From 2af4ef67e2d89b09cf7dca1bd07307aa3565888f Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 13 Nov 2019 21:34:24 +0100 Subject: [PATCH 1/3] add ipi3 --- ws2019/ipi/uebungen/binomial.cpp | 18 +++ ws2019/ipi/uebungen/binomial_fast.cpp | 25 ++++ ws2019/ipi/uebungen/ipi3.pdf | Bin 0 -> 113204 bytes ws2019/ipi/uebungen/ipi3.tex | 177 ++++++++++++++++++++++++++ ws2019/ipi/uebungen/uebung3.cpp | 14 ++ 5 files changed, 234 insertions(+) create mode 100644 ws2019/ipi/uebungen/binomial.cpp create mode 100644 ws2019/ipi/uebungen/binomial_fast.cpp create mode 100644 ws2019/ipi/uebungen/ipi3.pdf create mode 100644 ws2019/ipi/uebungen/ipi3.tex create mode 100644 ws2019/ipi/uebungen/uebung3.cpp diff --git a/ws2019/ipi/uebungen/binomial.cpp b/ws2019/ipi/uebungen/binomial.cpp new file mode 100644 index 0000000..337dcda --- /dev/null +++ b/ws2019/ipi/uebungen/binomial.cpp @@ -0,0 +1,18 @@ +#include "cpp_headers/fcpp.hh" + +// Berechnet den Binomialkoeffizienten (n ueber k) rekursiv +int binomial(int n, int k) { + return cond(k > n, + // Falls k > n: n ueber k = 0 + 0, + cond(k == 0 || n == k, + // Falls k = 0 oder n = k: n ueber k = 1 + 1, + // sonst rekursiv berechnen + binomial(n-1, k-1) + binomial(n-1, k))); +} + +int main(int argc, char **argv) { + return print(binomial(readarg_int(argc, argv, 1), + readarg_int(argc, argv, 2))); +} diff --git a/ws2019/ipi/uebungen/binomial_fast.cpp b/ws2019/ipi/uebungen/binomial_fast.cpp new file mode 100644 index 0000000..08a5ee5 --- /dev/null +++ b/ws2019/ipi/uebungen/binomial_fast.cpp @@ -0,0 +1,25 @@ +#include "cpp_headers/fcpp.hh" + +// Helper Funktion fuer Fakultaet +// berechnet Fakultaet linear iterativ +int fakIter(int produkt, int zaehler, int ende) { + return cond(zaehler>ende, + produkt, + fakIter(produkt*zaehler,zaehler+1,ende)); +} + +// Berechnet die Fakulaet einer Zahl +int fakultaet(int n) { + return fakIter(1,1,n); +} + +// Berechnet den Binomial Koeffizienten (n ueber k) durch seine explizite +// Darstellung +int binomial_fast(int n, int k) { + return fakultaet(n)/(fakultaet(k) * fakultaet(n-k)); +} + +int main(int argc, char **argv) { + return print(binomial_fast(readarg_int(argc, argv, 1), + readarg_int(argc, argv, 2))); +} diff --git a/ws2019/ipi/uebungen/ipi3.pdf b/ws2019/ipi/uebungen/ipi3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..29202baac2840a9991b1e8d4bf04c0b1ad79402a GIT binary patch literal 113204 zcmbrlWmFv6)~1U~VTHQ~3s!}@ySuvw5AIF^1b24`9!NrPcXti$5FofqvyXgzZ*TgZ zoN?<1qsHLRT5nBxrYtHYF$qR66FY!ve7a{Iz)A)pb1<<6@bLkd<;?9ZU98AhI9RyI z{_z1YOT4pnF?S+kmasK;F&8s8bucpr2nYb2U7XB~?EoH&N4mNWn;bZPV_(J#g2TD_ znpr>5jYILmE;6~*Vde^1-WkzEn@+!9oL@k_`*p1pZ?2M0JD@?$+TWB1g!Fa$D$-&o(qc9H(GTvX5Z^2aeSk%ju|~~o?%MLVf8GL$--W&eVjcB8rJ+X- zqCMi=jZ;<&@4dTt>C?l`!QrJYHA~^=)WIrySyUa_&iei;`>)ptUvlhIzTMs?ej9*# zCJ=yhdK#K>$4;7a5cYagNF_q>P*}Z>%HlVm=Qe7j;onTywuOnqp!bK(z#z_Rb!ycbsV|=Vy3Jx8w`kCL@)~7> zf}t+Yvb_5^S!^#FYw^JWSUPN>3h_O9QPj36(=Pd^YeN*Qn2$b`HY26nXS2&n4WrB7 zn%RWX;`NDXJ6LUbs(#&tWWyzcap`RBKxjU3XiZ*qb7_j}1|i}egxx(C4; zKJOib>8E_h?*>my5J3vogm0-MT72tnF5fw;-l)q%p@%_uP2w; zQx-E40rr^bhbZ84DJ8BWpyvbC-(f6+J_8DB_y;h&QwvPg`a6NYGN`3c)&+)5I{
