\dx*-2.8800},{\dy*-2.0184}) + -- ({\dx*-2.8600},{\dy*-1.8878}) + -- ({\dx*-2.8400},{\dy*-1.8057}) + -- ({\dx*-2.8200},{\dy*-1.7552}) + -- ({\dx*-2.8000},{\dy*-1.7264}) + -- ({\dx*-2.7800},{\dy*-1.7127}) + -- ({\dx*-2.7600},{\dy*-1.7099}) + -- ({\dx*-2.7400},{\dy*-1.7149}) + -- ({\dx*-2.7200},{\dy*-1.7255}) + -- ({\dx*-2.7000},{\dy*-1.7401}) + -- ({\dx*-2.6800},{\dy*-1.7575}) + -- ({\dx*-2.6600},{\dy*-1.7768}) + -- ({\dx*-2.6400},{\dy*-1.7972}) + -- ({\dx*-2.6200},{\dy*-1.8183}) + -- ({\dx*-2.6000},{\dy*-1.8397}) + -- ({\dx*-2.5800},{\dy*-1.8612}) + -- ({\dx*-2.5600},{\dy*-1.8825}) + -- ({\dx*-2.5400},{\dy*-1.9036}) + -- ({\dx*-2.5200},{\dy*-1.9245}) + -- ({\dx*-2.5000},{\dy*-1.9453}) + -- ({\dx*-2.4800},{\dy*-1.9663}) + -- ({\dx*-2.4600},{\dy*-1.9877}) + -- ({\dx*-2.4400},{\dy*-2.0099}) + -- ({\dx*-2.4200},{\dy*-2.0335}) + -- ({\dx*-2.4000},{\dy*-2.0591}) + -- ({\dx*-2.3800},{\dy*-2.0876}) + -- ({\dx*-2.3600},{\dy*-2.1200}) + -- ({\dx*-2.3400},{\dy*-2.1578}) + -- ({\dx*-2.3200},{\dy*-2.2024}) + -- ({\dx*-2.3000},{\dy*-2.2561}) + -- ({\dx*-2.2800},{\dy*-2.3216}) + -- ({\dx*-2.2600},{\dy*-2.4024}) + -- ({\dx*-2.2400},{\dy*-2.5032}) + -- ({\dx*-2.2200},{\dy*-2.6303}) + -- ({\dx*-2.2000},{\dy*-2.7928}) + -- ({\dx*-2.1800},{\dy*-3.0032}) + -- ({\dx*-2.1600},{\dy*-3.2809}) + -- ({\dx*-2.1400},{\dy*-3.6556}) + -- ({\dx*-2.1200},{\dy*-4.1772}) + -- ({\dx*-2.1000},{\dy*-4.9351}) + -- ({\dx*-2.0800},{\dy*-6.1082}) + -- ({\dx*-2.0600},{\dy*-8.1132}) + -- ({\dx*-2.0400},{\dy*-12.2003}) + -- ({\dx*-2.0200},{\dy*-24.6199}) + -- ({\dx*-2.0190},{\dy*-25.9316})} +\def\gammasinfour{({\dx*-3.9950},{\dy*8.4124}) + -- ({\dx*-3.9800},{\dy*2.2112}) + -- ({\dx*-3.9600},{\dy*1.2344}) + -- ({\dx*-3.9400},{\dy*0.9517}) + -- ({\dx*-3.9200},{\dy*0.8420}) + -- ({\dx*-3.9000},{\dy*0.8009}) + -- ({\dx*-3.8800},{\dy*0.7935}) + -- ({\dx*-3.8600},{\dy*0.8045}) + -- ({\dx*-3.8400},{\dy*0.8265}) + -- ({\dx*-3.8200},{\dy*0.8550}) + -- ({\dx*-3.8000},{\dy*0.8874}) + -- ({\dx*-3.7800},{\dy*0.9219}) + -- ({\dx*-3.7600},{\dy*0.9573}) + -- ({\dx*-3.7400},{\dy*0.9926}) + -- ({\dx*-3.7200},{\dy*1.0272}) + -- ({\dx*-3.7000},{\dy*1.0607}) + -- ({\dx*-3.6800},{\dy*1.0925}) + -- ({\dx*-3.6600},{\dy*1.1223}) + -- ({\dx*-3.6400},{\dy*1.1500}) + -- ({\dx*-3.6200},{\dy*1.1752}) + -- ({\dx*-3.6000},{\dy*1.1979}) + -- ({\dx*-3.5800},{\dy*1.2179}) + -- ({\dx*-3.5600},{\dy*1.2351}) + -- ({\dx*-3.5400},{\dy*1.2496}) + -- ({\dx*-3.5200},{\dy*1.2612}) + -- ({\dx*-3.5000},{\dy*1.2701}) + -- ({\dx*-3.4800},{\dy*1.2763}) + -- ({\dx*-3.4600},{\dy*1.2798}) + -- ({\dx*-3.4400},{\dy*1.2810}) + -- ({\dx*-3.4200},{\dy*1.2800}) + -- ({\dx*-3.4000},{\dy*1.2769}) + -- ({\dx*-3.3800},{\dy*1.2723}) + -- ({\dx*-3.3600},{\dy*1.2665}) + -- ({\dx*-3.3400},{\dy*1.2600}) + -- ({\dx*-3.3200},{\dy*1.2534}) + -- ({\dx*-3.3000},{\dy*1.2475}) + -- ({\dx*-3.2800},{\dy*1.2434}) + -- ({\dx*-3.2600},{\dy*1.2423}) + -- ({\dx*-3.2400},{\dy*1.2458}) + -- ({\dx*-3.2200},{\dy*1.2563}) + -- ({\dx*-3.2000},{\dy*1.2768}) + -- ({\dx*-3.1800},{\dy*1.3117}) + -- ({\dx*-3.1600},{\dy*1.3676}) + -- ({\dx*-3.1400},{\dy*1.4544}) + -- ({\dx*-3.1200},{\dy*1.5890}) + -- ({\dx*-3.1000},{\dy*1.8013}) + -- ({\dx*-3.0800},{\dy*2.1511}) + -- ({\dx*-3.0600},{\dy*2.7775}) + -- ({\dx*-3.0400},{\dy*4.0974}) + -- ({\dx*-3.0200},{\dy*8.1943}) + -- ({\dx*-3.0050},{\dy*33.1416})} +\def\gammasinfive{({\dx*-4.9990},{\dy*-8.3507}) + -- ({\dx*-4.9800},{\dy*-0.4942}) + -- ({\dx*-4.9600},{\dy*-0.3489}) + -- ({\dx*-4.9400},{\dy*-0.3421}) + -- ({\dx*-4.9200},{\dy*-0.3693}) + -- ({\dx*-4.9000},{\dy*-0.4094}) + -- ({\dx*-4.8800},{\dy*-0.4553}) + -- ({\dx*-4.8600},{\dy*-0.5037}) + -- ({\dx*-4.8400},{\dy*-0.5530}) + -- ({\dx*-4.8200},{\dy*-0.6021}) + -- ({\dx*-4.8000},{\dy*-0.6502}) + -- ({\dx*-4.7800},{\dy*-0.6969}) + -- ({\dx*-4.7600},{\dy*-0.7418}) + -- ({\dx*-4.7400},{\dy*-0.7846}) + -- ({\dx*-4.7200},{\dy*-0.8249}) + -- ({\dx*-4.7000},{\dy*-0.8626}) + -- ({\dx*-4.6800},{\dy*-0.8973}) + -- ({\dx*-4.6600},{\dy*-0.9291}) + -- ({\dx*-4.6400},{\dy*-0.9577}) + -- ({\dx*-4.6200},{\dy*-0.9829}) + -- ({\dx*-4.6000},{\dy*-1.0047}) + -- ({\dx*-4.5800},{\dy*-1.0230}) + -- ({\dx*-4.5600},{\dy*-1.0377}) + -- ({\dx*-4.5400},{\dy*-1.0488}) + -- ({\dx*-4.5200},{\dy*-1.0563}) + -- ({\dx*-4.5000},{\dy*-1.0600}) + -- ({\dx*-4.4800},{\dy*-1.0601}) + -- ({\dx*-4.4600},{\dy*-1.0566}) + -- ({\dx*-4.4400},{\dy*-1.0496}) + -- ({\dx*-4.4200},{\dy*-1.0390}) + -- ({\dx*-4.4000},{\dy*-1.0251}) + -- ({\dx*-4.3800},{\dy*-1.0080}) + -- ({\dx*-4.3600},{\dy*-0.9878}) + -- ({\dx*-4.3400},{\dy*-0.9647}) + -- ({\dx*-4.3200},{\dy*-0.9390}) + -- ({\dx*-4.3000},{\dy*-0.9110}) + -- ({\dx*-4.2800},{\dy*-0.8810}) + -- ({\dx*-4.2600},{\dy*-0.8495}) + -- ({\dx*-4.2400},{\dy*-0.8169}) + -- ({\dx*-4.2200},{\dy*-0.7841}) + -- ({\dx*-4.2000},{\dy*-0.7518}) + -- ({\dx*-4.1800},{\dy*-0.7215}) + -- ({\dx*-4.1600},{\dy*-0.6947}) + -- ({\dx*-4.1400},{\dy*-0.6742}) + -- ({\dx*-4.1200},{\dy*-0.6644}) + -- ({\dx*-4.1000},{\dy*-0.6730}) + -- ({\dx*-4.0800},{\dy*-0.7150}) + -- ({\dx*-4.0600},{\dy*-0.8253}) + -- ({\dx*-4.0400},{\dy*-1.1085}) + -- ({\dx*-4.0200},{\dy*-2.0855}) + -- ({\dx*-4.0010},{\dy*-41.6072})} +\def\gammasinsix{({\dx*-5.9998},{\dy*6.9477}) + -- ({\dx*-5.9800},{\dy*0.1349}) + -- ({\dx*-5.9600},{\dy*0.1629}) + -- ({\dx*-5.9400},{\dy*0.2134}) + -- ({\dx*-5.9200},{\dy*0.2691}) + -- ({\dx*-5.9000},{\dy*0.3260}) + -- ({\dx*-5.8800},{\dy*0.3829}) + -- ({\dx*-5.8600},{\dy*0.4391}) + -- ({\dx*-5.8400},{\dy*0.4940}) + -- ({\dx*-5.8200},{\dy*0.5472}) + -- ({\dx*-5.8000},{\dy*0.5985}) + -- ({\dx*-5.7800},{\dy*0.6477}) + -- ({\dx*-5.7600},{\dy*0.6945}) + -- ({\dx*-5.7400},{\dy*0.7387}) + -- ({\dx*-5.7200},{\dy*0.7800}) + -- ({\dx*-5.7000},{\dy*0.8184}) + -- ({\dx*-5.6800},{\dy*0.8537}) + -- ({\dx*-5.6600},{\dy*0.8856}) + -- ({\dx*-5.6400},{\dy*0.9142}) + -- ({\dx*-5.6200},{\dy*0.9392}) + -- ({\dx*-5.6000},{\dy*0.9606}) + -- ({\dx*-5.5800},{\dy*0.9783}) + -- ({\dx*-5.5600},{\dy*0.9923}) + -- ({\dx*-5.5400},{\dy*1.0024}) + -- ({\dx*-5.5200},{\dy*1.0086}) + -- ({\dx*-5.5000},{\dy*1.0109}) + -- ({\dx*-5.4800},{\dy*1.0094}) + -- ({\dx*-5.4600},{\dy*1.0039}) + -- ({\dx*-5.4400},{\dy*0.9947}) + -- ({\dx*-5.4200},{\dy*0.9816}) + -- ({\dx*-5.4000},{\dy*0.9648}) + -- ({\dx*-5.3800},{\dy*0.9443}) + -- ({\dx*-5.3600},{\dy*0.9203}) + -- ({\dx*-5.3400},{\dy*0.8929}) + -- ({\dx*-5.3200},{\dy*0.8621}) + -- ({\dx*-5.3000},{\dy*0.8283}) + -- ({\dx*-5.2800},{\dy*0.7914}) + -- ({\dx*-5.2600},{\dy*0.7519}) + -- ({\dx*-5.2400},{\dy*0.7098}) + -- ({\dx*-5.2200},{\dy*0.6655}) + -- ({\dx*-5.2000},{\dy*0.6193}) + -- ({\dx*-5.1800},{\dy*0.5717}) + -- ({\dx*-5.1600},{\dy*0.5230}) + -- ({\dx*-5.1400},{\dy*0.4741}) + -- ({\dx*-5.1200},{\dy*0.4260}) + -- ({\dx*-5.1000},{\dy*0.3804}) + -- ({\dx*-5.0800},{\dy*0.3405}) + -- ({\dx*-5.0600},{\dy*0.3135}) + -- ({\dx*-5.0400},{\dy*0.3204}) + -- ({\dx*-5.0200},{\dy*0.4657}) + -- ({\dx*-5.0002},{\dy*41.6531})} diff --git a/buch/papers/laguerre/images/gammaplot.pdf b/buch/papers/laguerre/images/gammaplot.pdf Binary files differnew file mode 100644 index 0000000..7c195f2 --- /dev/null +++ b/buch/papers/laguerre/images/gammaplot.pdf diff --git a/buch/papers/laguerre/images/gammaplot.tex b/buch/papers/laguerre/images/gammaplot.tex new file mode 100644 index 0000000..5a68f0a --- /dev/null +++ b/buch/papers/laguerre/images/gammaplot.tex @@ -0,0 +1,73 @@ +% +% gammaplot.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\input{gammapaths.tex} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\definecolor{mainColor}{HTML}{D72864} % OST pink + +\draw[->] (-6.1,0) -- (5.3,0) coordinate[label={$z$}]; +\draw[->] (0,-5.1) -- (0,6.4) coordinate[label={right:$\Gamma(z)$}]; + +\foreach \x in {-1,-2,-3,-4,-5,-6}{ + \draw (\x,-0.1) -- (\x,0.1); + \draw[line width=0.1pt] (\x,-5) -- (\x,6.2); +} +\foreach \x in {1,2,3,4,5}{ + \draw (\x,-0.1) -- (\x,0.1); + \node at (\x,0) [below] {$\x$}; +} +\foreach \y in {-5,-4,-3,-2,-1,1,2,3,4,5,6}{ + \draw (-0.1,\y) -- (0.1,\y); +} +\foreach \y in {1,2,3,4,5,6}{ + \node at (0,\y) [left] {$\y$}; +} +\foreach \y in {-1,-2,-3,-4,-5}{ + \node at (0,\y) [right] {$\y$}; +} +\foreach \x in {-1,-3,-5}{ + \node at (\x,0) [below left] {$\x$}; +} +\foreach \x in {-2,-4,-6}{ + \node at (\x,0) [above left] {$\x$}; +} + +\def\dx{1} +\def\dy{1} + +\begin{scope} +\clip (-6.1,-5) rectangle (4.3,6.2); + +% \draw[color=darkgreen,line width=1.4pt] \gammasinplus; +% \draw[color=darkgreen,line width=1.4pt] \gammasinone; +% \draw[color=darkgreen,line width=1.4pt] \gammasintwo; +% \draw[color=darkgreen,line width=1.4pt] \gammasinthree; +% \draw[color=darkgreen,line width=1.4pt] \gammasinfour; +% \draw[color=darkgreen,line width=1.4pt] \gammasinfive; +% \draw[color=darkgreen,line width=1.4pt] \gammasinsix; + +\draw[color=mainColor,line width=1.4pt] \gammaplus; +\draw[color=mainColor,line width=1.4pt] \gammaone; +\draw[color=mainColor,line width=1.4pt] \gammatwo; +\draw[color=mainColor,line width=1.4pt] \gammathree; +\draw[color=mainColor,line width=1.4pt] \gammafour; +\draw[color=mainColor,line width=1.4pt] \gammafive; +\draw[color=mainColor,line width=1.4pt] \gammasix; + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/laguerre/images/integrand.pdf b/buch/papers/laguerre/images/integrand.pdf Binary files differnew file mode 100644 index 0000000..76be412 --- /dev/null +++ b/buch/papers/laguerre/images/integrand.pdf diff --git a/buch/papers/laguerre/images/integrand_exp.pdf b/buch/papers/laguerre/images/integrand_exp.pdf Binary files differnew file mode 100644 index 0000000..5fe7a7b --- /dev/null +++ b/buch/papers/laguerre/images/integrand_exp.pdf diff --git a/buch/papers/laguerre/images/laguerre_poly.pdf 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a/buch/papers/laguerre/main.tex +++ b/buch/papers/laguerre/main.tex @@ -8,7 +8,25 @@ \begin{refsection} \chapterauthor{Patrik Müller} -{\large \color{red} TODO: Einleitung} +{\parindent0pt Die} Laguerre\--Polynome, +benannt nach Edmond Laguerre (1834 - 1886), +sind Lösungen der ebenfalls nach Laguerre benannten Differentialgleichung. +Laguerre entdeckte diese Polynome, als er Approximations\-methoden +für das Integral +% $\int_0^\infty \exp(-x) / x \, dx $ +\begin{align*} +\int_0^\infty \frac{e^{-x}}{x} \, dx +\end{align*} +suchte. +Darum möchten wir uns in diesem Kapitel, +ganz im Sinne des Entdeckers, +den Laguerre-Polynomen für Approximationen von Integralen mit +exponentiell-abfallenden Funktionen widmen. +Namentlich werden wir versuchen, mittels Laguerre-Polynomen und +der Gauss-Quadratur eine geeignete Approximation für die Gamma-Funktion zu finden. + +Laguerre-Polynome tauchen zudem auch in der Quantenmechanik im radialen Anteil +der Lösung für die Schrödinger-Gleichung eines Wasserstoffatoms auf. \input{papers/laguerre/definition} \input{papers/laguerre/eigenschaften} diff --git a/buch/papers/laguerre/packages.tex b/buch/papers/laguerre/packages.tex index 4ebc172..a80d091 100644 --- a/buch/papers/laguerre/packages.tex +++ b/buch/papers/laguerre/packages.tex @@ -6,4 +6,4 @@ % if your paper needs special packages, add package commands as in the % following example -\usepackage{derivative} +\DeclareMathOperator{\real}{Re}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/presentation.pdf b/buch/papers/laguerre/presentation/presentation.pdf Binary files differnew file mode 100644 index 0000000..3d00de3 --- /dev/null +++ b/buch/papers/laguerre/presentation/presentation.pdf diff --git a/buch/papers/laguerre/presentation/presentation.tex b/buch/papers/laguerre/presentation/presentation.tex new file mode 100644 index 0000000..3db69f5 --- /dev/null +++ b/buch/papers/laguerre/presentation/presentation.tex @@ -0,0 +1,134 @@ +\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer} + +% style +\mode<presentation>{ + \usetheme{Frankfurt} +} +%packages +\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-} +\usepackage[ngerman]{babel} +\usepackage{graphicx} +\usepackage{array} + +\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\usepackage{ragged2e} + +\usepackage{bm} % bold math +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{multirow} % multi row in tables +\usepackage{booktabs} %toprule midrule bottomrue in tables +\usepackage{scrextend} +\usepackage{textgreek} +\usepackage[rgb]{xcolor} + +\usepackage{ marvosym } % \Lightning + +\usepackage{multimedia} % embedded videos + +\usepackage{tikz} +\usepackage{pgf} +\usepackage{pgfplots} + +\usepackage{algorithmic} + +%citations +\usepackage[style=verbose,backend=biber]{biblatex} +\addbibresource{references.bib} + + +%math font +\usefonttheme[onlymath]{serif} + +%Beamer Template modifications +%\definecolor{mainColor}{HTML}{0065A3} % HSR blue +\definecolor{mainColor}{HTML}{D72864} % OST pink +\definecolor{invColor}{HTML}{28d79b} % OST pink +\definecolor{dgreen}{HTML}{38ad36} % Dark green + +%\definecolor{mainColor}{HTML}{000000} % HSR blue +\setbeamercolor{palette primary}{bg=white,fg=mainColor} +\setbeamercolor{palette secondary}{bg=orange,fg=mainColor} +\setbeamercolor{palette tertiary}{bg=yellow,fg=red} +\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic +\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points) +\setbeamercolor{section in toc}{fg=black} % TOC sections +\setbeamertemplate{section in toc}[sections numbered] +\setbeamertemplate{subsection in toc}{% + \hspace{1.2em}{$\bullet$}~\inserttocsubsection\par} + +\setbeamertemplate{itemize items}[circle] +\setbeamertemplate{description item}[circle] +\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true] +\beamertemplatenavigationsymbolsempty + +\setbeamercolor{footline}{fg=gray} +\setbeamertemplate{footline}{% + \hfill\usebeamertemplate***{navigation symbols} + \hspace{0.5cm} + \insertframenumber{}\hspace{0.2cm}\vspace{0.2cm} +} + +\usepackage{caption} +\captionsetup{labelformat=empty} + +%Title Page +\title{Laguerre-Polynome} +\subtitle{Anwendung: Approximation der Gamma-Funktion} +\author{Patrik Müller} +% \institute{OST Ostschweizer Fachhochschule} +% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}} +\date{\today} + +\input{../packages.tex} + +\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}% + +\newcommand*{\HL}{\textcolor{mainColor}} +\newcommand*{\RD}{\textcolor{red}} +\newcommand*{\BL}{\textcolor{blue}} +\newcommand*{\GN}{\textcolor{dgreen}} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + + +\makeatletter +\newcount\my@repeat@count +\newcommand{\myrepeat}[2]{% + \begingroup + \my@repeat@count=\z@ + \@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}% + \endgroup +} +\makeatother + +\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings} + +\begin{document} + +\begin{frame} + \titlepage +\end{frame} + +\begin{frame}{Inhaltsverzeichnis} + \tableofcontents +\end{frame} + +\input{sections/laguerre} + +\input{sections/gaussquad} + +\input{sections/gamma} + +\input{sections/gamma_approx} + +\appendix +\begin{frame} + % \centering + % \Large + % \textbf{Vielen Dank für die Aufmerksamkeit} +\end{frame} + +\end{document} diff --git a/buch/papers/laguerre/presentation/sections/gamma.tex b/buch/papers/laguerre/presentation/sections/gamma.tex new file mode 100644 index 0000000..7dca39b --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/gamma.tex @@ -0,0 +1,51 @@ +\section{Gamma-Funktion} + +\begin{frame}{Gamma-Funktion} +\begin{columns} + +\begin{column}{0.55\textwidth} +\begin{figure}[h] +\vspace{-16pt} +\centering +% \scalebox{0.51}{\input{../images/gammaplot.pdf}} +\includegraphics[width=1\textwidth]{../images/gammaplot.pdf} +% \caption{Gamma-Funktion} +\end{figure} +\end{column} + +\begin{column}{0.45\textwidth} +Verallgemeinerung der Fakultät +\begin{align*} +\Gamma(n) = (n-1)! +\end{align*} + +Integralformel +\begin{align*} +\Gamma(z) += +\int_0^\infty x^{z-1} e^{-x} \, dx +,\quad +\operatorname{Re} z > 0 +\end{align*} + +Funktionalgleichung +\begin{align*} +z \Gamma(z) += +\Gamma(z + 1) +\end{align*} + +Reflektionsformel +\begin{align*} +\Gamma(z) \Gamma(1 - z) += +\frac{\pi}{\sin \pi z} +, \quad +\text{für } +z \notin \mathbb{Z} +\end{align*} + +\end{column} +\end{columns} + +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/sections/gamma_approx.tex b/buch/papers/laguerre/presentation/sections/gamma_approx.tex new file mode 100644 index 0000000..ecd02ab --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/gamma_approx.tex @@ -0,0 +1,201 @@ +\section{Approximieren der Gamma-Funktion} + +\begin{frame}{Anwenden der Gauss-Laguerre-Quadratur auf $\Gamma(z)$} + +\begin{align*} +\Gamma(z) + & = +\int_0^\infty x^{z-1} e^{-x} \, dx +\uncover<2->{ +\approx +\sum_{i=1}^{n} f(x_i) A_i +} +\uncover<3->{ += +\sum_{i=1}^{n} x^{z-1} A_i +} +\\\\ +\uncover<4->{ + & \text{wobei } +A_i = \frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2} +\text{ und $x_i$ die Nullstellen von $L_n(x)$} +} +\end{align*} + +\end{frame} + +\begin{frame}{Fehlerabschätzung} +\begin{align*} +R_n(\xi) + & = +\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) +\\ + & = +(z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z - 2n - 1} +,\quad +0 < \xi < \infty +\end{align*} + +% \textbf{Probleme:} +\begin{itemize} +\item Funktion ist unbeschränkt +\item Maximum von $R_n$ gibt oberes Limit des Fehlers an +\uncover<2->{\item[$\Rightarrow$] Schwierig ein Maximum von $R_n(\xi)$ zu finden} +\end{itemize} +\end{frame} + +\begin{frame}{Einfacher Ansatz} + +\begin{figure}[h] +\centering +% \scalebox{0.91}{\input{../images/rel_error_simple.pgf}} +% \resizebox{!}{0.72\textheight}{\input{../images/rel_error_simple.pgf}} +\includegraphics[width=0.77\textwidth]{../images/rel_error_simple.pdf} +\caption{Relativer Fehler des einfachen Ansatzes für verschiedene reele Werte +von $z$ und Grade $n$ der Laguerre-Polynome} +\end{figure} + +\end{frame} + +\begin{frame}{Wieso sind die Resultate so schlecht?} + +\textbf{Beobachtungen} +\begin{itemize} +\item Wenn $z \in \mathbb{Z}$ relativer Fehler $\rightarrow 0$ +\item Gewisse Periodizität zu erkennen +\item Für grosse und kleine $z$ ergibt sich ein schlechter relativer Fehler +\item Es gibt Intervalle $[a,a+1]$ mit minimalem relativem Fehler +\item $a$ ist abhängig von $n$ +\end{itemize} + +\uncover<2->{ +\textbf{Ursache?} +\begin{itemize} +\item Vermutung: Integrand ist problematisch +} +\uncover<3->{ +\item[$\Rightarrow$] Analysieren von $f(x)$ und dem Integranden +} +\end{itemize} +\end{frame} + +\begin{frame}{$f(x) = x^z$} +\begin{figure}[h] +\centering +% \scalebox{0.91}{\input{../images/integrand.pgf}} +\includegraphics[width=0.8\textwidth]{../images/integrand.pdf} +% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\end{figure} +\end{frame} + +\begin{frame}{Integrand $x^z e^{-x}$} +\begin{figure}[h] +\centering +% \scalebox{0.91}{\input{../images/integrand_exp.pgf}} +\includegraphics[width=0.8\textwidth]{../images/integrand_exp.pdf} +% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\end{figure} +\end{frame} + +\begin{frame}{Neuer Ansatz?} + +\textbf{Vermutung} +\begin{itemize} +\item Es gibt Intervalle $[a(n), a(n)+1]$ in denen der relative Fehler minimal +ist +\item $a(n) > 0$ +\end{itemize} + +\uncover<2->{ +\textbf{Idee} +\begin{itemize} +\item[$\Rightarrow$] Berechnen von $\Gamma(z)$ im geeigneten Intervall und dann +mit Funktionalgleichung zurückverschieben +\end{itemize} +} + +\uncover<3->{ +\textbf{Wie finden wir $\boldsymbol{a(n)}$?} +\begin{itemize} +\item Minimieren des Fehlerterms mit zusätzlichem Verschiebungsterm +} +\uncover<4->{$\Rightarrow$ Schwierig das Maximum des Fehlerterms zu bestimmen} +\uncover<5->{\item Empirisch $a(n)$ bestimmen} +\uncover<6->{$\Rightarrow$ Sinnvoll, +da Gauss-Quadratur nur für kleine $n$ praktischen Nutzen hat} +\end{itemize} +\end{frame} + +\begin{frame}{Verschiebungsterm} +\begin{columns} +\begin{column}{0.625\textwidth} +\begin{figure}[h] +\centering +\includegraphics[width=1\textwidth]{../images/targets.pdf} +\caption{Optimaler Verschiebungsterm $m^*$ in Abhängigkeit von $z$ und $n$} +\end{figure} +\end{column} +\begin{column}{0.375\textwidth} +\begin{align*} +\Gamma(z) +\approx +\frac{1}{(z-m)_{m}} \sum_{i=1}^{n} x_i^{z + m - 1} A_i +\end{align*} +\end{column} +\end{columns} +\end{frame} + +\begin{frame}{Schätzen von $m^*$} +\begin{columns} +\begin{column}{0.65\textwidth} +\begin{figure} +\centering +\vspace{-12pt} +% \scalebox{0.7}{\input{../images/estimates.pgf}} +\includegraphics[width=1.0\textwidth]{../images/estimates.pdf} +% \caption{Integrand $x^z$ mit unterschiedlichen Werten für $z$} +\end{figure} +\end{column} +\begin{column}{0.34\textwidth} +\begin{align*} +\hat{m} +&= +\alpha n + \beta +\\ +&\approx +1.34093 n + 0.854093 +\\ +m^* +&= +\lceil \hat{m} - \operatorname{Re}z \rceil +\end{align*} +\end{column} +\end{columns} + +\end{frame} + +\begin{frame}{} +\begin{figure}[h] +\centering +% \scalebox{0.6}{\input{../images/rel_error_shifted.pgf}} +\includegraphics{../images/rel_error_shifted.pdf} +\caption{Relativer Fehler mit $n=8$, unterschiedlichen Verschiebungstermen $m$ und $z\in(0, 1)$} +\end{figure} +\end{frame} + +\begin{frame}{} +\begin{figure}[h] +\centering +% \scalebox{0.6}{\input{../images/rel_error_range.pgf}} +\includegraphics{../images/rel_error_range.pdf} +\caption{Relativer Fehler mit $n=8$, Verschiebungsterm $m^*$ und $z\in(-5, 5)$} +\end{figure} +\end{frame} + +\begin{frame}{Vergleich mit Lanczos-Methode} +Maximaler relativer Fehler für $n=6$ +\begin{itemize} + \item Lanczos-Methode $< 10^{-12}$ + \item Unsere Methode $\approx 10^{-6}$ +\end{itemize} +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/sections/gaussquad.tex b/buch/papers/laguerre/presentation/sections/gaussquad.tex new file mode 100644 index 0000000..4d973b8 --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/gaussquad.tex @@ -0,0 +1,67 @@ +\section{Gauss-Quadratur} + +\begin{frame}{Gauss-Quadratur} +\textbf{Idee} +\begin{itemize}[<+->] +\item Polynome können viele Funktionen approximieren +\item Wenn Verfahren gut für Polynome funktioniert, +sollte es auch für andere Funktionen funktionieren +\item Integrieren eines Interpolationspolynom +\item Interpolationspolynom ist durch Funktionswerte $f(x_i)$ bestimmt +$\Rightarrow$ Integral kann durch Funktionswerte berechnet werden +\item Evaluation der Funktionswerte an geeigneten Stellen +\end{itemize} +\end{frame} + +\begin{frame}{Gauss-Quadratur} +\begin{align*} +\int_{-1}^{1} f(x) \, dx +\approx +\sum_{i=1}^n f(x_i) A_i +\end{align*} + +\begin{itemize}[<+->] +\item Exakt für Polynome mit Grad $2n-1$ +\item Interpolationspolynome müssen orthogonal sein +\item Stützstellen $x_i$ sind Nullstellen des Polynoms +\item Fehler: +\begin{align*} +E += +\frac{f^{(2n)}(\xi)}{(2n)!} \int_{-1}^{1} l(x)^2 \, dx +,\quad +\text{wobei } +l(x) = \prod_{i=1}^n (x-x_i) +\end{align*} +\end{itemize} +\end{frame} + +\begin{frame}{Gauss-Laguerre-Quadratur} +\begin{itemize}[<+->] +\item Erweiterung des Integrationsintervall von $[-1, 1]$ auf $(a, b)$ +\item Hinzufügen einer Gewichtsfunktion +\item Bei uneigentlichen Integralen muss Gewichtsfunktion schneller als jedes +Integrationspolynom gegen $0$ gehen +\item[$\Rightarrow$] Für Laguerre-Polynome haben wir den Definitionsbereich +$(0, \infty)$ und die Gewichtsfunktion $w(x) = e^{-x}$ +\begin{align*} +\int_0^\infty & f(x) e^{-x} \, dx +\approx +\sum_{i=1}^n f(x_i) A_i +\\ + & \text{wobei } +A_i = \frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2} +\text{ und $x_i$ die Nullstellen von $L_n(x)$} +\end{align*} +\end{itemize} +\end{frame} + +\begin{frame}{Fehler der Gauss-Laguerre-Quadratur} +\begin{align*} +R_n += +\frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) +,\quad +0 < \xi < \infty +\end{align*} +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex new file mode 100644 index 0000000..f99214e --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -0,0 +1,91 @@ +\section{Laguerre-Polynome} + +\begin{frame}{Laguerre-Differentialgleichung} + +\begin{itemize} +\item Benannt nach Edmond Nicolas Laguerre (1834-1886) +\item Aus Artikel von 1879, +in dem er $\int_0^\infty \exp(-x)/x \, dx$ analysierte +\end{itemize} + +\begin{align*} +x y''(x) + (1 - x) y'(x) + n y(x) + & = +0 +, \quad +n \in \mathbb{N}_0 +, \quad +x \in \mathbb{R} +\end{align*} + +\end{frame} + +\begin{frame}{Lösen der Differentialgleichung} + +\begin{align*} +x y''(x) + (1 - x) y'(x) + n y(x) + & = +0 +\\ +\end{align*} + +\uncover<2->{ +\centering +\begin{tikzpicture}[remember picture,overlay] +%% use here too +\path[draw=mainColor, very thick,->](0, 1.1) to +node[anchor=west]{Potenzreihenansatz} (0, -0.8); +\end{tikzpicture} +} + +\begin{align*} +\uncover<3->{ +L_n(x) + & = +\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k +} +\end{align*} +\uncover<4->{ +\begin{itemize} + \item Die Lösungen der DGL sind die Laguerre-Polynome +\end{itemize} +} +\end{frame} + +\begin{frame} +\begin{figure}[h] +\centering +% \resizebox{0.74\textwidth}{!}{\input{../images/laguerre_poly.pgf}} +\includegraphics[width=0.7\textwidth]{../images/laguerre_poly.pdf} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\end{figure} +\end{frame} + +\begin{frame}{Orthogonalität} +\begin{itemize}[<+->] +\item Beweis: Umformen in Sturm-Liouville-Problem (siehe Paper) +\begin{alignat*}{5} +((p(x) &y'(x)))' + q(x) &y(x) +&= +\lambda &w(x) &y(x) +\\ +((x e^{-x} &y'(x)))' + 0 &y(x) +&= +n &e^{-x} &y(x) +\end{alignat*} +\item Definitionsbereich $(0, \infty)$ +\item Gewichtsfunktion $w(x) = e^{-x}$ +\end{itemize} + +\uncover<4->{ +\begin{align*} +\int_0^\infty e^{-x} L_n(x) L_m(x) \, dx += +0 +,\quad +n \neq m +,\quad +n, m \in \mathbb{N} +\end{align*} +} +\end{frame}
