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-rw-r--r--buch/papers/laguerre/definition.tex24
-rw-r--r--buch/papers/laguerre/eigenschaften.tex67
-rw-r--r--buch/papers/laguerre/gamma.tex291
-rw-r--r--buch/papers/laguerre/images/integrands.pgf2865
-rw-r--r--buch/papers/laguerre/images/integrands_exp.pgf1968
-rw-r--r--buch/papers/laguerre/images/laguerre_polynomes.pdfbin16239 -> 0 bytes
-rw-r--r--buch/papers/laguerre/images/laguerre_polynomes.pgf1838
-rw-r--r--buch/papers/laguerre/images/rel_error_mirror.pgf3054
-rw-r--r--buch/papers/laguerre/images/rel_error_simple.pgf2940
-rw-r--r--buch/papers/laguerre/images/rel_error_simple.pngbin0 -> 61966 bytes
-rw-r--r--buch/papers/laguerre/images/schaetzung.pgf1160
-rw-r--r--buch/papers/laguerre/images/targets.pdfbin0 -> 12940 bytes
-rw-r--r--buch/papers/laguerre/main.tex16
-rw-r--r--buch/papers/laguerre/quadratur.tex12
-rw-r--r--buch/papers/laguerre/references.bib9
-rw-r--r--buch/papers/laguerre/scripts/gamma_approx.ipynb80
-rw-r--r--buch/papers/laguerre/scripts/gamma_approx.py197
-rw-r--r--buch/papers/laguerre/scripts/integrand.py49
-rw-r--r--buch/papers/laguerre/scripts/laguerre_plot.py5
19 files changed, 14493 insertions, 82 deletions
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. For loading figures
+%% from other directories you can use the `import` package
+%% \usepackage{import}
+%%
+%% and then include the figures with
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+%%
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diff --git a/buch/papers/laguerre/images/integrands_exp.pgf b/buch/papers/laguerre/images/integrands_exp.pgf
new file mode 100644
index 0000000..de5078f
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diff --git a/buch/papers/laguerre/images/laguerre_polynomes.pdf b/buch/papers/laguerre/images/laguerre_polynomes.pdf
deleted file mode 100644
index 3976bc7..0000000
--- a/buch/papers/laguerre/images/laguerre_polynomes.pdf
+++ /dev/null
Binary files differ
diff --git a/buch/papers/laguerre/images/laguerre_polynomes.pgf b/buch/papers/laguerre/images/laguerre_polynomes.pgf
new file mode 100644
index 0000000..8df1baf
--- /dev/null
+++ b/buch/papers/laguerre/images/laguerre_polynomes.pgf
@@ -0,0 +1,1838 @@
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diff --git a/buch/papers/laguerre/images/rel_error_mirror.pgf b/buch/papers/laguerre/images/rel_error_mirror.pgf
new file mode 100644
index 0000000..de1cd53
--- /dev/null
+++ b/buch/papers/laguerre/images/rel_error_mirror.pgf
@@ -0,0 +1,3054 @@
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+%%
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diff --git a/buch/papers/laguerre/images/rel_error_simple.pgf b/buch/papers/laguerre/images/rel_error_simple.pgf
new file mode 100644
index 0000000..9368616
--- /dev/null
+++ b/buch/papers/laguerre/images/rel_error_simple.pgf
@@ -0,0 +1,2940 @@
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diff --git a/buch/papers/laguerre/images/rel_error_simple.png b/buch/papers/laguerre/images/rel_error_simple.png
new file mode 100644
index 0000000..8bcd8e0
--- /dev/null
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Binary files differ
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
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diff --git a/buch/papers/laguerre/images/targets.pdf b/buch/papers/laguerre/images/targets.pdf
new file mode 100644
index 0000000..22c2c5a
--- /dev/null
+++ b/buch/papers/laguerre/images/targets.pdf
Binary files differ
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()