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13,49s & 29,13s & 244,64s\\ \hline + $(2n)^{3}$ & 32s & 80s & 800s \\ \hline + $2^{2n}$ & 16s & 100s & 10000s\\ \hline + \end{tabular} + + \item + \begin{enumerate}[1.] + \item $1$ + \item $\log \log n$ + \item $\log n$ + \item $n^{\epsilon}$ + \item $n^{c}$ + \item $n^{\log n}$ + \item $c^{n}$ + \item $n^{n}$ + \item $c^{\left( c^{n} \right) }$ + \end{enumerate} + + \item Beweisen Sie folgende Behauptungen + \begin{enumerate} + \item $x^{a} = O(x^{b}) \iff a -b \le 0$ + \begin{proof} + Seien $a, b, x \in \R$ mit $x > 1 $ + + Zu zeigen: $\exists c \in \R$: $x^{a} \le c x^{b} \iff a \le b$ + + Wegen $x > 1$ ist $\ln(x) > 0$ und $\ln(x)$ streng monoton steigend, wähle $c \ge 1$, dann + folgt: + \begin{align*} + &a \le b \\ + \iff &\ln(x) \cdot a \le \ln(x) \cdot b \le \ln(c) + \ln(x) \cdot b \\ + \iff &\ln(x^{a}) \le \ln(c\cdot b^{x})\\ + \iff & x^{a} \le c x^{b} + .\end{align*} + \end{proof} + \item $\log_a(x) = \Theta(\log_b(x))$ $\forall a, b \in \R^{+}$ + \begin{proof} + Seien $a, b, x \in \R^{+}$. + + Zu zeigen: $\exists c \in \R$: $\log_a(x) = c \cdot \log_b(x)$. + \begin{align*} + &\log_a(x) = \log_a(x) \\ + \implies & \log_a(x) = \log_a(b) \cdot \frac{\log_a(x)}{\log_a(b)} \\ + \implies & \log_a(x) = \log_a(b) \cdot \log_b(x) + .\end{align*} + Mit $c := \log_a(b)$ folgt damit: + \[ + \log_a(x) = c \cdot \log_b(x) + .\] + \end{proof} + \item $a^{x} = O(b^{x}) \iff 0 \le a \le b$ + \begin{proof} + Seien $a, b, x \in \R$ mit $x \ge 0$ + + Zu zeigen: $\exists c \in \R$: $a^{x} \le c b^{x} \iff 0 \le a \le b$ + \begin{align*} + &0 \le a \le b \\ + \iff &a^{x} \le b^{x} \le b^{x+1} = b \cdot b^{x} + .\end{align*} + Mit $c := b$ folgt damit: + \[ + a^{x} \le c \cdot b^{x} \iff 0 \le a \le b + .\] + \end{proof} + \end{enumerate} +\end{enumerate} + +\end{aufgabe} + +\begin{aufgabe} Klassischer Euklidischer Algorithmus + + Sei $a, b \in \N_0, a+b > 0$ gegeben. + \[ + \text{ggT}(a, b) = \begin{cases} + a & b =0 \\ + \text{ggT}(b,a) & a < b \\ + \text{ggT}(a - b, b) & a \ge b \\ + \end{cases} + .\] + + \begin{enumerate} + \item Für $a \ge b > 0$ gilt: + \[ + \text{ggT}(a,b) = \text{ggT}(a -b, b) + .\] + \begin{proof} + Wegen $b \neq 0$ und $a \ge b$ folgt nach Definition: + \[ + \text{ggT}(a, b) = \text{ggT}(a -b, b) + .