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| -rw-r--r-- | buch/chapters/070-orthogonalitaet/images/jacobi.pdf | bin | 65748 -> 66426 bytes | |||
| -rw-r--r-- | buch/chapters/070-orthogonalitaet/images/jacobi.tex | 21 | ||||
| -rw-r--r-- | buch/chapters/070-orthogonalitaet/jacobi.tex | 15 |
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 Binary files differindex 8eb4f09..1504128 100644 --- a/buch/chapters/070-orthogonalitaet/images/jacobi.pdf +++ b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf 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. % % |