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diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index d111f6f..3e5d423 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -22,7 +22,7 @@ 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 +Zur Lösung von \eqref{laguerre:dgl} verwenden wir einen Potenzreihenansatz. Da wir bereits wissen, dass die Lösung orthogonale Polynome sind, erscheint dieser Ansatz sinnvoll. @@ -118,6 +118,17 @@ 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_polynomes.pgf}} +% \includegraphics[width=0.7\textwidth]{% +% papers/laguerre/images/laguerre_polynomes.eps% +% } +\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 @@ -142,16 +153,5 @@ 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..9b901ae 100644 --- a/buch/papers/laguerre/eigenschaften.tex +++ b/buch/papers/laguerre/eigenschaften.tex @@ -3,35 +3,44 @@ % % (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. -} +% \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} +% Die Laguerre-Polynome besitzen einige interessante Eigenschaften +% \rhead{Eigenschaften} -\subsection{Orthogonalität - \label{laguerre:subsection:orthogonal}} +% \subsection{Orthogonalität +% \label{laguerre:subsection:orthogonal}} +\section{Orthogonalität + \label{laguerre:section:orthogonal}} Im Abschnitt~\ref{laguerre:section:definition} haben wir behauptet, dass die Laguerre-Polynome orthogonale Polynome 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, +in dem wir diese Operatoren einander gleichsetzen. Aus der Beziehung \begin{align} S @@ -56,17 +65,19 @@ 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 +117,14 @@ 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} +Damit können wir schlussfolgern, dass die verallgemeinerten Laguerre-Polynome +orthogonal bezüglich des Skalarproduktes auf dem Intervall $(0, \infty)$ +mit der verallgemeinerten Laguerre\--Gewichtsfunktion $w(x)=x^\nu e^{-x}$ sind. +Die Laguerre-Polynome ($\nu=0$) sind somit orthognal im Intervall $(0, \infty)$ +mit der Gewichtsfunktion $w(x)=e^{-x}$. -\subsection{Drei-Terme Rekursion} +% \subsection{Rodrigues-Formel} -\subsection{Beziehung mit der Hypergeometrischen Funktion} +% \subsection{Drei-Terme Rekursion} +% \subsection{Beziehung mit der Hypergeometrischen Funktion} diff --git a/buch/papers/laguerre/gamma.tex b/buch/papers/laguerre/gamma.tex index e3838b0..da2fa93 100644 --- a/buch/papers/laguerre/gamma.tex +++ b/buch/papers/laguerre/gamma.tex @@ -19,52 +19,303 @@ 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 $e^{-t}$ ist genau die Gewichtsfunktion der Laguerre-Integration und +der Definitionsbereich passt ebenfalls genau für dieses Verfahren. +Zu erwähnen ist auch, dass für die verallgemeinerte Laguerre-Integration die +Gewichtsfunktion $t^\nu e^{-t}$ genau 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 $\operatorname{Re} z = 1/2$ 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{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 scheint diese Methode nicht geeignet für unser Vorhaben. -Fehlerterm: +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 auf Grund der vorherigen Punkte, +die zweite Variante weiterzuverfolgen. + +\subsubsection{Naiver Ansatz} + +\begin{figure} +\centering +\input{papers/laguerre/images/rel_error_simple.pgf} +\caption{Relativer Fehler des naiven Ansatzes +für verschiedene reele Werte von $z$ und Grade $n$ der Laguerre-Polynome} +\label{laguerre:fig:rel_error_simple} +\end{figure} + +Bevor wir die Gauss-Laguerre-Quadratur anwenden, +möchten wir als erstes 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} +, +\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. + +Wenden wir nun also naiv die Gauss-Laguerre-Quadratur auf die Gammafunktion 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 $z < 2n$, +was laut der Theorie der Gauss-Quadratur auch zu erwarten ist, +denn die Approximation via Gauss-Quadratur +ist exakt für zu integrierende Polynome mit Grad $< 2n-1$. +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} +\caption{Relativer Fehler des naiven Ansatz 