\] + \end{proof} + \item Der klassische Euklidische Algorithmus terminiert. + \begin{proof} + Seien $(a_n)_{n\in\N} \in \N_0$ und $(b_n)_{n\in\N} \in \N_0$ Folgen mit $a_1 = a$ und + $b_1 = b$. + + Sei $n \in \N$ beliebig. + + \begin{itemize} + \item Falls $b_n = 0$ terminiert der Algorithmus direkt. + \item + Falls $a_n < b_n$, folgt nach Definition: + \[ + a_{n+1} = b_n \text{ und } b_{n+1} = a_n < b_n + .\] Damit folgt: + \[ + b_{n+1} < b_n + .\] + \item Falls $a_n \ge b_n$ folgt nach Definition: + \[ + a_{n+1} = a_n - b_n \text{ und } b_{n+1} = b_n + .\] Wegen $a_{n+1} < a_n $ folgt, dass $\exists k \in \N$: $a_{n+k} < b_n$. Dann + tritt wieder der zweite Fall ein, d.h. + \[ + b_{n+k+1} < b_n + .\] + \end{itemize} + + Damit folgt, dass $(b_n)_{n\in\N}$ für fast alle $n \in \N$ streng monoton fällt. + Da $(a_n - b_n) \in \N$, folgt: + \[ + \exists k \in \N\text{: } b_k = 0 + .\] + Damit terminiert der \textit{klassische Euklidische Algorithmus} immer. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Binomialkoeffizient + \begin{enumerate}[a)] + \item Programm siehe \textit{binomial.cc} + + Für $n = 35$ und $k = 18$ benötigt das Programm mehr als 20 Sekunden für die Berechnung. + + Für $n = 34$ und $k = 18$ liefert das Programm $-2091005866$. Das liegt an der + begrenzten Größe des Datentyps \textbf{int}. Bei Überschreitung der maximalen Größe + beginnt der Wert erneut bei dem Minimalwert des Datentyps \textbf{int}. Deshalb können + wir dann negative Ergebnisse erhalten. + \item Der Rechenaufwand für die rekursive Berechnung des Binomialkoeffizienten ist: + \begin{align*} + A_{n, 0} = A_{n, n} = A_{n, k>n} = 1 \\ + A_{n,k} = A_{n-1, k-1} + A_{n-1, k} + .\end{align*} + \item Programm siehe \textit{binomial\_fast.cc} + + Die schnellere Variante hat die Komplexität $O(n)$, da die Komplexität der Fakultätsfunktion + $O(n)$ ist und diese einfach dreimal ausgeführt wird. + + Programm (c) ist deutlich schneller als Programm (a) für größere Zahlen $n$ und $k$. Aufgrund + der Verwendung der Fakultät, wird allerdings schneller die maximale Größe eines \textbf{int}'s + erreicht. Dadurch erhalten wir bereits für recht kleine Werte für $n$ und $k$ falsche Ergebnisse. + \item Ein effizienter Algorithmus berechnet jede Zeile linear iterativ, aus der vorhergehenden + Zeile, damit ist die Komplexität $O(n)$. + + Der Unterschied entsteht daraus, dass bei (a) einzelne Binomialkoeffizienten mehrfach + ausgerechnet werden müssen und bei (d) jeder Binomialkoeffizient genau einmal ausgerechnet wird. + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2019/ipi/uebungen/uebung3.cpp b/ws2019/ipi/uebungen/uebung3.cpp new file mode 100644 index 0000000..d1e6e05 --- /dev/null +++ b/ws2019/ipi/uebungen/uebung3.cpp @@ -0,0 +1,14 @@ +#include "cpp_headers/fcpp.hh" + +int binomial(int n, int k) { + return cond(k > n, + 0, + cond(k == 0 || n == k, + 1, + binomial(n-1, k-1) + binomial(n-1, k))); +} + +int main(int argc, char **argv) { + return print(binomial(readarg_int(argc, argv, 1), + readarg_int(argc, argv, 2))); +} From 144794829b5e6afae44895ee75e62d264bbf31bf Mon 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zho07LDa!