\ No newline at end of file diff --git a/buch/papers/laguerre/quadratur.tex b/buch/papers/laguerre/quadratur.tex index 60fad7f..a494362 100644 --- a/buch/papers/laguerre/quadratur.tex +++ b/buch/papers/laguerre/quadratur.tex @@ -5,35 +5,70 @@ % \section{Gauss-Quadratur \label{laguerre:section:quadratur}} - {\large \color{red} TODO: Einleitung und kurze Beschreibung Gauss-Quadratur} +Die Gauss-Quadratur ist ein numerisches Integrationsverfahren, +welches die Eigenschaften von orthogonalen Polynomen verwendet. +Herleitungen und Analysen der Gauss-Quadratur können im +Abschnitt~\ref{buch:orthogonal:section:gauss-quadratur} gefunden werden. +Als grundlegende Idee wird die Beobachtung, +dass viele Funktionen sich gut mit Polynomen approximieren lassen, +verwendet. +Stellt man also sicher, +dass ein Verfahren gut für Polynome funktioniert, +sollte es auch für andere Funktionen angemessene Resultate liefern. +Es wird ein Polynom verwendet, +welches an den Punkten $x_0 < x_1 < \ldots < x_n$ +die Funktionwerte~$f(x_i)$ annimmt. +Als Resultat kann das Integral via einer gewichteten Summe der Form \begin{align} -\int_a^b f(x) w(x) +\int_a^b f(x) w(x) \, dx \approx -\sum_{i=1}^N f(x_i) A_i +\sum_{i=1}^n f(x_i) A_i \label{laguerre:gaussquadratur} \end{align} +berechnet werden. +Die Gauss-Quadratur ist exakt für Polynome mit Grad $2n -1$, +wenn ein Interpolationspolynom von Grad $n$ gewählt wurde. \subsection{Gauss-Laguerre-Quadratur \label{laguerre:subsection:gausslag-quadratur}} -Die Gauss-Quadratur kann auch auf Skalarprodukte mit Gewichtsfunktionen -ausgeweitet werden. -In unserem Falle möchten wir die Gauss Quadratur auf die Laguerre-Polynome -$L_n$ ausweiten. -Diese sind orthogonal im Intervall $(0, \infty)$ bezüglich -der Gewichtsfunktion $e^{-x}$. -Gleichung~\eqref{laguerre:laguerrequadratur} lässt sich wiefolgt umformulieren: +Wir möchten nun die Gauss-Quadratur auf die Berechnung +von uneigentlichen Integralen erweitern, +spezifisch auf das Interval $(0, \infty)$. +Mit dem vorher beschriebenen Verfahren ist dies nicht direkt möglich. +Mit einer Transformation die das unendliche Intervall $(a, \infty)$ mit +\begin{align*} +x += +a + \frac{1 - t}{t} +\end{align*} +auf das Intervall $[0, 1]$ transformiert, +kann dies behoben werden. +Für unseren Fall gilt $a = 0$. +Das Integral eines Polynomes in diesem Intervall ist immer divergent. +Darum müssen wir das Polynom mit einer Funktion multiplizieren, +die schneller als jedes Polynom gegen $0$ geht, +damit das Integral immer noch konvergiert. +Die Laguerre-Polynome $L_n$ schaffen hier Abhilfe, +da ihre Gewichtsfunktion $w(x) = e^{-x}$ schneller +gegen $0$ konvergiert als jedes Polynom. +% In unserem Falle möchten wir die Gauss Quadratur auf die Laguerre-Polynome +% $L_n$ ausweiten. +% Diese sind orthogonal im Intervall $(0, \infty)$ bezüglich +% der Gewichtsfunktion $e^{-x}$. +Die Gleichung~\eqref{laguerre:gaussquadratur} lässt sich wie folgt +umformulieren: \begin{align} \int_{0}^{\infty} f(x) e^{-x} dx \approx -\sum_{i=1}^{N} f(x_i) A_i +\sum_{i=1}^{n} f(x_i) A_i \label{laguerre:laguerrequadratur} \end{align} \subsubsection{Stützstellen und Gewichte} Nach der Definition der Gauss-Quadratur müssen als Stützstellen die Nullstellen des verwendeten Polynoms genommen werden. -Das heisst für das Laguerre-Polynom $L_n$ müssen dessen Nullstellen $x_i$ und -als Gewichte $A_i$ werden die Integrale $l_i(x)e^{-x}$ verwendet werden. +Für das Laguerre-Polynom $L_n$ müssen demnach dessen Nullstellen $x_i$ und +als Gewichte $A_i$ die Integrale $l_i(x)e^{-x}$ verwendet werden. Dabei sind \begin{align*} l_i(x_j) @@ -41,39 +76,112 @@ l_i(x_j) \delta_{ij} = \begin{cases} -1 & i=j \\ -0 & \text{sonst.} +1 & i=j \\ +0 & \text{sonst} \end{cases} +% . \end{align*} -Laut \cite{abramowitz+stegun} sind die Gewichte also -\begin{align} +die Lagrangschen Interpolationspolynome. +Laut \cite{laguerre:hildebrand2013introduction} können die Gewichte mit +\begin{align*} A_i + & = +-\frac{C_{n+1} \gamma_n}{C_n \phi'_n(x_i) \phi_{n+1} (x_i)} +\end{align*} +berechnet werden. +$C_i$ entspricht dabei dem Koeffizienten von $x^i$ +des orthogonalen Polynoms $\phi_n(x)$, $\forall i =0,\ldots,n$ und +\begin{align*} +\gamma_n += +\int_0^\infty w(x) \phi_n^2(x)\,dx +\end{align*} +dem Normalisierungsfaktor. +Wir setzen nun $\phi_n(x) = L_n(x)$ und +nutzen den Vorzeichenwechsel der Laguerre-Koeffizienten aus, +damit erhalten wir +\begin{align*} +A_i + & = +-\frac{C_{n+1} \gamma_n}{C_n L'_n(x_i) L_{n+1} (x_i)} +\\ + & = \frac{C_n}{C_{n-1}} \frac{\gamma_{n-1}}{L_{n-1}(x_i) L'_n(x_i)} +. +\end{align*} +Für Laguerre-Polynome gilt +\begin{align*} +\frac{C_n}{C_{n-1}} += +-\frac{1}{n} +\quad \text{und} \quad +\gamma_n = -\frac{x_i}{(n + 1)^2 \left[ L_{n + 1}(x_i)\right]^2} +1 +. +\end{align*} +Daraus folgt +\begin{align} +A_i +&= +- \frac{1}{n L_{n-1}(x_i) L'_n(x_i)} +. +\label{laguerre:gewichte_lag_temp} +\end{align} +Nun kann die Rekursionseigenschaft der Laguerre-Polynome +\begin{align*} +x L'_n(x) +&= +n L_n(x) - n L_{n-1}(x) +\\ +&= (x - n - 1) L_n(x) + (n + 1) L_{n+1}(x) +\end{align*} +umgeformt werden und da $x_i$ die Nullstellen von $L_n(x)$ sind, +vereinfacht sich der Term zu +\begin{align*} +x_i L'_n(x_i) +&= +- n L_{n-1}(x_i) +\\ +&= + (n + 1) L_{n+1}(x_i) +. +\end{align*} +Setzen wir das nun in \eqref{laguerre:gewichte_lag_temp} ein, +ergibt sich +\begin{align} +\nonumber +A_i +&= +\frac{1}{x_i \left[ L'_n(x_i) \right]^2} +\\ +&= +\frac{x_i}{(n+1)^2 \left[ L_{n+1}(x_i) \right]^2} . \label{laguerre:quadratur_gewichte} \end{align} \subsubsection{Fehlerterm} +Die Gauss-Laguerre-Quadratur mit $n$ Stützstellen berechnet Integrale +von Polynomen bis zum Grad $2n - 1$ exakt. +Für beliebige Funktionen kann eine Fehlerabschätzung angegeben werden. Der Fehlerterm $R_n$ folgt direkt aus der Approximation \begin{align*} -\int_0^{\infty} f(x) e^{-x} dx +\int_0^{\infty} f(x) e^{-x} \, dx = \sum_{i=1}^n f(x_i) A_i + R_n \end{align*} -un \cite{abramowitz+stegun} gibt in als +und \cite{laguerre:abramowitz+stegun} gibt ihn als \begin{align} R_n -= + & = +\frac{f^{(2n)}(\xi)}{(2n)!} \int_0^\infty l(x)^2 e^{-x}\,dx +\\ + & = \frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) ,\quad 0 < \xi < \infty -\label{lagurre:lag_error} +\label{laguerre:lag_error} \end{align} an. - -{ -\large \color{red} -TODO: -Noch mehr Text / bessere Beschreibungen in allen Abschnitten -} +Der Fehler ist also abhängig von der $2n$-ten Ableitung +der zu integrierenden Funktion. diff --git a/buch/papers/laguerre/references.bib b/buch/papers/laguerre/references.bib index 6956ade..d21009b 100644 --- a/buch/papers/laguerre/references.bib +++ b/buch/papers/laguerre/references.bib @@ -3,8 +3,17 @@ % % (c) 2020 Autor, Hochschule Rapperswil % +@book{laguerre:hildebrand2013introduction, + title={Introduction to Numerical Analysis: Second Edition}, + author={Hildebrand, F.B.}, + isbn={9780486318554}, + series={Dover Books on Mathematics}, + year={2013}, + publisher={Dover Publications}, + pages = {389} +} -@book{abramowitz+stegun, +@book{laguerre:abramowitz+stegun, added-at = {2008-06-25T06:25:58.000+0200}, address = {New York}, author = {Abramowitz, Milton and Stegun, Irene A.}, @@ -19,4 +28,23 @@ timestamp = {2008-06-25T06:25:58.000+0200}, title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, year = 1972 +} + +@article{laguerre:Cassity1965AbcissasCA, + title={Abcissas, coefficients, and error term for the generalized Gauss-Laguerre quadrature formula using the zero ordinate}, + author={C. Ronald Cassity}, + journal={Mathematics of Computation}, + year={1965}, + volume={19}, + pages={287-296} +} + +@online{laguerre:lanczos, + title = {Lanczos Approximation}, + author={Eric W. Weisstein}, + url = {https://mathworld.wolfram.com/LanczosApproximation.html}, + date = {2022-07-18}, + year = {2022}, + month = {7}, + day = {18} }
\ No newline at end of file diff --git a/buch/papers/laguerre/scripts/estimates.py b/buch/papers/laguerre/scripts/estimates.py new file mode 100644 index 0000000..21551f3 --- /dev/null +++ b/buch/papers/laguerre/scripts/estimates.py @@ -0,0 +1,49 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + + import gamma_approx as ga + import targets + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + N = 200 + ns = np.arange(2, 13) + step = 1 / (N - 1) + x = np.linspace(step, 1 - step, N + 1) + + bests = targets.find_best_loc(N, ns=ns) + mean_m = np.mean(bests, -1) + + intercept, bias = np.polyfit(ns, mean_m, 1) + fig, axs = plt.subplots( + 2, num=1, sharex=True, clear=True, constrained_layout=True, figsize=(4.5, 3.6) + ) + xl = np.array([ns[0] - 0.5, ns[-1] + 0.5]) + axs[0].plot(xl, intercept * xl + bias, label=r"$\hat{m}$") + axs[0].plot(ns, mean_m, "x", label=r"$\overline{m}$") + axs[1].plot( + ns, ((intercept * ns + bias) - mean_m), "-x", label=r"$\hat{m} - \overline{m}$" + ) + axs[0].set_xlim(*xl) + axs[0].set_xticks(ns) + axs[0].set_yticks(np.arange(np.floor(mean_m[0]), np.ceil(mean_m[-1]) + 0.1, 2)) + # axs[0].set_title("Schätzung von Mittelwert") + # axs[1].set_title("Fehler") + axs[-1].set_xlabel(r"$n$") + for ax in axs: + ax.grid(1) + ax.legend() + # fig.savefig(f"{ga.img_path}/estimates.pgf") + fig.savefig(f"{ga.img_path}/estimates.pdf") + + print(f"Intercept={intercept:.6g}, Bias={bias:.6g}") + predicts = np.ceil(intercept * ns[:, None] + bias - np.real(x)) + print(f"Error: {np.mean(np.abs(bests - predicts))}") diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb deleted file mode 100644 index 44f3abd..0000000 --- a/buch/papers/laguerre/scripts/gamma_approx.ipynb +++ /dev/null @@ -1,395 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Gauss-Laguerre Quadratur für die Gamma-Funktion\n", - "\n", - "$$\n", - " \\Gamma(z)\n", - " = \n", - " \\int_0^\\infty t^{z-1}e^{-t}dt\n", - "$$\n", - "\n", - "$$\n", - " \\int_0^\\infty f(x) e^{-x} dx \n", - " \\approx \n", - " \\sum_{i=1}^{N} f(x_i) w_i\n", - " \\qquad\\text{ wobei }\n", - " w_i = \\frac{x_i}{(n+1)^2 [L_{n+1}(x_i)]^2}\n", - "$$\n", - "und $x_i$ sind Nullstellen des Laguerre Polynoms $L_n(x)$\n", - "\n", - "Der Fehler ist gegeben als\n", - "\n", - "$$\n", - " E \n", - " =\n", - " \\frac{(n!)^2}{(2n)!} f^{(2n)}(\\xi) \n", - " = \n", - " \\frac{(-2n + z)_{2n}}{(z-m)_m} \\frac{(n!)^2}{(2n)!} \\xi^{z + m - 2n - 1}\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from cmath import exp, pi, sin, sqrt\n", - "import scipy.special\n", - "\n", - "EPSILON = 1e-07\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "lanczos_p = [\n", - " 676.5203681218851,\n", - " -1259.1392167224028,\n", - " 771.32342877765313,\n", - " -176.61502916214059,\n", - " 12.507343278686905,\n", - " -0.13857109526572012,\n", - " 9.9843695780195716e-6,\n", - " 1.5056327351493116e-7,\n", - "]\n", - "\n", - "\n", - "def drop_imag(z):\n", - " if abs(z.imag) <= EPSILON:\n", - " z = z.real\n", - " return z\n", - "\n", - "\n", - "def lanczos_gamma(z):\n", - " z = complex(z)\n", - " if z.real < 0.5:\n", - " y = pi / (sin(pi * z) * lanczos_gamma(1 - z)) # Reflection formula\n", - " else:\n", - " z -= 1\n", - " x = 0.99999999999980993\n", - " for (i, pval) in enumerate(lanczos_p):\n", - " x += pval / (z + i + 1)\n", - " t = z + len(lanczos_p) - 0.5\n", - " y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x\n", - " return drop_imag(y)\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "zeros, weights = np.polynomial.laguerre.laggauss(8)\n", - "# zeros = np.array(\n", - "# [\n", - "# 1.70279632305101000e-1,\n", - "# 9.03701776799379912e-1,\n", - "# 2.25108662986613069e0,\n", - "# 4.26670017028765879e0,\n", - "# 7.04590540239346570e0,\n", - "# 1.07585160101809952e1,\n", - "# 1.57406786412780046e1,\n", - "# 2.28631317368892641e1,\n", - "# ]\n", - "# )\n", - "\n", - "# weights = np.array(\n", - "# [\n", - "# 3.69188589341637530e-1,\n", - "# 4.18786780814342956e-1,\n", - "# 1.75794986637171806e-1,\n", - "# 3.33434922612156515e-2,\n", - "# 2.79453623522567252e-3,\n", - "# 9.07650877335821310e-5,\n", - "# 8.48574671627253154e-7,\n", - "# 1.04800117487151038e-9,\n", - "# ]\n", - "# )\n", - "\n", - "\n", - "def pochhammer(z, n):\n", - " return np.prod(z + np.arange(n))\n", - "\n", - "\n", - "def find_shift(z, target):\n", - " factor = 1.0\n", - " steps = int(np.floor(target - np.real(z)))\n", - " zs = z + steps\n", - " if steps > 0:\n", - " factor = 1 / pochhammer(z, steps)\n", - " elif steps < 0:\n", - " factor = pochhammer(zs, -steps)\n", - " return zs, factor\n", - "\n", - "\n", - "def laguerre_gamma(z, x, w, target=11):\n", - " # res = 0.0\n", - " z = complex(z)\n", - " if z.real < 1e-3:\n", - " res = pi / (\n", - " sin(pi * z) * laguerre_gamma(1 - z, x, w, target)\n", - " ) # Reflection formula\n", - " else:\n", - " z_shifted, correction_factor = find_shift(z, target)\n", - " res = np.sum(x ** (z_shifted - 1) * w)\n", - " res *= correction_factor\n", - " res = drop_imag(res)\n", - " return res\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "def eval_laguerre(x, target=12):\n", - " return np.array([laguerre_gamma(xi, zeros, weights, target) for xi in x])\n", - "\n", - "\n", - "def eval_lanczos(x):\n", - " return np.array([lanczos_gamma(xi) for xi in x])\n", - "\n", - "\n", - "def eval_mean_laguerre(x, targets):\n", - " return np.mean([eval_laguerre(x, target) for target in targets], 0)\n", - "\n", - "\n", - "def calc_rel_error(x, y):\n", - " return (y - x) / x\n", - "\n", - "\n", - "def evaluate(x, target=12):\n", - " lanczos_gammas = eval_lanczos(x)\n", - " laguerre_gammas = eval_laguerre(x, target)\n", - " rel_error = calc_rel_error(lanczos_gammas, laguerre_gammas)\n", - " return rel_error\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Test with real values" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Empirische Tests zeigen:\n", - "- $n=4 \\Rightarrow m=6$\n", - "- $n=5 \\Rightarrow m=7$ oder $m=8$\n", - "- $n=6 \\Rightarrow m=9$\n", - "- $n=7 \\Rightarrow m=10$\n", - "- $n=8 \\Rightarrow m=11$ oder $m=12$\n", - "- $n=9 \\Rightarrow m=13$\n", - "- $n=10 \\Rightarrow m=14$\n", - "- $n=11 \\Rightarrow m=15$ oder $m=16$\n", - "- $n=12 \\Rightarrow m=17$\n", - "- $n=13 \\Rightarrow m=18 \\Rightarrow $ Beginnt numerisch instabil zu werden \n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "zeros, weights = np.polynomial.laguerre.laggauss(12)\n", - "targets = np.arange(16, 21)\n", - "mean_targets = ((16, 17),)\n", - "x = np.linspace(EPSILON, 1 - EPSILON, 101)\n", - "_, axs = plt.subplots(\n", - " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", - ")\n", - "\n", - "lanczos = eval_lanczos(x)\n", - "for mean_target in mean_targets:\n", - " vals = eval_mean_laguerre(x, mean_target)\n", - " rel_error_mean = calc_rel_error(lanczos, vals)\n", - " axs[0].plot(x, rel_error_mean, label=mean_target)\n", - " axs[1].semilogy(x, np.abs(rel_error_mean), label=mean_target)\n", - "\n", - "mins = []\n", - "maxs = []\n", - "for target in targets:\n", - " rel_error = evaluate(x, target)\n", - " mins.append(np.min(np.abs(rel_error[(0.1 <= x) & (x <= 0.9)])))\n", - " maxs.append(np.max(np.abs(rel_error)))\n", - " axs[0].plot(x, rel_error, label=target)\n", - " axs[1].semilogy(x, np.abs(rel_error), label=target)\n", - "# axs[0].set_ylim(*(np.array([-1, 1]) * 