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 $[a, a+1]$, $a \in \mathbb{Z}$, +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/integrands.pgf} +\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/integrands_exp.pgf} +\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 $a$, +welches das Intervall $[a,a+1]$ definiert, +in dem der relative Fehler minimal ist, +grösser als $0$ und abhängig von $n$ ist. \subsubsection{Finden der optimalen Berechnungsstelle} +% 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 $[a, a+1]$, +$a \in \mathbb{Z}$, +evaluiert und dann mit der +Funktionalgleichung \eqref{laguerre:gamma_funktional} zurückverschiebt. +Aus Gründen der Übersichtlichkeit möchten wir das Problem nur für reele Zahlen +formulieren. + +Die optimale Stelle $a \leq z^* \leq a+1$, +mit $z^* \in \mathbb{R}$ wollen wir finden, +in dem wir den Fehlerterm \eqref{laguerre:lag_error} anpassen +und in einem nächsten Schritt minimieren. +Zudem nehmen wir an, +dass $z < z^*$ ist. +Wir fügen einen Verschiebungsterm um $m \in \mathbb{N}$ Stellen ein, +daraus folgt \begin{align*} -R_n +R_{n,m}(\xi) = \frac{(z - 2n)_{2n}}{(z - m)_m} \frac{(n!)^2}{(2n)!} \xi^{z + m - 2n - 1} -. +,\quad +\text{für } +\xi \in (0, \infty) +, +\end{align*} +wobei $z^* = z + m - 1$ ist. +Das Optimierungsproblem daraus lässt sich als +\begin{align*} +\operatorname*{argmin}_m \max_\xi R_{n,m}(\xi) \end{align*} +formulieren. +Allerdings ist die Funktion $R_{n,m}(\xi)$ unbeschränkt. +Dazu müssten wir $\xi$ versuchen unter Kontroller zu bringen, +was ein äussersts schwieriges Unterfangen zu sein scheint. +Da die Gauss-Quadratur aber sowieso nur wirklich Sinn macht für kleine $n$, +können die Intervalle $[a(n), a(n)+1]$ empirisch gesucht werden. + + +\begin{align*} +m^* += +\lceil \alpha n + \beta + \lfloor z \rfloor - z \rceil - \lfloor z \rfloor +\end{align*} + +\begin{figure} +\centering +\includegraphics{papers/laguerre/images/targets.pdf} +\caption{$a$ in Abhängigkeit von $z$ und $n$} +\label{laguerre:fig:targets} +\end{figure} + + +\begin{figure} +\centering +\input{papers/laguerre/images/schaetzung.pgf} +\caption{Schätzung Mittelwert von $m$ und Fehler} +\label{laguerre:fig:schaetzung} +\end{figure} +% 2. Die Fehlerabschätzung ist problematisch, +% weil die Funktion R_n(\xi) unbeschränkt ist. +% Daher kann man nicht einfach nach dem Maximum von R_n(\xi) suchen. +% Man muss zunächst irgendwie das \xi unter Kontrolle bringen. +% Das scheint mir äusserst schwierig zu sein. + +% Ich möchte daher folgendes anregen: +% Im Sinne der Formulierung des Problems, +% wie im Punkt 1 oben könnten Sie für verschiedene n +% nach den optimalen Intervallen [a(n),a(n)+1] suchen, +% und versuchen, einen empirischen Zusammenhang (Faustregel) +% zwischen n und a(n) zu formulieren. +% Das ist etwa gleich gut, +% da ja der Witz der Gauss-Integration ist, +% dass man eben nur sehr kleine n überhaupt in Betracht zieht, +% d.h. man braucht keine exakte Gesetzmässigkeit für a(n). + { \large \color{red} @@ -72,5 +323,3 @@ TODO: Geeignete Minimierung für Fehler finden, so dass sie mit den emprisich bestimmen optimalen Punkten übereinstimmen. } - -\subsection{Resultate} diff --git a/buch/papers/laguerre/images/integrands.pgf b/buch/papers/laguerre/images/integrands.pgf new file mode 100644 index 0000000..c48ff96 --- /dev/null +++ b/buch/papers/laguerre/images/integrands.pgf @@ -0,0 +1,2865 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. 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\usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \usepackage{fontspec} +%% \setmainfont{DejaVuSerif.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setsansfont{DejaVuSans.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setmonofont{DejaVuSansMono.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{3.000000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% 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the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \usepackage{fontspec} +%% \setmainfont{DejaVuSerif.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setsansfont{DejaVuSans.