{g9nH>Vn;VK=&_rRu+1@;J^)Z}qpyJUs#h7?Hsz3a7wxHV8Sv4nypsEhn zEsWAPDftXB2C$#M%tNX)EBdo-oi@bXTJ??s<+Slf1US`6R2qyNdTb^08p>04nR7X? zIl<^P#Z1oc^2$eZcXdC}OO!EQ%%dZ36Jq@vJ_Rf0W%38huys;UMckph!PU1uLI6xN78L6cxud#k8dFLHpJ@zWN>skv3Eb$Ce(UBbQND+u-Jd zjj5KfK4(5z>Tu(E(UUXEFH0R?)g~>`hacA8*$Q?Iy~9CT=(6{?kzUbYG5}xfaw#NG zDFxdhPRT(0jQ~6$fR%%q5N!<& N7@vZIsf`)mzW@|e*ogoD literal 0 HcmV?d00001 diff --git a/ws2019/ana/lectures/analysis8.tex b/ws2019/ana/lectures/analysis8.tex new file mode 100644 index 0000000..4b6de9e --- /dev/null +++ b/ws2019/ana/lectures/analysis8.tex @@ -0,0 +1,228 @@ +\documentclass{../../../lecture} + +\begin{document} + +\begin{bem}[Organisatorisches] + Nächsten Mittwoch (20. November) findet keine Vorlesung und auch keine Plenarübung statt. Aber Mittwoch (27.11.) Vorlesung + statt Plenarübung im großen Hörsaal Chemie +\end{bem} + +\begin{satz}[Wichtiger Satz] + Der Körper $\R$ ist vollständig, d.h. jede + Cauchy Folge in $\R$ hat einen Limes +\end{satz} + +\begin{proof} + Sei $(a_n)_{n\in\N}$ eine Cauchy Folge reeller Zahlen, d.h. $a_n \in \R$. + \[ + a_n \in \R: \forall n \in \N: \exists \text{ C.F. } (a_{n, m}) + .\] + \[ + a_{n,m} \in \Q \forall n, m \in \N, a_n = \lim_{m \to \infty} a_{n,m} + .\] + + $\forall n \in \N$ wähle Index $k_n \in \N$ mit + \[ + |a_n - a_{n,kn}| < \frac{1}{n} + .\] + + $k_n$ existiert, weil $\lim_{m \to \infty} a_{n,m} = a_n$ und damit $|a_n - a_{n,m}| \to 0$ also $\exists \epsilon |a_n - a_{n.m}| < \epsilon < \frac{1}{n}$ + und Archimedisches Axiom. + + Ziel: zu zeigen $\left( a_{n, k_{n}} \right)_{n \in \N}$ rationaler Zahlen ist Cauchy Folge. + + Sei $\epsilon > 0$. Dann $\exists n_{\epsilon} \in \N$ s.d. $\forall n, m \ge n_\epsilon$ : + \begin{align*} + |a_n - a_m| < \frac{1}{3} \epsilon, |a_n - a_{n,k_n}| < \frac{1}{3} \epsilon \\ + (a_n)_{n \in \N} \text{C.F.} \\ + |a_m - a_{m,k_m}| < \frac{1}{3} \epsilon \text{(AA)} + .\end{align*} + + und folglich + \[ + |a_{n, k_n} - a_{m, k_m}| \le |a_{n, k_n} - a_n + a_n - a_m + a_m - a_{m, k_m}| \le |a_{n, k_n} - a_n| + | a_n - a_m| + |a_m - a_{m, k_m}| < \epsilon + .\] + $\implies (a_{n, k_n})_{n \in \N}$ Cauchy Folge + $\implies$ Nach Konstruktion der $\R$ folgt, dass $\exists $ ,,limes' $a \in \R$, s.d. + \[ + \forall \epsilon > 0, \exists n_\epsilon \in \N, \forall n \ge n_\epsilon: |a_{n, k_n} - a| < \epsilon + .\] + Dann gilt für die Folge $(a_n)_{n\in\N}$: + \[ + |a_n - a| \le |a_n - a_{n, k_n}| + |a_{n, k_n} - a| \le \frac{1}{n} + |a_{n, k_n} - a| \to 0 + .