3.5e-8))\n", - "\n", - "axs[0].set_xlim(x[0], x[-1])\n", - "axs[1].set_ylim(np.min(mins), 1.04*np.max(maxs))\n", - "for ax in axs:\n", - " ax.legend()\n", - " ax.grid(which=\"both\")\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "targets = (16, 17)\n", - "xmax = 15\n", - "x = np.linspace(-xmax + EPSILON, xmax - EPSILON, 1000)\n", - "\n", - "mean_lag = eval_mean_laguerre(x, targets)\n", - "lanczos = eval_lanczos(x)\n", - "rel_error = calc_rel_error(lanczos, mean_lag)\n", - "rel_error_simple = evaluate(x, targets[-1])\n", - "# rel_error = evaluate(x, target)\n", - "\n", - "_, axs = plt.subplots(\n", - " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", - ")\n", - "axs[0].plot(x, rel_error, label=targets)\n", - "axs[1].semilogy(x, np.abs(rel_error), label=targets)\n", - "axs[0].plot(x, rel_error_simple, label=targets[-1])\n", - "axs[1].semilogy(x, np.abs(rel_error_simple), label=targets[-1])\n", - "axs[0].set_xlim(x[0], x[-1])\n", - "# axs[0].set_ylim(*(np.array([-1, 1]) * 4.2e-8))\n", - "# axs[1].set_ylim(1e-10, 5e-8)\n", - "for ax in axs:\n", - " ax.legend()\n", - "\n", - "x2 = np.linspace(-5 + EPSILON, 5, 4001)\n", - "_, ax = plt.subplots(constrained_layout=True, figsize=(8, 6))\n", - "ax.plot(x2, eval_mean_laguerre(x2, targets))\n", - "ax.set_xlim(x2[0], x2[-1])\n", - "ax.set_ylim(-7.5, 25)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Test with complex values" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "targets = (16, 17)\n", - "vals = np.linspace(-5 + EPSILON, 5, 100)\n", - "x, y = np.meshgrid(vals, vals)\n", - "mesh = x + 1j * y\n", - "input = mesh.flatten()\n", - "\n", - "mean_lag = eval_mean_laguerre(input, targets).reshape(mesh.shape)\n", - "lanczos = eval_lanczos(input).reshape(mesh.shape)\n", - "rel_error = np.abs(calc_rel_error(lanczos, mean_lag))\n", - "\n", - "lag = eval_laguerre(input, targets[-1]).reshape(mesh.shape)\n", - "rel_error_simple = np.abs(calc_rel_error(lanczos, lag))\n", - "# rel_error = evaluate(x, target)\n", - "\n", - "fig, axs = plt.subplots(\n", - " 2,\n", - " 2,\n", - " sharex=True,\n", - " sharey=True,\n", - " clear=True,\n", - " constrained_layout=True,\n", - " figsize=(12, 10),\n", - ")\n", - "_c = axs[0, 1].pcolormesh(x, y, np.log10(np.abs(lanczos - mean_lag)), shading=\"gouraud\")\n", - "_c = axs[0, 0].pcolormesh(x, y, np.log10(np.abs(lanczos - lag)), shading=\"gouraud\")\n", - "fig.colorbar(_c, ax=axs[0, :])\n", - "_c = axs[1, 1].pcolormesh(x, y, np.log10(rel_error), shading=\"gouraud\")\n", - "_c = axs[1, 0].pcolormesh(x, y, np.log10(rel_error_simple), shading=\"gouraud\")\n", - "fig.colorbar(_c, ax=axs[1, :])\n", - "_ = axs[0, 0].set_title(\"Absolute Error\")\n", - "_ = axs[1, 0].set_title(\"Relative Error\")\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "z = 0.5\n", - "ns = [4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12] # np.arange(4, 13)\n", - "ms = np.arange(6, 18)\n", - "xi = np.logspace(0, 2, 201)[:, None]\n", - "lanczos = eval_lanczos([z])[0]\n", - "\n", - "_, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(12, 8))\n", - "ax.grid(1)\n", - "for n, m in zip(ns, ms):\n", - " zeros, weights = np.polynomial.laguerre.laggauss(n)\n", - " c = scipy.special.factorial(n) ** 2 / scipy.special.factorial(2 * n)\n", - " e = np.abs(\n", - " scipy.special.poch(z - 2 * n, 2 * n)\n", - " / scipy.special.poch(z - m, m)\n", - " * c\n", - " * xi ** (z - 2 * n + m - 1)\n", - " )\n", - " ez = np.sum(\n", - " scipy.special.poch(z - 2 * n, 2 * n)\n", - " / scipy.special.poch(z - m, m)\n", - " * c\n", - " * zeros[:, None] ** (z - 2 * n + m - 1),\n", - " 0,\n", - " )\n", - " lag = eval_laguerre([z], m)[0]\n", - " err = np.abs(lanczos - lag)\n", - " # print(m+z,ez)\n", - " # for zi,ezi in zip(z[0], ez):\n", - " # print(f\"{m+zi}: {ezi}\")\n", - " # ax.semilogy(xi, e, color=color)\n", - " lines = ax.loglog(xi, e, label=str(n))\n", - " ax.axhline(err, color=lines[0].get_color())\n", - " # ax.set_xticks(np.arange(xi[-1] + 1))\n", - " # ax.set_ylim(1e-8, 1e5)\n", - "_ = ax.legend()\n", - "# _ = ax.legend([f\"z={zi}\" for zi in z[0]])\n", - "# _ = [ax.axvline(x) for x in zeros]\n" - ] - } - ], - "metadata": { - "interpreter": { - "hash": "767d51c1340bd893661ea55ea3124f6de3c7a262a8b4abca0554b478b1e2ff90" - }, - "kernelspec": { - "display_name": "Python 3.8.10 64-bit", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.10" - }, - "orig_nbformat": 4 - }, - "nbformat": 4, - "nbformat_minor": 2 -} diff --git a/buch/papers/laguerre/scripts/gamma_approx.py b/buch/papers/laguerre/scripts/gamma_approx.py new file mode 100644 index 0000000..5b09e59 --- /dev/null +++ b/buch/papers/laguerre/scripts/gamma_approx.py @@ -0,0 +1,116 @@ +from pathlib import Path + +import numpy as np +import scipy.special + +EPSILON = 1e-7 +root = str(Path(__file__).parent) +img_path = f"{root}/../images" +fontsize = "medium" +cmap = "plasma" + + +def _prep_zeros_and_weights(x, w, n): + if x is None or w is None: + return np.polynomial.laguerre.laggauss(n) + return x, w + + +def drop_imag(z): + if abs(z.imag) <= EPSILON: + z = z.real + return z + + +def pochhammer(z, n): + return np.prod(z + np.arange(n)) + + +def find_optimal_shift(z, n): + mhat = 1.34093 * n + 0.854093 + steps = int(np.floor(mhat - np.real(z))) + return steps + + +def get_shifting_factor(z, steps): + factor = 1.0 + if steps > 0: + factor = 1 / pochhammer(z, steps) + elif steps < 0: + factor = pochhammer(z + steps, -steps) + return factor + + +def laguerre_gamma_shifted(z, x=None, w=None, n=8, target=11): + x, w = _prep_zeros_and_weights(x, w, n) + n = len(x) + + z += 0j + steps = int(np.floor(target - np.real(z))) + z_shifted = z + steps + correction_factor = get_shifting_factor(z, steps) + + res = np.sum(x ** (z_shifted - 1) * w) + res *= correction_factor + res = drop_imag(res) + return res + + +def laguerre_gamma_opt_shifted(z, x=None, w=None, n=8): + if z == 0.0: + return np.infty + x, w = _prep_zeros_and_weights(x, w, n) + n = len(x) + + z += 0j + opt_shift = find_optimal_shift(z, n) + correction_factor = get_shifting_factor(z, opt_shift) + z_shifted = z + opt_shift + + res = np.sum(x ** (z_shifted - 1) * w) + res *= correction_factor + res = drop_imag(res) + return res + + +def laguerre_gamma_simple(z, x=None, w=None, n=8): + if z == 0.0: + return np.infty + x, w = _prep_zeros_and_weights(x, w, n) + z += 0j + res = np.sum(x ** (z - 1) * w) + res = drop_imag(res) + return res + + +def laguerre_gamma_mirror(z, x=None, w=None, n=8): + if z == 0.0: + return np.infty + x, w = _prep_zeros_and_weights(x, w, n) + z += 0j + if z.real < 1e-3: + return np.pi / ( + np.sin(np.pi * z) * laguerre_gamma_simple(1 - z, x, w) + ) # Reflection formula + return laguerre_gamma_simple(z, x, w) + + +def eval_laguerre_gamma(z, x=None, w=None, n=8, func="simple", **kwargs): + x, w = _prep_zeros_and_weights(x, w, n) + if func == "simple": + f = laguerre_gamma_simple + elif func == "mirror": + f = laguerre_gamma_mirror + elif func == "optimal_shifted": + f = laguerre_gamma_opt_shifted + else: + f = laguerre_gamma_shifted + return np.array([f(zi, x, w, n, **kwargs) for zi in z]) + + +def calc_rel_error(x, y): + mask = np.abs(x) != np.infty + rel_error = np.zeros_like(y) + rel_error[mask] = (y[mask] - x[mask]) / x[mask] + rel_error[~mask] = 0.0 + return rel_error diff --git a/buch/papers/laguerre/scripts/integrand.py b/buch/papers/laguerre/scripts/integrand.py new file mode 100644 index 0000000..e970721 --- /dev/null +++ b/buch/papers/laguerre/scripts/integrand.py @@ -0,0 +1,42 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Plot for integrand of gamma function with shifting terms.""" + +if __name__ == "__main__": + import os + from pathlib import Path + + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + EPSILON = 1e-12 + xlims = np.array([-3, 3]) + + root = str(Path(__file__).parent) + img_path = f"{root}/../images" + os.makedirs(img_path, exist_ok=True) + + t = np.logspace(*xlims, 1001)[:, None] + + z = np.array([-4.5, -2, -1, -0.5, 0.0, 0.5, 1, 2, 4.5]) + r = t ** z + + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(4, 2.4)) + ax.semilogx(t, r) + ax.set_xlim(*(10.0 ** xlims)) + ax.set_ylim(1e-3, 40) + ax.set_xlabel(r"$x$") + # ax.set_ylabel(r"$x^z$") + ax.grid(1, "both") + labels = [f"$z={zi: 3.1f}$" for zi in np.squeeze(z)] + ax.legend(labels, ncol=2, loc="upper left", fontsize="small") + fig.savefig(f"{img_path}/integrand.pdf") diff --git a/buch/papers/laguerre/scripts/integrand_exp.py b/buch/papers/laguerre/scripts/integrand_exp.py new file mode 100644 index 0000000..e649b26 --- /dev/null +++ b/buch/papers/laguerre/scripts/integrand_exp.py @@ -0,0 +1,46 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Plot for integrand of gamma function with shifting terms.""" + +if __name__ == "__main__": + import os + from pathlib import Path + + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + EPSILON = 1e-12 + xlims = np.array([-3, 3]) + + root = str(Path(__file__).parent) + img_path = f"{root}/../images" + os.makedirs(img_path, exist_ok=True) + + t = np.logspace(*xlims, 1001)[:, None] + + z = np.array([-1, -0.5, 0.0, 0.5, 1, 2, 3, 4, 4.5]) + e = np.exp(-t) + r = t ** z * e + + fig, ax = plt.subplots(num=2, clear=True, constrained_layout=True, figsize=(4, 2.4)) + ax.semilogx(t, r) + # ax.plot(t,np.exp(-t)) + ax.set_xlim(10 ** (-2), 20) + ax.set_ylim(1e-3, 10) + ax.set_xlabel(r"$x$") + # ax.set_ylabel(r"$x^z e^{-x}$") + ax.grid(1, "both") + labels = [f"$z={zi: 3.1f}$" for zi in np.squeeze(z)] + ax.legend(labels, ncol=2, loc="upper left", fontsize="small") + # fig.savefig(f"{img_path}/integrand_exp.pgf") + fig.savefig(f"{img_path}/integrand_exp.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py deleted file mode 100644 index b9088d0..0000000 --- a/buch/papers/laguerre/scripts/laguerre_plot.py +++ /dev/null @@ -1,100 +0,0 @@ -#!/usr/bin/env python3 -# -*- coding:utf-8 -*- -"""Some plots for Laguerre Polynomials.""" - -import os -from pathlib import Path - -import matplotlib.pyplot as plt -import numpy as np -import scipy.special as ss - - -def get_ticks(start, end, step=1): - ticks = np.arange(start, end, step) - return ticks[ticks != 0] - - -N = 1000 -step = 5 -t = np.linspace(-1.05, 10.5, N)[:, None] -root = str(Path(__file__).parent) -img_path = f"{root}/../images" -os.makedirs(img_path, exist_ok=True) - - -# fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7)) -# ax = fig.add_subplot(axes_class=AxesZero) -fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4)) -for n in np.arange(0, 8): - k = np.arange(0, n + 1)[None] - L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1) - ax.plot(t, L, label=f"n={n}") - -ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True) -ax.set_xticks(get_ticks(0, t[-1], step)) -ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0])) -ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large") - -ylim = 13 -ax.set_yticks(np.arange(-ylim, ylim), minor=True) -ax.set_yticks(np.arange(-step * (ylim // step), ylim, step)) -ax.set_ylim(-ylim, ylim) -ax.set_ylabel(r"$y$", y=0.95, labelpad=-18, rotation=0, fontsize="large") - -ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large") - -# set the x-spine -ax.spines[["left", "bottom"]].set_position("zero") -ax.spines[["right", "top"]].set_visible(False) -ax.xaxis.set_ticks_position("bottom") -hlx = 0.4 -dx = t[-1, 0] - t[0, 0] -dy = 2 * ylim -hly = dy / dx * hlx -dps = fig.dpi_scale_trans.inverted() -bbox = ax.get_window_extent().transformed(dps) -width, height = bbox.width, bbox.height - -# manual arrowhead width and length -hw = 1.0 / 60.0 * dy -hl = 1.0 / 30.0 * dx -lw = 0.5 # axis line width -ohg = 0.0 # arrow overhang - -# compute matching arrowhead length and width -yhw = hw / dy * dx * height / width -yhl = hl / dx * dy * width / height - -# draw x and y axis -ax.arrow( - t[-1, 0] - hl, - 0, - hl, - 0.0, - fc="k", - ec="k", - lw=lw, - head_width=hw, - head_length=hl, - overhang=ohg, - length_includes_head=True, - clip_on=False, -) - -ax.arrow( - 0, - ylim - yhl, - 0.0, - yhl, - fc="k", - ec="k", - lw=lw, - head_width=yhw, - head_length=yhl, - overhang=ohg, - length_includes_head=True, - clip_on=False, -) - -fig.savefig(f"{img_path}/laguerre_polynomes.pdf") diff --git a/buch/papers/laguerre/scripts/laguerre_poly.py b/buch/papers/laguerre/scripts/laguerre_poly.py new file mode 100644 index 0000000..9700ab4 --- /dev/null +++ b/buch/papers/laguerre/scripts/laguerre_poly.py @@ -0,0 +1,108 @@ +import numpy as np + + +def get_ticks(start, end, step=1): + ticks = np.arange(start, end, step) + return ticks[ticks != 0] + + +if __name__ == "__main__": + import os + from pathlib import Path + + import matplotlib as mpl + import matplotlib.pyplot as plt + import scipy.special as ss + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + N = 1000 + step = 5 + t = np.linspace(-1.05, 10.5, N)[:, None] + root = str(Path(__file__).parent) + img_path = f"{root}/../images" + os.makedirs(img_path, exist_ok=True) + + # fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7)) + # ax = fig.add_subplot(axes_class=AxesZero) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4)) + for n in np.arange(0, 8): + k = np.arange(0, n + 1)[None] + L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1) + ax.plot(t, L, label=f"$n={n}$") + + ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True) + ax.set_xticks(get_ticks(0, t[-1], step)) + ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0])) + ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large") + + ylim = 13 + ax.set_yticks(get_ticks(-ylim, ylim), minor=True) + ax.set_yticks(get_ticks(-step * (ylim // step), ylim, step)) + ax.set_ylim(-ylim, ylim) + ax.set_ylabel(r"$y$", y=0.95, labelpad=-18, rotation=0, fontsize="large") + + ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large") + + # set the x-spine + ax.spines[["left", "bottom"]].set_position("zero") + ax.spines[["right", "top"]].set_visible(False) + ax.xaxis.set_ticks_position("bottom") + hlx = 0.4 + dx = t[-1, 0] - t[0, 0] + dy = 2 * ylim + hly = dy / dx * hlx + dps = fig.dpi_scale_trans.inverted() + bbox = ax.get_window_extent().transformed(dps) + width, height = bbox.width, bbox.height + + # manual arrowhead width and length + hw = 1.0 / 60.0 * dy + hl = 1.0 / 30.0 * dx + lw = 0.5 # axis line width + ohg = 0.0 # arrow overhang + + # compute matching arrowhead length and width + yhw = hw / dy * dx * height / width + yhl = hl / dx * dy * width / height + + # draw x and y axis + ax.arrow( + t[-1, 0] - hl, + 0, + hl, + 0.0, + fc="k", + ec="k", + lw=lw, + head_width=hw, + head_length=hl, + overhang=ohg, + length_includes_head=True, + clip_on=False, + ) + + ax.arrow( + 0, + ylim - yhl, + 0.0, + yhl, + fc="k", + ec="k", + lw=lw, + head_width=yhw, + head_length=yhl, + overhang=ohg, + length_includes_head=True, + clip_on=False, + ) + + # fig.savefig(f"{img_path}/laguerre_poly.pgf") + fig.savefig(f"{img_path}/laguerre_poly.