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setmonofont{DejaVuSansMono.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.000000in}{4.000000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% 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packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \usepackage{fontspec} +%% \setmainfont{DejaVuSerif.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setsansfont{DejaVuSans.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setmonofont{DejaVuSansMono.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% 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For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \usepackage{fontspec} +%% \setmainfont{DejaVuSerif.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setsansfont{DejaVuSans.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setmonofont{DejaVuSansMono.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{2.500000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% 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b/buch/papers/laguerre/images/rel_error_simple.png diff --git a/buch/papers/laguerre/images/schaetzung.pgf b/buch/papers/laguerre/images/schaetzung.pgf new file mode 100644 index 0000000..873a10c --- /dev/null +++ b/buch/papers/laguerre/images/schaetzung.pgf @@ -0,0 +1,1160 @@ +%% Creator: Matplotlib, PGF backend +%% +%% To include the figure in your LaTeX document, write +%% \input{<filename>.pgf} +%% +%% Make sure the required packages are loaded in your preamble +%% \usepackage{pgf} +%% +%% Also ensure that all the required font packages are loaded; for instance, +%% the lmodern package is sometimes necessary when using math font. +%% \usepackage{lmodern} +%% +%% Figures using additional raster images can only be included by \input if +%% they are in the same directory as the main LaTeX file. For loading figures +%% from other directories you can use the `import` package +%% \usepackage{import} +%% +%% and then include the figures with +%% \import{<path to file>}{<filename>.pgf} +%% +%% Matplotlib used the following preamble +%% \usepackage{fontspec} +%% \setmainfont{DejaVuSerif.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setsansfont{DejaVuSans.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% \setmonofont{DejaVuSansMono.ttf}[Path=\detokenize{/home/mup/.local/lib/python3.8/site-packages/matplotlib/mpl-data/fonts/ttf/}] +%% +\begingroup% +\makeatletter% +\begin{pgfpicture}% +\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{4.000000in}}% +\pgfusepath{use as bounding box, clip}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetmiterjoin% +\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.000000pt}% 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+\pgfpathlineto{\pgfqpoint{0.556162in}{2.276777in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.556162in}{2.276777in}}{\pgfqpoint{4.402168in}{1.681553in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{0.756261in}{2.276777in}}% +\pgfpathlineto{\pgfqpoint{0.756261in}{3.958330in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{% +\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}% +\pgfusepath{stroke,fill}% +}% +\begin{pgfscope}% +\pgfsys@transformshift{0.756261in}{2.276777in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfpathrectangle{\pgfqpoint{0.556162in}{2.276777in}}{\pgfqpoint{4.402168in}{1.681553in}}% +\pgfusepath{clip}% +\pgfsetrectcap% +\pgfsetroundjoin% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{1.556655in}{2.276777in}}% +\pgfpathlineto{\pgfqpoint{1.556655in}{3.958330in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsetbuttcap% +\pgfsetroundjoin% +\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetfillcolor{currentfill}% +\pgfsetlinewidth{0.803000pt}% +\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% 