\] + $\implies a = \lim_{n \to \infty} a_n$ + + $\Q$ ist dicht in $\R$, d.h. + \[ + \forall a \in \R \text{ gilt } \forall \epsilon > 0 \exists q_\epsilon + \in \Q \text{ s.d. } |a - q_\epsilon| \le \epsilon + .\] + + Nach Konstruktion von $\R$ folgt: + \begin{align*} + \forall \text{C.F.} (a_n)_{n\in\N} a_n \in \Q: \\ + \exists a \in \R: a = \lim_{n \to \infty} a_n + .\end{align*} + $\implies \forall \epsilon > 0$ $\exists n_\epsilon$ $|a_n - a| < \epsilon \forall n \ge n_\epsilon$ +\end{proof} + +\begin{bem}[Archimedisches Axiom] + \begin{align*} + \forall a \in \R: \exists n \in \N: \text{s.d.} n - a > 0 \\ + \implies \forall \epsilon > 0 \exists n \in \N \text{s.d. } n - \frac{1}{\epsilon} > 0 \\ + \implies \frac{1}{n} < \epsilon + .\end{align*} +\end{bem} + +\subsection{Wichtige Aussage} + +$\R$ ist vollständig, da alle C.F. in $\R$ haben einen Grenzwert in $\R$. + +\subsection{Weitere Möglichkeiten, die Vollständigkeit von $\R$ zu charakterisieren} + +\begin{definition}[Maximum, Minimum, obere/untere Schranke, Supremum, Infimum] + Sei $M \subset \R, M \neq \emptyset$. + + Maximum: + \[ + \text{max} M := b \in M: b \ge x, \forall x \in M + .\] + Minimum: + \[ + \text{min} M := a \in M: x \ge a, \forall x \in M + .\] + Obere Schranke: + \[ + b \in \R, \text{ s.d. } b \ge x, \forall x \in M + .\] + Untere Schranke: + \[ + a \in \R, \text{ s.d. } x \ge a, \forall x \in M + .\] + Eine Menge $M$ heißt nach oben (unten) beschränkt, falls eine obere (untere) Schranke von $M$ existiert. + + Supremum: kleinste obere Schranke + + Infimum: größte untere Schranke +\end{definition} + +\begin{bsp} + \begin{itemize} + \item $\N$ ist von unten beschränkt, z.B. mit 0. $\text{min} \N = 1$, von oben unbeschränkt. + \item $M = \{x \in \R \mid x^2 < 2\} $: obere Schranke ist $\sqrt{2} $ und untere Schranke ist + $ - \sqrt{2} $, aber $M$ besitzt kein Maximum bzw. Minimum. + $\sqrt{2}$ ist Supremum und $-\sqrt{2} $ ist Infimum + \end{itemize} +\end{bsp} + +\begin{bem} + Falls $b = \text{sup} M \iff$: + \begin{enumerate} + \item $b$ ist eine obere Schranke von $M$, d.h. $\forall x \in M: x \le b$ und + \item Jede Zahl $c < b$ ist keine obere Schranke von M, d.h. $\forall c \in M, c < b, \exists x \in M$: $c < x$ + (oder $\forall \epsilon > 0 \exists x \in M\colon x > b - \epsilon$) + \end{enumerate} + + Analog für $a = \text{inf}M$ +\end{bem} + +\begin{bsp} + \[ + I = (a, b) := \{x \in \R \mid a < x < b\} + .\] dann gilt $\text{sup} I = b, \text{inf} I = a$. +\end{bsp} + +\begin{bem} + Das Supremum (Infimum) muss nicht zur Menge $M$ gehören, aber falls + $\text{sup}M \in M$, dann $\text{sup}M = \text{max} M$. +\end{bem} + +\begin{satz}[Vollständigkeit in $\R$ Nr. 2] + $\R$ vollständig $\iff$ jede nichtleere beschränkte Teilmenge $M \in \R$ besitzt ein Supremum bzw. Infimum +\end{satz} + +\begin{definition}[Intervalle] + \[ + [a, b] := \{x \in \R \mid a \le x \le b\} \text{ abgeschlossenes Intervall} + .