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_complex.py b/buch/papers/laguerre/scripts/rel_error_complex.py new file mode 100644 index 0000000..4a714fa --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_complex.py @@ -0,0 +1,43 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + xmax = 4 + vals = np.linspace(-xmax + ga.EPSILON, xmax, 100) + x, y = np.meshgrid(vals, vals) + mesh = x + 1j * y + input = mesh.flatten() + + lanczos = scipy.special.gamma(mesh) + lag = ga.eval_laguerre_gamma(input, n=8, func="optimal_shifted").reshape(mesh.shape) + rel_error = np.abs(ga.calc_rel_error(lanczos, lag)) + + fig, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(3.5, 2.1)) + _c = ax.pcolormesh( + x, y, rel_error, shading="gouraud", cmap=ga.cmap, norm=mpl.colors.LogNorm() + ) + cbr = fig.colorbar(_c, ax=ax) + cbr.minorticks_off() + # ax.set_title("Relative Error") + ax.set_xlabel("Re($z$)") + ax.set_ylabel("Im($z$)") + minor_ticks = np.arange(-xmax, xmax + ga.EPSILON) + ticks = np.arange(-xmax, xmax + ga.EPSILON, 2) + ax.set_xticks(ticks) + ax.set_xticks(minor_ticks, minor=True) + ax.set_yticks(ticks) + ax.set_yticks(minor_ticks, minor=True) + fig.savefig(f"{ga.img_path}/rel_error_complex.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_mirror.py b/buch/papers/laguerre/scripts/rel_error_mirror.py new file mode 100644 index 0000000..7348d5e --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_mirror.py @@ -0,0 +1,38 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + xmin = -15 + xmax = 15 + ns = np.arange(2, 12, 2) + ylim = np.array([-11, 1]) + x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, 400) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.5)) + for n in ns: + gamma_lag = ga.eval_laguerre_gamma(x, n=n, func="mirror") + rel_err = ga.calc_rel_error(gamma, gamma_lag) + ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$") + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(*(10.0 ** ylim)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON, 5)) + ax.set_xticks(np.arange(xmin, xmax), minor=True) + ax.set_yticks(10.0 ** np.arange(*ylim, 2)) + ax.set_xlabel(r"$z$") + # ax.set_ylabel("Relativer Fehler") + ax.legend(ncol=1, loc="upper left", fontsize=ga.fontsize) + ax.grid(1, "both") + # fig.savefig(f"{ga.img_path}/rel_error_mirror.pgf") + fig.savefig(f"{ga.img_path}/rel_error_mirror.pdf") diff --git a/buch/papers/laguerre/scripts/rel_error_range.py b/buch/papers/laguerre/scripts/rel_error_range.py new file mode 100644 index 0000000..ece3b6d --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_range.py @@ -0,0 +1,41 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + N = 1201 + xmax = 6 + xmin = -xmax + ns = np.arange(2, 12, 2) + ylim = np.array([-11, -1.2]) + + x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, N) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2)) + for n in ns: + gamma_lag = ga.eval_laguerre_gamma(x, n=n, func="optimal_shifted") + rel_err = ga.calc_rel_error(gamma, gamma_lag) + ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$") + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(*(10.0 ** ylim)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON, 2)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON), minor=True) + ax.set_yticks(10.0 ** np.arange(*ylim, 2)) + ax.set_yticks(10.0 ** np.arange(*ylim, 1), "", minor=True) + ax.set_xlabel(r"$z$") + # ax.set_ylabel("Relativer Fehler") + ax.legend(ncol=1, loc="upper left", fontsize=ga.fontsize) + ax.grid(1, "both") + # fig.savefig(f"{ga.img_path}/rel_error_range.pgf") + fig.savefig(f"{ga.img_path}/rel_error_range.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_shifted.py b/buch/papers/laguerre/scripts/rel_error_shifted.py new file mode 100644 index 0000000..f53c89b --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_shifted.py @@ -0,0 +1,40 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + n = 8 # order of Laguerre polynomial + N = 200 # number of points in interval + + step = 1 / (N - 1) + x = np.linspace(step, 1 - step, N + 1) + targets = np.arange(10, 14) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.1)) + for target in targets: + gamma_lag = ga.eval_laguerre_gamma(x, target=target, n=n, func="shifted") + rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lag)) + ax.semilogy(x, rel_error, label=f"$m={target}$", linewidth=3) + gamma_lgo = ga.eval_laguerre_gamma(x, n=n, func="optimal_shifted") + rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lgo)) + ax.semilogy(x, rel_error, "m", linestyle=":", label="$m^*$", linewidth=3) + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(5e-9, 5e-8) + ax.set_xlabel(r"$z$") + ax.set_xticks(np.linspace(0, 1, 6)) + ax.set_xticks(np.linspace(0, 1, 11), minor=True) + ax.grid(1, "both") + ax.legend(ncol=1, fontsize=ga.fontsize) + # fig.savefig(f"{ga.img_path}/rel_error_shifted.pgf") + fig.savefig(f"{ga.img_path}/rel_error_shifted.pdf") + # plt.show() diff --git a/buch/papers/laguerre/scripts/rel_error_simple.py b/buch/papers/laguerre/scripts/rel_error_simple.py new file mode 100644 index 0000000..686500b --- /dev/null +++ b/buch/papers/laguerre/scripts/rel_error_simple.py @@ -0,0 +1,41 @@ +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + import numpy as np + import scipy.special + + import gamma_approx as ga + + # mpl.rc("text", usetex=True) + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + # mpl.rcParams.update({"font.family": "serif", "font.serif": "TeX Gyre Termes"}) + + # Simple / naive + xmin = -5 + xmax = 30 + ns = np.arange(2, 12, 2) + ylim = np.array([-11, 6]) + x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, 400) + gamma = scipy.special.gamma(x) + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.5)) + for n in ns: + gamma_lag = ga.eval_laguerre_gamma(x, n=n) + rel_err = ga.calc_rel_error(gamma, gamma_lag) + ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$") + ax.set_xlim(x[0], x[-1]) + ax.set_ylim(*(10.0 ** ylim)) + ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON, 5)) + ax.set_xticks(np.arange(xmin, xmax), minor=True) + ax.set_yticks(10.0 ** np.arange(*ylim, 2)) + ax.set_xlabel(r"$z$") + # ax.set_ylabel("Relativer Fehler") + ax.legend(ncol=3, fontsize=ga.fontsize) + ax.grid(1, "both") + # fig.savefig(f"{ga.img_path}/rel_error_simple.pgf") + fig.savefig(f"{ga.img_path}/rel_error_simple.pdf") diff --git a/buch/papers/laguerre/scripts/targets.py b/buch/papers/laguerre/scripts/targets.py new file mode 100644 index 0000000..3bc7f52 --- /dev/null +++ b/buch/papers/laguerre/scripts/targets.py @@ -0,0 +1,58 @@ +import numpy as np +import scipy.special + +import gamma_approx as ga + + +def find_best_loc(N=200, a=1.375, b=0.5, ns=None): + if ns is None: + ns = np.arange(2, 13) + bests = [] + step = 1 / (N - 1) + x = np.linspace(step, 1 - step, N + 1) + gamma = scipy.special.gamma(x) + for n in ns: + zeros, weights = np.polynomial.laguerre.laggauss(n) + est = np.ceil(b + a * n) + targets = np.arange(max(est - 2, 0), est + 3) + rel_error = [] + for target in targets: + gamma_lag = ga.eval_laguerre_gamma(x, target=target, x=zeros, w=weights, func="shifted") + rel_error.append(np.abs(ga.calc_rel_error(gamma, gamma_lag))) + rel_error = np.stack(rel_error, -1) + best = np.argmin(rel_error, -1) + targets[0] + bests.append(best) + return np.stack(bests, 0) + + +if __name__ == "__main__": + import matplotlib as mpl + import matplotlib.pyplot as plt + + mpl.rcParams.update( + { + "mathtext.fontset": "stix", + "font.family": "serif", + "font.serif": "TeX Gyre Termes", + } + ) + + N = 200 + ns = np.arange(2, 13) + + bests = find_best_loc(N, ns=ns) + + fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(3.5, 2.1)) + v = ax.imshow(bests, cmap=ga.cmap, aspect="auto", interpolation="nearest") + plt.colorbar(v, ax=ax, label=r"$m^*$") + ticks = np.arange(0, N + 1, N // 5) + ax.set_xlim(0, 1) + ax.set_xticks(ticks) + ax.set_xticklabels([f"{v:.2f}" for v in ticks / N]) + ax.set_xticks(np.arange(0, N + 1, N // 20), minor=True) + ax.set_yticks(np.arange(len(ns))) + ax.set_yticklabels(ns) + ax.set_xlabel(r"$z$") + ax.set_ylabel(r"$n$") + # fig.savefig(f"{ga.img_path}/targets.pgf") + fig.savefig(f"{ga.img_path}/targets.pdf") diff --git a/buch/papers/lambertw/Bilder/Strategie.png b/buch/papers/lambertw/Bilder/Strategie.png Binary files differnew file mode 100644 index 0000000..e78abd3 --- /dev/null +++ b/buch/papers/lambertw/Bilder/Strategie.png diff --git a/buch/papers/lambertw/Bilder/VerfolgungskurveBsp.png b/buch/papers/lambertw/Bilder/VerfolgungskurveBsp.png Binary files differnew file mode 100644 index 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font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="618" y="1288">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="618" y="1288">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="670" y="1288">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="670" y="1288">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="670" y="1288">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="721" y="1288">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="721" y="1288">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="721" y="1288">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="773" y="1288">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="773" y="1288">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="773" y="1288">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="825" y="1288">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="825" y="1288">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="825" y="1288">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="876" y="1288">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="876" y="1288">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="876" y="1288">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="934" y="1288">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="934" y="1288">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="934" y="1288">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="980" y="1288">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="980" y="1288">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="980" y="1288">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1032" y="1288">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1032" y="1288">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1032" y="1288">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1083" y="1288">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1083" y="1288">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1083" y="1288">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1135" y="1288">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1135" y="1288">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1135" y="1288">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1187" y="1288">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1187" y="1288">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1187" y="1288">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1238" y="1288">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1238" y="1288">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1238" y="1288">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1290" y="1288">1.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1290" y="1288">1.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1290" y="1288">1.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1342" y="1288">1.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1342" y="1288">1.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1342" y="1288">1.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1393" y="1288">1.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1393" y="1288">1.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1393" y="1288">1.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1451" y="1288">2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1451" y="1288">2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1451" y="1288">2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1497" y="1288">2.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1497" y="1288">2.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1497" y="1288">2.