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author | LordMcFungus <mceagle117@gmail.com> | 2021-03-22 18:05:11 +0100 |
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committer | GitHub <noreply@github.com> | 2021-03-22 18:05:11 +0100 |
commit | 76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7 (patch) | |
tree | 11b2d41955ee4bfa0ae5873307c143f6b4d55d26 /vorlesungen/slides/3/fibonacci.tex | |
parent | more chapter structure (diff) | |
parent | add title image (diff) | |
download | SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.tar.gz SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.zip |
Merge pull request #1 from AndreasFMueller/master
update
Diffstat (limited to 'vorlesungen/slides/3/fibonacci.tex')
-rw-r--r-- | vorlesungen/slides/3/fibonacci.tex | 71 |
1 files changed, 71 insertions, 0 deletions
diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex new file mode 100644 index 0000000..3d01020 --- /dev/null +++ b/vorlesungen/slides/3/fibonacci.tex @@ -0,0 +1,71 @@ +% +% fibonacci.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\begin{frame}[t] +\frametitle{Fibonacci} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{block}{Fibonacci-Rekursion} +$x_i$ Fibonacci-Zahlen\uncover<2->{, d.~h.~$x_{n+1\mathstrut}=x_{n\mathstrut}+x_{n-1\mathstrut}$} +\[ +\uncover<3->{ +v_n += +\begin{pmatrix} +x_{n+1}\\ +x_n +\end{pmatrix}} +\uncover<4->{ +\quad\Rightarrow\quad +v_n = +\underbrace{ +\begin{pmatrix} +1&1\\ +1&0 +\end{pmatrix} +}_{\displaystyle=\Phi} +v_{n-1}} +\uncover<5->{ +\quad\Rightarrow\quad +v_n += +\Phi^n +v_0}\uncover<6->{, +\; +v_0 = \begin{pmatrix} 1\\0\end{pmatrix}} +\] +\end{block} +\vspace{-5pt} +\uncover<7->{% +\begin{block}{Rekursionsformel für $\Phi$} +\vspace{-12pt} +\begin{align*} +v_{n}&=v_{n-1}+v_{n-2} +&&\uncover<8->{\Rightarrow& +\Phi^n v_0 &= \Phi^{n-1} v_0 + \Phi^{n-2}v_0} +&&\uncover<9->{\Rightarrow& +\Phi^{n-2}(\Phi^2-\Phi-I)v_0&=0} +\\ +\end{align*} +\vspace{-22pt}% + +\uncover<10->{$\Phi$ ist $\chi_\Phi(X)=m_\Phi(X) = X^2-X-1$, irreduzibel} +\end{block}} + +\uncover<11->{% +\begin{block}{Faktorisierung} +\vspace{-12pt} +\[ +(X-\Phi)(X-(I-\Phi)) +\uncover<12->{= +X^2-X +\Phi(I-\Phi)} +\uncover<13->{= +X^2-X -(\underbrace{\Phi^2-\Phi}_{\displaystyle=I}) +} +\] +\end{block}} + +\end{frame} |