\] + \[ + (a, b) := \{x \in \R \mid a < x < b\} \text{ offenes Intervall} + .\] + \[ + (a, b] := \{x \in \R \mid a < x \le b\} \text{ halboffenes Interval} + .\] + \[ + [a, b) \text{ analog} + .\] +\end{definition} + +\begin{definition}[Intervallschachtelung] + ist eine Folge von abgeschlossenen Intervallen $I_n := [a_n, b_n] := \{ x \in \R \mid a_n \le x \le b_n\}, n \in \N $ + mit Eigenschaften. + 1) $I_{n+1} \subset I_n n \in \N$ (bedeutet $a_n \le a_{n+1} \le b_{n+1} \le b_n$ + 2) $\forall \epsilon > 0, \exists I_{n}$ mit der Länge + \[ + |b_n - a_n| < \epsilon \text{ d.h. } |b_n - a_n| \to 0, n \to \infty + .\] +\end{definition} + +\begin{satz}[Vollständigkeit in $\R$ Nr. 3] + Vollständigkeit in $\R$ $\iff$ Intervallschachtelungseigenschaft d.h. für jede Intervallschachtelung. + \[ + (I_n)_{n\in\N} \in \R, \exists c \in \R + \] so dass + \[ + {c} = \bigcap_{n = 1}^{\infty} I_n := \{x \in \R | x \in I_n \forall n \in \N\} + .\] + Diese Aussage ist verwandt mit dem Axiom vom Dedekindischen Schnitt +\end{satz} + +\begin{satz}[Trennungseigenschaft] + Seien $A, B \subset \R, A \neq \emptyset, B \neq \emptyset$ mit + $a < b \forall a \in A, b \in B$ + + Dann existiert immer ein $c \in \R$, welches A und B trennt: + \[ + \forall a \in A, b \in B \text{ gilt } a \le c \le b + .\] + Dies ist ebenfalls $\iff$ zur Vollständigkeit in $\R$ +\end{satz} + +\begin{lemma}[Existenz der $k$-ten Wurzel einer positiven reellen Zahl] + $\forall a \in \R^{+}$ $\forall k \in \N$: existiert eine positive $k$-te Wurzel. + Das heißt die Lösung der Gleichung + \[ + x^{k} = a + .\] ist $\sqrt[k]{a}$ (Bezeichnung). +\end{lemma} + +\begin{proof} + 1) Die Eindeutigkeit der $\sqrt[k]{a}$ (falls sie existiert) + + Seien $x_1, x_2 \in \R$ zwei $k$-te Wurzeln des $a \in R^{+}$ : + \[ + x_1^{k} = a = x_2^{k} + .\] + Dann gilt: + \[ + 0 = x_1^{k} - x_2^{k} = (x_1 - x_2) \underbrace{\sum_{m=0}^{k-1} x_1^{k-1-m} x_2^{m}}_{> 0} + .\] mit dieser Hilfsformel + \[ + x^{n} - y^{n} = (x - y) \sum_{k=0}^{n-1} x^{n-1-k}y^{k} + .\] $\implies$ $x_1 - x_2 = 0 \implies x_1 = x_2$ + + 2) Existenz: + $a = 1 \implies \sqrt[k]{1} = 1$ ($1^{k} = 1$ ) + Sei $a > 1$ und Annahme, dass $\exists$ Wurzel für $0 < a' < 1$ + Dann definiere: + \[ + \sqrt[k]{a} := \frac{1}{\sqrt[k]{\frac{1}{a}} } + .\] + \[ + \left( \sqrt[k]{a} \right) ^{k} = \left( \frac{1}{\sqrt[k]{a'} } \right)^{k} = \frac{1}{\sqrt[k]{a'}^{k} } = \frac{1}{a'} = a + .\] + Es bleibt zu zeigen: $\exists \sqrt[k]{a} $ für $0 < a < 1$. +\end{proof} + +\end{document}