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1549" y="1288">2.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1549" y="1288">2.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1549" y="1288">2.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1600" y="1288">2.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1600" y="1288">2.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1600" y="1288">2.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1652" y="1288">2.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1652" y="1288">2.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1652" y="1288">2.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1704" y="1288">2.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1704" y="1288">2.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1704" y="1288">2.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1755" y="1288">2.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1755" y="1288">2.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1755" y="1288">2.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1807" y="1288">2.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1807" y="1288">2.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1807" y="1288">2.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1859" y="1288">2.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1859" y="1288">2.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1859" y="1288">2.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1910" y="1288">2.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1910" y="1288">2.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1910" y="1288">2.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1968" y="1288">3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1968" y="1288">3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1968" y="1288">3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2014" y="1288">3.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="2014" y="1288">3.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2014" y="1288">3.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2066" y="1288">3.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="2066" y="1288">3.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2066" y="1288">3.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2117" y="1288">3.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="2117" y="1288">3.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2117" y="1288">3.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1226">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1226">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1226">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1174">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1174">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1174">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1122">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1122">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1122">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1071">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1071">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1071">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1019">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1019">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1019">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="967">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="967">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="967">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="916">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="916">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="916">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="864">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="864">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="864">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="812">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="812">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="812">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="405" y="760">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="405" y="760">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="405" y="760">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="709">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="709">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="709">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="657">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="657">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="657">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="605">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="605">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="605">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="554">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="554">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="554">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="502">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="502">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="502">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="450">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="450">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="450">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" 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LaTeX Error: File `papers/lambertw/teil0.tex' not found. + +Type X to quit or <RETURN> to proceed, +or enter new name. (Default extension: tex) + +Enter file name: +! Emergency stop. +<read *> + +l.30 \input{papers/lambertw/teil0.tex} + ^^M +*** (cannot \read from terminal in nonstop modes) + + +Here is how much of TeX's memory you used: + 36 strings out of 478371 + 593 string characters out of 5852527 + 296836 words of memory out of 5000000 + 18242 multiletter control sequences out of 15000+600000 + 403598 words of font info for 28 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 23i,1n,32p,120b,183s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/main.tex b/buch/papers/lambertw/main.tex index c125c33..68b7a5d 100644 --- a/buch/papers/lambertw/main.tex +++ b/buch/papers/lambertw/main.tex @@ -3,10 +3,10 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:lambertw}} +\chapter{Verfolgungskurven\label{chapter:lambertw}} \lhead{Thema} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{David Hugentobler und Yanik Kuster} Ein paar Hinweise für die korrekte Formatierung des Textes \begin{itemize} @@ -28,9 +28,10 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren \end{itemize} \input{papers/lambertw/teil0.tex} -\input{papers/lambertw/teil1.tex} -\input{papers/lambertw/teil2.tex} -\input{papers/lambertw/teil3.tex} +%\input{papers/lambertw/teil1.tex} +%\input{papers/lambertw/teil2.tex} +%\input{papers/lambertw/teil3.tex} +\input{papers/lambertw/teil4.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/lambertw/packages.tex b/buch/papers/lambertw/packages.tex index 6581a5a..366de78 100644 --- a/buch/papers/lambertw/packages.tex +++ b/buch/papers/lambertw/packages.tex @@ -8,3 +8,5 @@ % following example %\usepackage{packagename} +\usepackage{graphicx} +\usepackage{float}
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+Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! 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+Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! 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+Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 13--16 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 o + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.17 \begin{table} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 \begin{tabular} + {|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 18. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 18. + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +! Undefined control sequence. +<recently read> \text + +l.20 \text + {}&\text{Geschwindigkeit}&\text{Abstand}&\text{Richtung}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.20 \text{}&\text + {Geschwindigkeit}&\text{Abstand}&\text{Richtung}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no G in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.20 \text{}&\text{Geschwindigkeit}&\text + {Abstand}&\text{Richtung}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no A in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +! Undefined control sequence. +l.20 ...text{Geschwindigkeit}&\text{Abstand}&\text + {Richtung}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no R in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +! Undefined control sequence. +<recently read> \text + +l.22 \text + {Strategie 1} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +! Undefined control sequence. +l.23 & \text + {konstant} & \text{-} & \text{direkt auf Ziel hinzu}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.23 & \text{konstant} & \text + {-} & \text{direkt auf Ziel hinzu}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no - in font nullfont! +! Undefined control sequence. +l.23 & \text{konstant} & \text{-} & \text + {direkt auf Ziel hinzu}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +! Undefined control sequence. +l.25 \text + {Strategie 2} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 2 in font nullfont! +! Undefined control sequence. +l.26 & \text + {-} & \text{konstant} & \text{direkt auf Ziel hinzu}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no - in font nullfont! +! Undefined control sequence. +l.26 & \text{-} & \text + {konstant} & \text{direkt auf Ziel hinzu}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.26 & \text{-} & \text{konstant} & \text + {direkt auf Ziel hinzu}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +! Undefined control sequence. +l.28 \text + {Strategie 3} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 3 in font nullfont! +! Undefined control sequence. +l.29 & \text + {konstant} & \text{-} & \text{etwas voraus Zielen}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.29 & \text{konstant} & \text + {-} & \text{etwas voraus Zielen}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no - in font nullfont! +! Undefined control sequence. +l.29 & \text{konstant} & \text{-} & \text + {etwas voraus Zielen}\\ +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.33 \end{table} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +Overfull \hbox (20.00006pt too wide) in paragraph at lines 18--34 +[][] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.42 I + n der Tabelle \eqref{lambertw:Strategien} sind drei mögliche Strategi... + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no I in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.42 In der Tabelle \eqref + {lambertw:Strategien} sind drei mögliche Strategi... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no k in font nullfont! +! Undefined control sequence. +l.45 In der Grafik \eqref + {lambertw:pursuerDGL2} ist das Problem dargestellt. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 42--48 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 42--48 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 42--48 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 42--48 +[]$ + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 42--48 +[]$ + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 42--48 +[]$ + [] + +! Undefined control sequence. +l.51 \quad|A\in\mathbb + {R}>0 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +Overfull \hbox (122.89459pt too wide) detected at line 52 +\OMS/cmsy/m/n/10 j[]j \/cmr/m/n/10 = \OML/cmm/m/it/10 konst \/cmr/m/n/10 = \OML +/cmm/m/it/10 A \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 A \OMS/cmsy/m/n/10 2 \OML/cmm +/m/it/10 R > \/cmr/m/n/10 0 + [] + +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! + +Overfull \hbox (81.8452pt too wide) detected at line 58 +[] \OMS/cmsy/m/n/10 j[]j \/cmr/m/n/10 = [] + [] + +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no P in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.65 \begin{align} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +Missing character: There is no - in font nullfont! +! Missing $ inserted. +<inserted text> + $ +l.67 ...}{|\overrightarrow{Z}-\overrightarrow{V}|} + \cdot +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\frac #1#2->{\begingroup #1\endgroup \over #2} + +l.67 ...}{|\overrightarrow{Z}-\overrightarrow{V}|} + \cdot +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced alignment tab character &. +l.69 & + = +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.73 & + = +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.75 \end{align} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +! Missing $ inserted. +<inserted text> + $ +l.75 \end{align} + +I've inserted something that you may have forgotten. +(See the <inserted text> above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. +<inserted text> + } +l.75 \end{align} + +I've inserted something that you may have forgotten. +(See the <inserted text> above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no K in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no V in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! 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Undefined control sequence. +l.79 \subsection + {Ziel +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.79 \subsection{Z + iel +You're in trouble here. 