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-rw-r--r--buch/chapters/070-orthogonalitaet/images/jacobi.pdfbin65748 -> 66426 bytes
-rw-r--r--buch/chapters/070-orthogonalitaet/images/jacobi.tex21
-rw-r--r--buch/chapters/070-orthogonalitaet/jacobi.tex15
3 files changed, 32 insertions, 4 deletions
diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.pdf b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
index 8eb4f09..1504128 100644
--- a/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
Binary files differ
diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.tex b/buch/chapters/070-orthogonalitaet/images/jacobi.tex
index 7d2853f..2a53e75 100644
--- a/buch/chapters/070-orthogonalitaet/images/jacobi.tex
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.tex
@@ -11,7 +11,7 @@
\usepackage{csvsimple}
\usetikzlibrary{arrows,intersections,math}
\begin{document}
-\def\skala{1}
+\def\skala{1.07}
\input{jacobipaths.tex}
\begin{tikzpicture}[>=latex,thick,scale=\skala]
@@ -36,10 +36,14 @@
\draw[color=blue] \legendrek;
\draw[color=red] \legendreweight;
\end{scope}
- \draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+ \draw[->] (-6.1,0) -- (6.3,0) coordinate[label={$x$}];
\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
\node at (6,2.3) [left] {$a=0$};
\node at (-6,2.3) [right] {$b=0$};
+ \draw ({-1*\dx},-0.1) -- ({-1*\dx},0.1);
+ \draw ({1*\dx},-0.1) -- ({1*\dx},0.1);
+ \node at ({1*\dx},-0.1) [below] {$1$};
+ \node at ({-0.97*\dx},-0.1) [below left] {$-1$};
\end{scope}
\begin{scope}[yshift=-5.6cm]
@@ -63,10 +67,15 @@
\draw[color=blue] \aoneo;
\draw[color=red] \aoneweight;
\end{scope}
- \draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+ \draw[->] (-6.1,0) -- (6.3,0) coordinate[label={$x$}];
\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
+ \fill[color=white,opacity=0.8] (4.9,2.1) rectangle (5.9,2.5);
\node at (6,2.3) [left] {$a=1$};
\node at (-6,2.3) [right] {$b=0$};
+ \draw ({-1*\dx},-0.1) -- ({-1*\dx},0.1);
+ \draw ({1*\dx},-0.1) -- ({1*\dx},0.1);
+ \node at ({1*\dx},-0.1) [below] {$1$};
+ \node at ({-0.97*\dx},-0.1) [below left] {$-1$};
\end{scope}
\begin{scope}[yshift=-11.2cm]
@@ -91,10 +100,14 @@
\draw[color=blue] \aminusoneo;
\draw[color=red] \aminusoneweight;
\end{scope}
- \draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+ \draw[->] (-6.1,0) -- (6.3,0) coordinate[label={$x$}];
\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
\node at (6,2.3) [left] {$a=-1$};
\node at (-6,2.3) [right] {$b=0$};
+ \draw ({-1*\dx},-0.1) -- ({-1*\dx},0.1);
+ \draw ({1*\dx},-0.1) -- ({1*\dx},0.1);
+ \node at ({1*\dx},-0.1) [below] {$1$};
+ \node at ({-0.97*\dx},-0.1) [below left] {$-1$};
\end{scope}
diff --git a/buch/chapters/070-orthogonalitaet/jacobi.tex b/buch/chapters/070-orthogonalitaet/jacobi.tex
index 3dcf523..8ccf6a3 100644
--- a/buch/chapters/070-orthogonalitaet/jacobi.tex
+++ b/buch/chapters/070-orthogonalitaet/jacobi.tex
@@ -177,10 +177,25 @@ Pol vom Grad $-\alpha$ an der Stelle $x=1$.
Funktionen müssen daher also ein Nullstelle mindestens vom Grad
$\frac12(1-\alpha)$ haben.
+\begin{figure}
+\centering
+\includegraphics[width=\textwidth]{chapters/070-orthogonalitaet/images/jacobi.pdf}
+\caption{Jacobi-Polynome vom Grad $1$ bis $14$ für verschiedene Werte
+der Parameter $a$ und $b$.
+Je grösser $a$, desto weniger Gewicht bekommen die Funktionswerte am
+rechten Rand und desto grösser werden die Funktionswerte.
+Für negative $a$ müssen die Polynome dagegen eine Nullstelle am
+rechten Rand haben.
+\label{buch:orthogonal:fig:jacobi-parameter}}
+\end{figure}
+
%
%
%
\subsection{Jacobi-Polynome niedrigen Grades}
+In Abbildung~\ref{buch:orthogonal:fig:jacobi-parameter}
+ist die Abhängigkeit der Jacobi-Polynome von den Parametern $a$ und $b$
+illustriert.
%
%