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  1. +56 −0 HA/u03/u03_gruppe_c&j.tex
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56 HA/u03/u03_gruppe_c&j.tex
@@ -59,8 +59,64 @@
\maketitle
\section{Schnelle Matrizenmultiplikation}
+\begin{enumerate}
+\item Da bei der Multiplikation zweier $n \times n$~-Matrizen jedes
+ Element der ersten mit $n$ Elementen der zweiten Matrix
+ multipliziert werden muss, werden insgesamt $n^2 \cdot n = n^3$
+ Multiplikationen sowie $n \cdot n \cdot n-1 = (n-1)n^2$
+ Additionen durchgeführt.
+
+\item
+\end{enumerate}
\section{Rekursionsgleichungen}
+\begin{enumerate}
+\item \begin{align}
+ T(n) = T(9n/10)+n \\
+ a = 1, b = 10, f(n) = n \\
+ \log _b a = log _{10} 1 = 0 \\
+ \Rightarrow f(n) = n = \Theta(n^{\log _b a}) \\
+ \Rightarrow T(n) = \Theta(\log n) \tag{Fall 2 des Master-Theorems}
+ \end{align}
+\item \begin{align}
+ T(n) = T(n-a) + T(a) + n, a \geq 1 \\
+ \end{align}
+ MT nicht anwendbar.
+\item \begin{align}
+ T(n) = T(\sqrt{n}) + 1 \\
+ \end{align}
+\item \begin{align}
+ T(n) = 2T(n/4) + \sqrt{n} \\
+ a = 2, b = 4, f(n) = \sqrt{n} \\
+ \log _b a = \log _4 2 = \frac{1}{2} \\
+ \Rightarrow \exists \varepsilon > 0, f(n) = \sqrt{n} = O(n^{\log
+ _b a+\varepsilon}), \exists c < 1, 2f(n/4) \leq cf(n) \\
+ \Rightarrow T(n) = \Theta(f(n)) \tag{Fall 3 des Master-Theorems}
+ \end{align}
+\item \begin{align}
+ T(n) = 7T(n/3) + n^2 \\
+ a = 7, b = 3, f(n) = n^2 \\
+ \log _b a = \log _3 7 \approx 1.77 \\
+ \Rightarrow \exists \varepsilon > 0, f(n) = n^2 = O(n^{\log _b a-\varepsilon}) \\
+ \Rightarrow T(n) = \Theta(n^{log _3 7}) \tag{Fall 1 des Master-Theorems}
+ \end{align}
+\item \begin{align}
+ T(n) = 2T(n/2) + n \log n \\
+ a = 2, b = 2, f(n) = n \log n \\
+ \log _b a = \log _2 2 = 1
+ \end{align}
+ MT nicht anwendbar, da $\nexists \varepsilon > 0, f(n) = n \log
+ n = \Omega(n^{\log _b a+\varepsilon}), \nexists \varepsilon > 0,
+ f(n) = n \log n = O(n^{\log _b a-\varepsilon}), f(n) \neq
+ \Theta(n^{\log _b a})$
+\item \begin{align}
+ T(n) = T(n-1) + \frac{1}{n} \\
+ a = 1, b = 1, f(n) = \frac{1}{n} \\
+ \log _b a = \log _1 1 = 1 \\
+ \Rightarrow f(n) = \frac{1}{n} = \Theta(n^{\log _b a}) \\
+ \Rightarrow T(n) = \Theta(n \log n) \tag{Fall 2 des Master-Theorems}
+ \end{align}
+\end{enumerate}
\section{Implementierung}

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