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Emergency stop. +<*> ./teil0.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 34 strings out of 478927 + 671 string characters out of 2852535 + 298175 words of memory out of 3000000 + 17993 multiletter control sequences out of 15000+600000 + 403430 words of font info for 27 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 23i,13n,32p,801b,95s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/teil0.tex b/buch/papers/lambertw/teil0.tex index 2b83d59..73fe187 100644 --- a/buch/papers/lambertw/teil0.tex +++ b/buch/papers/lambertw/teil0.tex @@ -3,20 +3,101 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{lambertw:section:teil0}} +\section{Was sind Verfolgungskurven? +\label{lambertw:section:teil0}} \rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{lambertw:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. + +Verfolgungskurven tauchen oft auf bei fragen wie, welchen Pfad begeht ein Hund während er einer Katze nachrennt. Ein solches Problem hat im Kern immer ein Verfolger und sein Ziel. Der Verfolger versucht sein Ziel zu ergattern und das Ziel versucht zu entkommen. Der Pfad, der der Verfolger während der Verfolgung begeht, wird Verfolgungskurve genannt. Um diese Kurve zu bestimmen, kann das Verfolgungsproblem als DGL formuliert werden. Diese DGL entspringt der Verfolgungsstrategie des Verfolgers. + + +\subsection{Verfolger und Verfolgungsstrategie +\label{lambertw:subsection:Verfolger}} +Wie bereits erwähnt, wird der Verfolger durch seine Verfolgungsstrategie definiert. Wir nehmen an, dass sich der Verfolger stur an eine Verfolgungsstrategie hält. Dabei gibt es viele mögliche Strategien, die der Verfolger wählen könnte. Die möglichen Strategien entstehen durch Festlegung einzelner Parameter, die der Verfolger kontrollieren kann. Der Verfolger hat nur einen direkten Einfluss auf seinen Geschwindigkeitsvektor. Mit diesem kann er neben Richtung und Betrag auch den Abstand zwischen Verfolger und Ziel kontrollieren. Wenn zwei dieser drei Parameter durch die Strategie definiert werden, ist der dritte nicht mehr frei. Daraus folgt, dass eine Strategie zwei dieser drei Parameter festlegen muss, um den Verfolger komplett zu beschreiben. + +\begin{table} + \centering + \begin{tabular}{|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + \hline + \text{}&\text{Geschwindigkeit}&\text{Abstand}&\text{Richtung}\\ + \hline + \text{Strategie 1} + & \text{konstant} & \text{-} & \text{direkt auf Ziel hinzu}\\ + + \text{Strategie 2} + & \text{-} & \text{konstant} & \text{direkt auf Ziel hinzu}\\ + + \text{Strategie 3} + & \text{konstant} & \text{-} & \text{etwas voraus Zielen}\\ + \hline + \end{tabular} + \caption{mögliche Verfolgungsstrategien} + \label{lambertw:Strategien} +\end{table} + + + + +%\begin{figure} +% \centering +% \includegraphics{.\papers\lambertw\Bilder\pursuerDGL2.pdf} +% \label{pursuer:pursuerDGL2} +%\end{figure} + +In der Tabelle \eqref{lambertw:Strategien} sind drei mögliche Strategien aufgezählt. +Folgend wird nur noch auf die Strategie 1 eingegangen. +Bei dieser Strategie ist die Geschwindigkeit konstant und der Verfolger bewegt sich immer direkt auf sein Ziel hinzu. +In der Grafik \eqref{lambertw:pursuerDGL2} ist das Problem dargestellt. +Wobei $\overrightarrow{V}$ der Ortsvektor des Verfolgers, $\overrightarrow{Z}$ der Ortsvektor des Ziels und $\overrightarrow{\dot{V}}$ der Geschwindigkeitsvektor des Verfolgers ist. +Die konstante Geschwindigkeit kann man mit der Gleichung +\begin{equation} + |\overrightarrow{\dot{V}}| + = konst = A + \quad|A\in\mathbb{R}>0 +\end{equation} +darstellen. Der Geschwindigkeitsvektor wiederum kann mit der Gleichung +\begin{equation} + \frac{\overrightarrow{Z}-\overrightarrow{V}}{|\overrightarrow{Z}-\overrightarrow{V}|}\cdot|\overrightarrow{\dot{V}}| + = + \overrightarrow{\dot{V}} +\end{equation} +beschrieben werden. +Durch die Subtraktion der Ortsvektoren $\overrightarrow{V}$ und $\overrightarrow{Z}$ entsteht ein Vektor der vom Punkt $V$ auf $Z$ zeigt. +Da die Länge dieses Vektors beliebig sein kann, wird durch Division mit dem Betrag, die Länge auf eins festgelegt. +Aus dem Verfolgungsproblem ist auch ersichtlich, dass die Punkte $V$ und $Z$ nicht am gleichen Ort starten und so eine Division durch Null ausgeschlossen ist. +Wenn die Punkte $V$ und $Z$ trotzdem am gleichen Ort starten, ist die Lösung trivial. +Nun wird die Gleichung mit deren rechten Seite skalar multipliziert, um das Gleichungssystem von zwei auf eine Gleichung zu reduzieren. +\begin{align} + \label{pursuer:pursuerDGL} + \frac{\overrightarrow{Z}-\overrightarrow{V}}{|\overrightarrow{Z}-\overrightarrow{V}|}\cdot + \overrightarrow{\dot{V}} + &= + |\overrightarrow{\dot{V}}|^2 + \\ + \frac{\overrightarrow{Z}-\overrightarrow{V}}{|\overrightarrow{Z}-\overrightarrow{V}|}\cdot \frac{\overrightarrow{\dot{V}}}{|\overrightarrow{\dot{V}}|} + &= + 1 +\end{align} +Diese DGL ist der Kern des Verfolgungsproblems, insofern der Verfolger die Strategie 1 verwendet. + + +\subsection{Ziel +\label{lambertw:subsection:Ziel}} +Als nächstes gehen wir auf das Ziel ein. +Wie der Verfolger wird auch unser Ziel sich strikt an eine Fluchtstrategie halten, welche von Anfang an bekannt ist. +Diese Strategie kann als Parameterdarstellung der Position nach der Zeit beschrieben werden. +Zum Beispiel könnte ein Ziel auf einer Geraden flüchten, welches auf einer Ebene mit der Parametrisierung +\begin{equation} + \vec{r}(t) + = + \begin{Bmatrix} + 0\\ + t + \end{Bmatrix} +\end{equation} +beschrieben werden könnte. +Mit dieser Gleichung ist das Ziel auch schon vollumfänglich definiert. +Die Fluchtkurve kann eine beliebige Form haben, jedoch wird die zu lösende DGL immer komplexer. + + diff --git a/buch/papers/lambertw/teil1.log b/buch/papers/lambertw/teil1.log new file mode 100644 index 0000000..d2eb5c6 --- /dev/null +++ b/buch/papers/lambertw/teil1.log @@ -0,0 +1,3259 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (MiKTeX 21.8) (preloaded format=pdflatex 2021.9.21) 5 APR 2022 23:32 +entering extended mode +**./teil1.tex +(teil1.tex +LaTeX2e <2021-06-01> patch level 1 +L3 programming layer <2021-08-27> +! 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Undefined control sequence. +l.25 ... wird der Sachverhalt in der Grafik \eqref + {pursuer_grafik1} aufgezeigt. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +! Missing $ inserted. +<inserted text> + $ +l.25 ... Sachverhalt in der Grafik \eqref{pursuer_ + grafik1} aufgezeigt. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 25. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 25. +! Extra }, or forgotten $. +l.25 ...halt in der Grafik \eqref{pursuer_grafik1} + aufgezeigt. +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. +<inserted text> + $ +l.27 + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 25--27 +[] + [] + + +Overfull \hbox (332.97824pt too wide) in paragraph at lines 25--27 +[]\OML/cmm/m/it/10 rafik\/cmr/m/n/10 1\OML/cmm/m/it/10 aufgezeigt:DerPunktPistd +erVerfolgerundderPunktAistseinZiel:$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.28 U + m dies mathematisch beschreiben zu können, wird der Richtungsvektor +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no U in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! 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+Missing character: There is no r in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 28--29 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 28--29 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (64.58458pt too wide) detected at line 33 +[] \/cmr/m/n/10 = [] + [] + +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no d in font nullfont! 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+ +Overfull \hbox (5.00002pt too wide) in paragraph at lines 33--38 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (15.71458pt too wide) in paragraph at lines 33--38 +[]$ + [] + + +Overfull \hbox (15.40556pt too wide) in paragraph at lines 33--38 +[]$ + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 33--38 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 33--38 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 33--38 +\/cmr/m/n/10 o + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.39 N + un wird die Gleichung mit deren rechten Seite skalar multipliziert, um... + +You're in trouble here. 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Undefined control sequence. +l.49 \subsection + {Beispiel} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.49 \subsection{B + eispiel} +You're in trouble here. 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Undefined control sequence. +l.50 ...ieses Problem wurde bereits die DGL \eqref + {eq:PursuerDGL} hergeleitet. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.53 \begin{equation} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 53--53 +[] + [] + + +Overfull \hbox (135.67946pt too wide) detected at line 57 +[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )[][] = [][][] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.59 \begin{equation} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 59--59 +[] + [] + + +Overfull \hbox (61.57726pt too wide) detected at line 65 +[] \OMS/cmsy/m/n/10 [] \/cmr/m/n/10 = 1 = \OML/cmm/m/it/10 v[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.79 S + ed ut perspiciatis unde omnis iste natus error sit voluptatem +You're in trouble here. 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+Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.98 U + t enim ad minima veniam, quis nostrum exercitationem ullam corporis +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! 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Undefined control sequence. +l.110 animi, id est laborum et dolorum fuga \eqref + {000tempmlate:equation1}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.112 E + t harum quidem rerum facilis est et expedita distinctio +You're in trouble here. 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Emergency stop. +<*> ./teil1.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 35 strings out of 478927 + 720 string characters out of 2852535 + 294175 words of memory out of 3000000 + 17993 multiletter control sequences out of 15000+600000 + 403738 words of font info for 28 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 23i,12n,32p,307b,98s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/teil1.tex b/buch/papers/lambertw/teil1.tex index 7b545c3..cc4a62a 100644 --- a/buch/papers/lambertw/teil1.tex +++ b/buch/papers/lambertw/teil1.tex @@ -3,9 +3,116 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 +\section{Ziel \label{lambertw:section:teil1}} \rhead{Problemstellung} + + + +%\begin{figure}[H] +% \centering +% \includegraphics[width=0.5\textwidth]{.\Bilder\something.pdf} +% \label{pursuer:grafik1} +%\end{figure} + + + +Je nach Verfolgungsstrategie die der Verfolger verwendet, entsteht eine andere DGL. +Für dieses konkrete Beispiel wird einfachheitshalber die simpelste Strategie gewählt. +Bei dieser Strategie bewegt sich der Verfolger immer direkt auf sein Ziel hinzu. +Womit der Geschwindigkeitsvektor des Verfolgers zu jeder Zeit direkt auf das Ziel zeigt. + +Um die DGL dieses Problems herzuleiten wird der Sachverhalt in der Grafik \eqref{pursuer:grafik1} aufgezeigt. +Der Punkt $P$ ist der Verfolger und der Punkt $A$ ist sein Ziel. + +Um dies mathematisch beschreiben zu können, wird der Richtungsvektor +\begin{equation} + \frac{A-P}{|A-P|} + = + \frac{\dot{P}}{|\dot{P}|} +\end{equation} +benötigt. Durch die Subtraktion der Ortsvektoren $\overrightarrow{OP}$ und $\overrightarrow{OA}$ entsteht ein Vektor der vom Punkt $P$ auf $A$ zeigt. +Da die Länge dieses Vektors beliebig sein kann, wird durch Division mit dem Betrag, die Länge auf eins festgelegt. +Aus dem Verfolgungsproblem ist auch ersichtlich, dass die Punkte $A$ und $P$ nicht am gleichen Ort starten und so eine Division durch Null ausgeschlossen ist. +Wenn die Punkte $A$ und $P$ trotzdem am gleichen Ort starten, ist die Lösung trivial. + +Nun wird die Gleichung mit deren rechten Seite skalar multipliziert, um das Gleichungssystem von zwei auf eine Gleichung zu reduzieren. +\begin{equation} + \label{pursuer:pursuerDGL} + \frac{A-P}{|A-P|}\cdot \frac{\dot{P}}{|\dot{P}|} + = + 1 +\end{equation} +Diese DGL ist der Kern des Verfolgungsproblems, insofern sich der Verfolger immer direkt auf sein Ziel zubewegt. + + +\subsection{Beispiel} +Das Verfolgungsproblem wird mithilfe eines konkreten Beispiels veranschaulicht. Dafür wird die einfachste Strategie verwendet, bei der sich der Verfolger direkt auf sein Ziel hinzu bewegt. Für dieses Problem wurde bereits die DGL \eqref{pursuer:pursuerDGL} hergeleitet. + +Um dieses Beispiel einfach zu halten, wird für den Verfolger und das Ziel jeweils eine konstante Geschwindigkeit von eins gewählt. Das Ziel wiederum startet im Ursprung und bewegt sich linear auf der positiven Y-Achse. + +\begin{align} + v_P^2 + &= + \dot{P}\cdot\dot{P} + = + 1 + \\[5pt] + v_A + &= + 1 + \\[5pt] + A + &= + \begin{pmatrix} + 0 \\ + v_A\cdot t + \end{pmatrix} + = + \begin{pmatrix} + 0 \\ + t + \end{pmatrix} + \\[5pt] + P + &= + \begin{pmatrix} + x \\ + y + \end{pmatrix} +\end{align} + +Die Anfangsbedingungen dieses Problems sind. + +\begin{align} + y(t)\bigg|_{t=0} + &= + y_0 + \\[5pt] + x(t)\bigg|_{t=0} + &= + x_0 \\[5pt] + \frac{\,dy}{\,dx}(t)\bigg|_{t=0} + &= + \frac{y_A(t) -y_P(t)}{x_A(t)-x_P(t)}\bigg|_{t=0} +\end{align} + +Mit den vorangegangenen Definitionen kann nun die DGL \eqref{pursuer:pursuerDGL} gelöst werden. +Dafür wird als erstes das Skalarprodukt ausgerechnet. + +\begin{equation} + \dfrac{-x\cdot\dot{x}+(t-y)\cdot\dot{y}}{\sqrt{x^2+(t-y)^2}} = 1 +\end{equation} + + + + + + + + + + Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae diff --git a/buch/papers/lambertw/teil2.log b/buch/papers/lambertw/teil2.log new file mode 100644 index 0000000..00ab7e3 --- /dev/null +++ b/buch/papers/lambertw/teil2.log @@ -0,0 +1,1580 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (MiKTeX 21.8) (preloaded format=pdflatex 2021.9.21) 18 JUL 2022 16:36 +entering extended mode +**./teil2.tex +(teil2.tex +LaTeX2e <2021-06-01> patch level 1 +L3 programming layer <2021-08-27> +! 