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+\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{1.556655in}{2.701268in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{1.956852in}{2.840483in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{2.357049in}{2.979182in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{2.757246in}{3.116851in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.157443in}{3.255550in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.557640in}{3.394249in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{3.957837in}{3.531918in}% +\pgfsys@useobject{currentmarker}{}% +\end{pgfscope}% +\begin{pgfscope}% +\pgfsys@transformshift{4.358034in}{3.670617in}% 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0000000..22c2c5a --- /dev/null +++ b/buch/papers/laguerre/images/targets.pdf diff --git a/buch/papers/laguerre/main.tex b/buch/papers/laguerre/main.tex index 00e3b43..9f836ef 100644 --- a/buch/papers/laguerre/main.tex +++ b/buch/papers/laguerre/main.tex @@ -8,7 +8,21 @@ \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 Approximationsmethoden +für das Integral $\int_0^\infty exp(-x)\, dx$ suchte. +Darum möchten wir in diesem Kapitel uns, +ganz im Sinne des Entdeckers, +den Laguerre-Polynomen für Approximationen von Integralen mit +exponentiell-abfallenden Funktionen widmen. +Namentlich werden wir versuchen, +eine geeignete Approximation für die Gamma-Funktion zu finden +mittels Laguerre-Polynomen und der Gauss-Quadratur. + +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/quadratur.tex b/buch/papers/laguerre/quadratur.tex index 60fad7f..b5ad316 100644 --- a/buch/papers/laguerre/quadratur.tex +++ b/buch/papers/laguerre/quadratur.tex @@ -7,7 +7,7 @@ \label{laguerre:section:quadratur}} {\large \color{red} TODO: Einleitung und kurze Beschreibung Gauss-Quadratur} \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 \label{laguerre:gaussquadratur} @@ -21,7 +21,7 @@ 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: +Gleichung~\eqref{laguerre:laguerrequadratur} lässt sich wie folgt umformulieren: \begin{align} \int_{0}^{\infty} f(x) e^{-x} dx \approx @@ -33,7 +33,7 @@ Gleichung~\eqref{laguerre:laguerrequadratur} lässt sich wiefolgt umformulieren: 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. +als Gewichte $A_i$ die Integrale $l_i(x)e^{-x}$ verwendet werden. Dabei sind \begin{align*} l_i(x_j) @@ -57,18 +57,18 @@ A_i \subsubsection{Fehlerterm} 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{abramowitz+stegun} gibt ihn als \begin{align} R_n = \frac{(n!)^2}{(2n)!} f^{(2n)}(\xi) ,\quad 0 < \xi < \infty -\label{lagurre:lag_error} +\label{laguerre:lag_error} \end{align} an. diff --git a/buch/papers/laguerre/references.bib b/buch/papers/laguerre/references.bib index 6956ade..e12e218 100644 --- a/buch/papers/laguerre/references.bib +++ b/buch/papers/laguerre/references.bib @@ -19,4 +19,13 @@ timestamp = {2008-06-25T06:25:58.000+0200}, title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, year = 1972 +} + +@article{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} }
\ No newline at end of file diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb index 44f3abd..a8280aa 100644 --- a/buch/papers/laguerre/scripts/gamma_approx.ipynb +++ b/buch/papers/laguerre/scripts/gamma_approx.ipynb @@ -136,14 +136,17 @@ "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", + " # 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", + " 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" ] @@ -207,9 +210,9 @@ "metadata": {}, "outputs": [], "source": [ - "zeros, weights = np.polynomial.laguerre.laggauss(12)\n", - "targets = np.arange(16, 21)\n", - "mean_targets = ((16, 17),)\n", + "zeros, weights = np.polynomial.laguerre.laggauss(8)\n", + "targets = np.arange(9, 14)\n", + "mean_targets = ((9, 10),)\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", @@ -226,7 +229,7 @@ "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", + " mins.append(np.min(np.abs(rel_error[(0.05 <= x) & (x <= 0.95)])))\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", @@ -365,6 +368,59 @@ "# _ = ax.legend([f\"z={zi}\" for zi in z[0]])\n", "# _ = [ax.axvline(x) for x in zeros]\n" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "bests = []\n", + "N = 200\n", + "step = 1 / (N - 1)\n", + "a = 11 / 8\n", + "b = 1 / 2\n", + "x = np.linspace(step, 1 - step, N + 1)\n", + "ns = np.arange(2, 13)\n", + "for n in ns:\n", + " zeros, weights = np.polynomial.laguerre.laggauss(n)\n", + " est = np.ceil(b + a * n)\n", + " targets = np.arange(max(est - 2, 0), est + 3)\n", + " rel_errors = np.stack([np.abs(evaluate(x, target)) for target in targets], -1)\n", + " best = np.argmin(rel_errors, -1) + targets[0]\n", + " bests.append(best)\n", + "bests = np.stack(bests, 0)\n", + "\n", + "fig, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(5, 3))\n", + "v = ax.imshow(bests, cmap=\"inferno\", aspect=\"auto\")\n", + "plt.colorbar(v, ax=ax, label=r'$m$')\n", + "ticks = np.arange(0, N + 1, 10)\n", + "ax.set_xlim(0, 1)\n", + "ax.set_xticks(ticks, [f\"{v:.2f}\" for v in ticks / N])\n", + "ax.set_xticks(np.arange(N + 1), minor=True)\n", + "ax.set_yticks(np.arange(len(ns)), ns)\n", + "ax.set_xlabel(r\"$z$\")\n", + "ax.set_ylabel(r\"$n$\")\n", + "# for best in bests:\n", + "# print(\", \".join([f\"{int(b):2d}\" for b in best]))\n", + "# print(np.unique(bests, return_counts=True))\n", + "\n", + "targets = np.mean(bests, -1)\n", + "intercept, bias = np.polyfit(ns, targets, 1)\n", + "_, axs2 = plt.subplots(2, sharex=True, clear=True, constrained_layout=True)\n", + "xl = np.array([1, ns[-1] + 1])\n", + "axs2[0].plot(ns, intercept * ns + bias)\n", + "axs2[0].plot(ns, targets, \"x\")\n", + "axs2[1].plot(ns, ((intercept * ns + bias) - targets), \"-x\")\n", + "print(np.mean(bests, -1))\n", + "print(f\"Intercept={intercept:.6g}, Bias={bias:.6g}\")\n", + "\n", + "\n", + "predicts = np.ceil(intercept * ns[:, None] + bias - x)\n", + "print(np.sum(np.abs(bests-predicts)))\n", + "# for best in predicts:\n", + "# print(\", \".join([f\"{int(b):2d}\" for b in best]))\n" + ] } ], "metadata": { diff --git a/buch/papers/laguerre/scripts/gamma_approx.py b/buch/papers/laguerre/scripts/gamma_approx.py new file mode 100644 index 0000000..90843b1 --- /dev/null +++ b/buch/papers/laguerre/scripts/gamma_approx.py @@ -0,0 +1,197 @@ +from pathlib import Path + +import matplotlib as mpl +import matplotlib.pyplot as plt +import numpy as np +import scipy.special + +EPSILON = 1e-7 +root = str(Path(__file__).parent) +img_path = f"{root}/../images" + + +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_shift(z, target): + factor = 1.0 + steps = int(np.floor(target - np.real(z))) + zs = z + steps + if steps > 0: + factor = 1 / pochhammer(z, steps) + elif steps < 0: + factor = pochhammer(zs, -steps) + return zs, factor + + +def laguerre_gamma_shift(z, x=None, w=None, n=8, target=11): + x, w = _prep_zeros_and_weights(x, w, n) + + z += 0j + z_shifted, correction_factor = find_shift(z, target) + 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): + 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): + 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 + else: + f = laguerre_gamma_shift + return np.array([f(zi, x, w, n, **kwargs) for zi in z]) + + +def calc_rel_error(x, y): + return (y - x) / x + + +ns = np.arange(2, 12, 2) + +# Simple / naive +xmin = -5 +xmax = 30 +ylim = np.array([-11, 6]) +x = np.linspace(xmin + EPSILON, xmax - 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 = eval_laguerre_gamma(x, n=n) + rel_err = 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 + EPSILON, 5)) +ax.set_xticks(np.arange(xmin, xmax), minor=True) +ax.set_yticks(10.0 ** np.arange(*ylim, 2)) +ax.set_yticks(10.0 ** np.arange(*ylim, 2)) +ax.set_xlabel(r"$z$") +ax.set_ylabel("Relativer Fehler") +ax.legend(ncol=3, fontsize="small") +ax.grid(1, "both") +fig.savefig(f"{img_path}/rel_error_simple.pgf") + + +# Mirrored +xmin = -15 +xmax = 15 +ylim = np.array([-11, 1]) +x = np.linspace(xmin + EPSILON, xmax - EPSILON, 400) +gamma = scipy.special.gamma(x) +fig2, ax2 = plt.subplots(num=2, clear=True, constrained_layout=True, figsize=(5, 2.5)) +for n in ns: + gamma_lag = eval_laguerre_gamma(x, n=n, func="mirror") + rel_err = calc_rel_error(gamma, gamma_lag) + ax2.semilogy(x, np.abs(rel_err), label=f"$n={n}$") +ax2.set_xlim(x[0], x[-1]) +ax2.set_ylim(*(10.0 ** ylim)) +ax2.set_xticks(np.arange(xmin, xmax + EPSILON, 5)) +ax2.set_xticks(np.arange(xmin, xmax), minor=True) +ax2.set_yticks(10.0 ** np.arange(*ylim, 2)) +# locmin = mpl.ticker.LogLocator(base=10.0,subs=0.1*np.arange(1,10),numticks=100) +# ax2.yaxis.set_minor_locator(locmin) +# ax2.yaxis.set_minor_formatter(mpl.ticker.NullFormatter()) +ax2.set_xlabel(r"$z$") +ax2.set_ylabel("Relativer Fehler") +ax2.legend(ncol=1, loc="upper left", fontsize="small") +ax2.grid(1, "both") +fig2.savefig(f"{img_path}/rel_error_mirror.pgf") + + +# Move to target +bests = [] +N = 200 +step = 1 / (N - 1) +a = 11 / 8 +b = 1 / 2 +x = np.linspace(step, 1 - step, N + 1) +gamma = scipy.special.gamma(x)[:, None] +ns = np.arange(2, 13) +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) + gamma_lag = np.stack( + [ + eval_laguerre_gamma(x, target=target, x=zeros, w=weights, func="shifted") + for target in targets + ], + -1, + ) + rel_error = np.abs(calc_rel_error(gamma, gamma_lag)) + best = np.argmin(rel_error, -1) + targets[0] + bests.append(best) +bests = np.stack(bests, 0) + +fig3, ax3 = plt.subplots(num=3, clear=True, constrained_layout=True, figsize=(5, 3)) +v = ax3.imshow(bests, cmap="inferno", aspect="auto", interpolation="nearest") +plt.colorbar(v, ax=ax3, label=r"$m$") +ticks = np.arange(0, N + 1, N // 5) +ax3.set_xlim(0, 1) +ax3.set_xticks(ticks, [f"{v:.2f}" for v in ticks / N]) +ax3.set_xticks(np.arange(0, N + 1, N // 20), minor=True) +ax3.set_yticks(np.arange(len(ns)), ns) +ax3.set_xlabel(r"$z$") +ax3.set_ylabel(r"$n$") +fig3.savefig(f"{img_path}/targets.pdf") + +targets = np.mean(bests, -1) +intercept, bias = np.polyfit(ns, targets, 1) +fig4, axs4 = plt.subplots( + 2, num=4, sharex=True, clear=True, constrained_layout=True, figsize=(5, 4) +) +xl = np.array([ns[0] - 0.5, ns[-1] + 0.5]) +axs4[0].plot(xl, intercept * xl + bias, label=r"$\hat{m}$") +axs4[0].plot(ns, targets, "x", label=r"$\bar{m}$") +axs4[1].plot( + ns, ((intercept * ns + bias) - targets), "-x", label=r"$\hat{m} - \bar{m}$" +) +axs4[0].set_xlim(*xl) +# axs4[0].set_title("Schätzung von Mittelwert") +# axs4[1].set_title("Fehler") +axs4[-1].set_xlabel(r"$z$") +for ax in axs4: + ax.grid(1) + ax.legend() +fig4.savefig(f"{img_path}/schaetzung.pgf") + +print(f"Intercept={intercept:.6g}, Bias={bias:.6g}") +predicts = np.ceil(intercept * ns[:, None] + bias - x) +print(f"Error: {int(np.sum(np.abs(bests-predicts)))}") + +# plt.show() diff --git a/buch/papers/laguerre/scripts/integrand.py b/buch/papers/laguerre/scripts/integrand.py new file mode 100644 index 0000000..0cf43d1 --- /dev/null +++ b/buch/papers/laguerre/scripts/integrand.py @@ -0,0 +1,49 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Plot for integrand of gamma function with shifting terms.""" + +import os +from pathlib import Path + +import matplotlib.pyplot as plt +import numpy as np + +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=(5, 3)) +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}/integrands.pgf") + +z2 = np.array([-1, -0.5, 0.0, 0.5, 1, 2, 3, 4, 4.5]) +e = np.exp(-t) +r2 = t ** z2 * e + +fig2, ax2 = plt.subplots(num=2, clear=True, constrained_layout=True, figsize=(5, 3)) +ax2.semilogx(t, r2) +# ax2.plot(t,np.exp(-t)) +ax2.set_xlim(10 ** (-2), 20) +ax2.set_ylim(1e-3, 10) +ax2.set_xlabel(r"$x$") +ax2.set_ylabel(r"$x^z e^{-x}$") +ax2.grid(1, "both") +labels =[f"$z={zi: 3.1f}$" for zi in np.squeeze(z2)] +ax2.legend(labels, ncol=2, loc="upper left", fontsize="small") +fig2.savefig(f"{img_path}/integrands_exp.pgf") +# plt.show() diff --git a/buch/papers/laguerre/scripts/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py index b9088d0..1be3552 100644 --- a/buch/papers/laguerre/scripts/laguerre_plot.py +++ b/buch/papers/laguerre/scripts/laguerre_plot.py @@ -29,7 +29,7 @@ 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.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)) @@ -97,4 +97,5 @@ ax.arrow( clip_on=False, ) -fig.savefig(f"{img_path}/laguerre_polynomes.pdf") +fig.savefig(f"{img_path}/laguerre_polynomes.pgf") +# plt.show() |