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+Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! 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Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.24 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? -\subsection{De finibus bonorum et malorum -\label{lambertw:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\subsection{Strategie 1 +\label{lambertw:subsection:Strategie1}} + + +\subsection{Strategie 2 +\label{lambertw:subsection:Strategie2}} diff --git a/buch/papers/lambertw/teil3.log b/buch/papers/lambertw/teil3.log new file mode 100644 index 0000000..018dcba --- /dev/null +++ b/buch/papers/lambertw/teil3.log @@ -0,0 +1,1580 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (MiKTeX 21.8) (preloaded format=pdflatex 2021.9.21) 5 APR 2022 21:53 +entering extended mode +**./teil3.tex +(teil3.tex +LaTeX2e <2021-06-01> patch level 1 +L3 programming layer <2021-08-27> +! Undefined control sequence. +l.6 \section + {Teil 3 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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+Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! 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Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.24 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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+Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! 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+Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! 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+Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! 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Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.24 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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+Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! 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Emergency stop. +<*> teil4.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 16 strings out of 478371 + 382 string characters out of 5852527 + 296836 words of memory out of 5000000 + 18226 multiletter control sequences out of 15000+600000 + 403430 words of font info for 27 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 13i,0n,12p,83b,18s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/teil4.tex b/buch/papers/lambertw/teil4.tex new file mode 100644 index 0000000..6184369 --- /dev/null +++ b/buch/papers/lambertw/teil4.tex @@ -0,0 +1,252 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Beispiel Verfolgungskurve +\label{lambertw:section:teil4}} +\rhead{Beispiel Verfolgungskurve} +In diesem Abschnitt wird rechnerisch das Beispiel einer Verfolgungskurve mit der Verfolgungsstrategie 1 beschreiben. + +Das zu verfolgende Ziel \(\overrightarrow{Z}\) wandert auf einer Gerade mit konstanter Geschwindigkeit \(v = 1\), wobei diese Gerade der \(y\)-Achse entspricht. Der Verfolger \(\overrightarrow{V}\) startet auf einem beliebigen Punkt im ersten Quadrant und bewegt sich auch mit konstanter Geschwindigkeit. Diese Anfangspunkte oder Anfangsbedingungen können wie folgt formuliert werden: +\begin{equation} + \overrightarrow{Z} + = + \left( \begin{array}{c} 0 \\ v \cdot t \end{array} \right) + = + \left( \begin{array}{c} 0 \\ t \end{array} \right) + ; + \overrightarrow{V} + = + \left( \begin{array}{c} x \\ y \end{array} \right) + \label{lambertw:Anfangspunkte} +\end{equation} +Wenn man diese Startpunkte in die Gleichung der Verfolgungskurve \eqref{lambertw:pursuerDGL} einfügt ergibt sich folgender Ausdruck: +\begin{equation} + \frac{\left( \begin{array}{c} 0-x \\ t-y \end{array} \right)}{\sqrt{x^2 + (t-y)^2}} + \circ + \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) + = + 1 + \label{lambertw:eqMitAnfangspunkte} +\end{equation} +Macht man den linken Term Bruchfrei und löst das Skalarprodukt auf, dann ergibt sich folgende DGL: +\[ + \left( \begin{array}{c} 0-x \\ t-y \end{array} \right) + \circ + \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) + = \sqrt{x^2 + (t-y)^2}\\ +\] +\begin{equation} + -x \cdot \dot{x} + (t-y) \cdot \dot{y} + = \sqrt{x^2 + (t-y)^2} + \label{lambertw:eq1BspVerfolgKurve} +\end{equation} +Im nächsten Schritt quadriert man beide Seiten, erweitert den neu entstandenen quadratischen Term, bringt alles auf die linke Seite und klammert gemeinsames aus. +\begin{align*} + ((t-y) \dot{y} - x \dot{x})^2 + &= x^2 + (t-y)^2 \\ + x^2 \dot{x}^2 - 2x(t-y) \dot{x} \dot{y} + (t-y)^2 \dot{y} + &= x^2 + (t-y)^2 \\ + \dot{x}^2 x^2 - x^2 - 2x(t-y) \dot{x} \dot{y} + \dot{y}^2 (t-y)^2 - (t-y)^2 + &= 0 \\ + (\dot{x}^2 - 1) \cdot x^2 - 2x(t-y) \dot{x} \dot{y} + (\dot{y}^2 - 1) \cdot (t-y)^2 + &= 0 +\end{align*} +Der letzte Ausdruck kann mittels folgender Beziehung \(\dot{x}^2 + \dot{y}^2 = 1\) vereinfacht werden, anschliessend wird die Gleichung mit \(-1\) multipliziert: +\[ + \underbrace{(\dot{x}^2 - 1)}_{\mathclap{-\dot{y}^2}} \cdot x^2 - 2x(t-y) \dot{x} \dot{y} + \underbrace{(\dot{y}^2 - 1)}_{\mathclap{-\dot{x}^2}} \cdot (t-y)^2 + = 0 +\] +\begin{align*} + - \dot{y}^2 \cdot x^2 - 2x(t-y) \dot{x} \dot{y} - \dot{x}^2 \cdot (t-y)^2 + &= 0 \\ + \dot{y}^2 \cdot x^2 + 2x(t-y) \dot{x} \dot{y} + \dot{x}^2 \cdot (t-y)^2 + &= 0 +\end{align*} +Im letzten Ausdruck erkennt man das Muster einer binomischen Formel, was den Ausdruck wesentlich vereinfacht: +\begin{align*} + x^2 \dot{y}^2 + 2 \cdot x \dot{y} \cdot (t-y) \dot{x} + (t-y)^2 \dot{x}^2 + &= 0 \\ + (x \dot{y} + (t-y) \dot{x})^2 + &= 0 +\end{align*} +Wenn man nun beidseitig die Quadratwurzel zieht, dann ergibt sich im Vergleich zu \eqref{lambertw:eq1BspVerfolgKurve} eine wesentlich einfachere DGL: +\begin{equation} + x \dot{y} + (t-y) \dot{x} + = 0 + \label{lambertw:equation5} +\end{equation} +Um die Ableitung nach der Zeit wegzubringen, wird beidseitig mit \(\dot{x}\) dividiert, wobei \(\frac{\dot{y}}{\dot{x}} = \frac{dy}{dt}/\frac{dx}{dt} = \frac{dy}{dx}\) entspricht. +\[ + x \frac{\dot{y}}{\dot{x}} + (t-y) \frac{\dot{x}}{\dot{x}} + = 0 +\] +Nach dem Kürzen und Vereinfachen ergibt sich folgende DGL: +\begin{equation} + x y^{\prime} + t - y + = 0 + \label{lambertw:DGLmitT} +\end{equation} +Hier wäre es passend wenn man die Abhängigkeit nach \(t\) komplett wegbringen könnte. Um dies zu erreichen muss man auf die Definition der Bogenlänge aus Analysis 2 zurückgreifen: +\begin{equation} + s + = + v \cdot t + = + t + = + \int_{x_0}^{x_{end}}\sqrt{1+y^{\prime\, 2}} \: dx + \label{lambertw:eqZuBogenlaenge} +\end{equation} +Nicht gerade auffällig ist die Richtung in welche hier integriert wird. Wenn der Verfolger sich wie vorgesehen am Anfang im ersten Quadranten befindet, dann muss sich dieser nach links bewegen, was nicht der üblichen Integrationsrichtung entspricht. Um eine Integration wie üblich von links nach rechts ausführen zu können, müssen die Integrationsgenerzen vertauscht werden, was in einem Vorzeichenwechsel resultiert. Wenn man nun \eqref{lambertw:eqZuBogenlaenge} in die DGL \eqref{lambertw:DGLmitT} einfügt, dann ergibt sich folgender Ausdruck: +\begin{equation} + x y^{\prime} - \int\sqrt{1+y^{\prime\, 2}} \: dx - y + = 0 + \label{lambertw:DGLohneT} +\end{equation} +Um das Integral los zu werden, leitet man den vorherigen Ausdruck \eqref{lambertw:DGLohneT} nach \(x\) ab: +\begin{align*} + y^{\prime}+ xy^{\prime\prime} - \sqrt{1+y^{\prime\, 2}} - y^{\prime} + &= 0 \\ + xy^{\prime\prime} - \sqrt{1+y^{\prime\, 2}} + &= 0 +\end{align*} +Mittels der Substitution \(y^{\prime} = u\) kann vorherige DGL in eine erster Ordnung umgewandelt werden: +\begin{equation*} + xu^{\prime} - \sqrt{1+u^2} + = 0 + \label{lambertw:DGLmitU} +\end{equation*} +Welche mittels Separation gelöst werden kann: +\begin{align*} + arsinh(u) + C_L + &= + ln(x) + C_R \\ + arsinh(u) + &= + ln(x) + C \\ + u + &= + sinh(ln(x) + C) +\end{align*} +In dem man die Substitution rückgängig macht, erhält man eine weitere DGL erster Ordnung die bereits separiert ist: +\begin{equation} + y^{\prime} + = + sinh(ln(x) + C) +\end{equation} +Diese kann mit den selben Methoden gelöst werden, diesmal in Kombination mit der exponentiellen Definition der \(sinh\)-Funktion: +\begin{align*} + y + &= + \int sinh(ln(x) + C) \\ + &= + \int \frac{1}{2} (e^{ln(x)+C} - e^{-(ln(x)+C)}) \\ + &= + \frac{e^C}{4} x^2 - \frac{ln(x)}{2 \cdot e^C} + C_1 \\ + &= + C_1 + C_2 x^2 - \frac{ln(x)}{8 \cdot C_2} +\end{align*} + +\begin{figure} + \centering + \includegraphics{papers/lambertw/Bilder/VerfolgungskurveBsp.png} + \caption[Graph der Verfolgungskurve]{Graph der Verfolgungskurve wobei, ({\color{red}rot}) die Funktion \ensuremath{y(x)} ist, ({\color{darkgreen}grün}) der quadratische Teil und ({\color{blue}blau}) dem \ensuremath{ln(x)}-Teil entspricht. + \label{lambertw:BildFunkLoes} + } +\end{figure} + +Das Resultat, wie ersichtlich, ist folgende Funktion \eqref{lambertw:funkLoes} welche mittels Anfangsbedingungen parametrisiert werden kann: +\begin{equation} + {\color{red}{y(x)}} + = + C_1 + C_2 {\color{darkgreen}{x^2}} {\color{blue}{-}} \frac{\color{blue}{ln(x)}}{8 \cdot C_2} + \label{lambertw:funkLoes} +\end{equation} +Für die Koeffizienten \(C_1\) und \(C_2\) ergibt sich ein Anfangswertproblem, welches für deren Bestimmung gelöst werden muss. Zuerst soll aber eine qualitative Intuition, oder Idee für das Aussehen der Funktion \(\bf{y(x)}\) geschaffen werden: +\begin{itemize} + \item + Für grosse \(x\)-Werte welche in der Regel in der Nähe von \(x_0\) sein sollten, ist der quadratisch Term in der Funktion dominant und somit für immer kleiner werdende \(x\) geht der Verfolger in Richtung \(y\)-Achse wobei seine Steigung stetig sinkt, was Sinn macht wenn der Verfolgte entlang der \(y\)-Achse steigt. + \item + Für \(x\)-Werte in der Nähe von \(0\) ist das asymptotische Verhalten des Logarithmus dominant, dies macht auch Sinn da sich der Verfolgte auf der \(y\)-Achse bewegt und der Verfolger im nachgeht. + \item + Aufgrund des Monotoniewechsels in der Kurve muss es auch ein Minimum aufweisen. Es stellt sich nun die Frage: Wo befindet sich dieser Punkt? Durch eine logische Überlegung kann eine Abschätzung darüber getroffen werden und zwar, dass dieser dann entsteht, wenn \(A\) und \(P\) die gleiche \(y\)-Koordinaten besitzen. In diesem Moment ändert die Richtung der \(y\)-Komponente der Geschwindigkeit und somit auch sein Vorzeichen. +\end{itemize} +Alle diese Eigenschafte stimmen mit dem überein, was man von einer Kurve dieser Art erwarten würde, siehe \ref{lambertw:BildFunkLoes}. Nun stellt sich die Frage wie die Kurve wirklich aussieht, dies wird durch das Einsetzen folgender Anfangsbedingungen erreicht: +\begin{equation} + y(x)\big \vert_{t=0} + = + y(x_0) + = + y_0 + \:;\: + \frac{dy}{dx}\bigg \vert_{t=0} + = + y^{\prime}(x_0) + = + \frac{y_0}{x_0} +\end{equation} +Leitet man die Funktion \eqref{lambertw:funkLoes} nach x ab und setzt die Anfangsbedingungen ein, dann ergibt sich folgendes Gleichungssystem: +\begin{subequations} + \begin{align} + y_0 + &= + C_1 + C_2 x^2_0 - \frac{ln(x_0)}{8 \cdot C_2} \\ + \frac{y_0}{x_0} + &= + 2 \cdot C_2 x_0 - \frac{ln(x_0)}{8 \cdot C_2} + \end{align} +\end{subequations} +... Mit folgenden Formeln geht es weiter: +\begin{align*} + \eta + &= + \left(\frac{x}{x_0}\right)^2 + \:;\: + r_0 + = + \sqrt{x_0^2+y_0^2} \\ + y + &= + \frac{1}{4}\left(\left(y_0+r_0\right)\eta+\left(r_0-y_0\right)ln\left(\eta\right)-r_0+3y_0\right) \\ + y^\prime + &= + \frac{1}{2}\left(\left(y_0+r_0\right)\frac{x}{x_0^2}+\left(r_0-y_0\right)\frac{1}{x}\right) \\ + -4t + &= + \left(y_0+r_0\right)\left(\eta-1\right)+\left(r_0-y_0\right)ln\left(\eta\right) \\ + -4t+\left(y_0+r_0\right) + &= + \left(y_0+r_0\right)\eta+\left(r_0-y_0\right)ln\left(\eta\right) \\ + e^{-4t+\left(y_0+r_0\right)} + &= + e^{\left(y_0+r_0\right)\eta}\cdot\eta^{\left(r_0-y_0\right)} \\ + e^{\frac{-4t}{r_0-y_0}+\frac{y_0+r_0}{r_0-y_0}} + &= + e^{\frac{y_0+r_0}{r_0-y_0}\eta}\cdot\eta\ \\ + \chi + &= + \frac{y_0+r_0}{r_0-y_0}; \cdot\chi \\ + \chi\cdot e^{\chi-\frac{4t}{r_0-y_0}} + &= + \chi\eta\cdot e^{\chi\eta} \\ + W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right) + &= + \chi\eta \\ + \frac{W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right)}{\chi} + &= + \eta \\ + \frac{W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right)}{\chi} + &= + \left(\frac{x}{x_0}\right)^2 \\ + x\left(t\right) + &= + \sqrt{\frac{W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right)}{\chi}} +\end{align*} +\begin{equation} + y(t) + = + \frac{1}{4}\left(\left(y_0+r_0\right)\frac{W\left(\chi\cdot e^{\chi\ -\ \frac{4t}{r_0-y_0}}\right)}{\chi}+\left(r_0-y_0\right)\cdot\mathrm{ln}\ \left(\frac{W\left(\chi\cdot e^{\chi\ -\ \frac{4t}{r_0-y_0}}\right)}{\chi}\right)-r_0+3y_0\right) + \label{lambertw:funkNachT} +\end{equation} |
