From 5c40908b44c170950343c8feaf142a3270c4d17f Mon Sep 17 00:00:00 2001 From: Peter Gerwinski <peter.gerwinski@hs-bochum.de> Date: Sun, 2 Dec 2018 12:36:40 +0100 Subject: [PATCH] =?UTF-8?q?Skript=20erg=C3=A4nzt,=20Reihenfolge=20angepa?= =?UTF-8?q?=C3=9Ft?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- script/hp-2018ws.pdf | Bin 636456 -> 639081 bytes script/hp-2018ws.tex | 1249 ++++++++++++++-------------- script/io-ports-and-interrupts.pdf | 1 + 3 files changed, 639 insertions(+), 611 deletions(-) create mode 120000 script/io-ports-and-interrupts.pdf diff --git a/script/hp-2018ws.pdf b/script/hp-2018ws.pdf index 41bf0c4421619d5d7c6370638d2a114b89920d01..27a7a4deadea401f65f577789974dc6ca49b934a 100644 GIT binary patch delta 171877 zcmZ4SM(t&T`UHDM)9DjGv8mPfhQ;RJb`z+%zW&BORR`J6cAjjGT)Tg|NFV9w7UB3) zG|BH$=#@Z=pt48t_ai5EExw_~9C5Al*37$eZ|<JFtcPpTG_OltElXl1A8>kbcG875 zmp)M^BcY{gnjg|Vla_w&G-*icT<@*p9i*kM>dCl3WTlY5PKWWK;#`rJJtYf0>$#0) 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\begin{center} @@ -93,7 +91,7 @@ \strut\vfill - Stand: 4.\ November 2018 + Stand: 2.\ Dezember 2018 Soweit nicht anders angegeben:\\ Text und Bilder: Copyright \copyright\ 2012, 2013, 2015, 2016, 2018\quad Peter Gerwinski\\ @@ -1493,6 +1491,13 @@ Diese sind sicherlich bequeme Möglichkeiten, zusätzliche \lstinline{if}s und zusätzliche Wahrheitswert-Variable zu vermeiden, verschleiern aber langfristig den Ablauf der Befehle. + Statt eine Schleife mit \lstinline{break} zu verlassen + oder Teile des Schleifeninneren mit \lstinline{continue} zu überspringen, + ist es besser, die Schleifenbedingung + und \lstinline{if}-Anweisungen innerhalb der Schleife so zu formulieren, + daß Sie kein \lstinline{break} oder \lstinline{continue} mehr benötigen. + Dadurch versteht man auch selbst besser, was das Programm eigentlich tut. + Das Programm wird übersichtlicher und oft sogar kürzer. In besonderem Maße gilt dies für die \lstinline{goto}-Anweisung. Hier ist nicht erkennbar, ob der Sprung nach oben geht (Schleife) oder nach unten (Verzweigung). @@ -1509,8 +1514,7 @@ \begin{hint} Verwenden Sie vorzeitiges \lstinline{return} mit Bedacht. - Verwenden Sie \lstinline{break} und \lstinline{continue} nur dann,\\ - wenn Sie genau wissen, was Sie tun. + Vermeiden Sie die Verwendung von \lstinline{break} und \lstinline{continue}. Verwenden Sie kein \lstinline{goto}. \end{hint} @@ -2999,7 +3003,8 @@ return 0; } \end{lstlisting} - + + \goodbreak \gitfile{hp}{script}{answer.c}: \begin{lstlisting} int answer (void) @@ -3007,7 +3012,8 @@ return 42; } \end{lstlisting} - + + \goodbreak \gitfile{hp}{script}{answer.h}: \begin{lstlisting} extern int answer (void); @@ -3495,475 +3501,46 @@ kann man das Problem systematisch angehen und ist nicht auf Herumraten angewiesen. - \section{Algorithmen} - - \subsection{Differentialgleichungen} - - Eine mathematische Gleichung mit einer gesuchten Zahl $x$, z.\,B. - \begin{displaymath} - x + 2 = -x - \end{displaymath} - läßt sich leicht nach der Unbekannten $x$ auflösen. - (In diesem Fall lautet die Lösung: $x = -1$.) - - Wesentlich schwieriger ist es, - eine mathematische Gleichung mit einer gesuchten Funktion $x(t)$ zu lösen, z.\,B.: - \begin{displaymath} - x'(t) = -x(t)\qquad\mbox{mit}\qquad x(0) = 1 - \end{displaymath} - Um hier auf die Lösung $x(t) = e^{-t}$ zu kommen, - sind bereits weitreichende mathematische Kenntnisse erforderlich. - - Eine derartige Gleichung, die einen Zusammenhang zwischen der gesuchten Funktion - und ihren Ableitungen vorgibt, heißt \newterm{Differentialgleichung}. - Viele physikalisch-technische Probleme werden durch Differentialgleichungen beschrieben. - - \goodbreak - - \subsubsection{Beispiel: Pendelschwingung} - - \begin{minipage}{9.4cm} - \setlength{\parskip}{\smallskipamount} - Das Verhalten eines Fadenpendels (mathematisches Pendel) - wird durch seine Auslenkung $\phi$ als Funktion der Zeit $t$ beschrieben. - Wie kann man $\phi(t)$ berechnen? - - Wie aus anderen Veranstaltungen (Grundlagen der Physik, Mechanik) her bekannt sein sollte, - wirkt auf ein Fadenpendel, das um den Winkel $\phi(t)$ ausgelenkt ist, - die tangentiale Kraft $F = -m \cdot g \cdot \sin\phi(t)$. - Gemäß der Formel $F = m \cdot a$ bewirkt diese Kraft - eine tangentiale Beschleunigung $a = -g \cdot \sin\phi(t)$. - (Das Minuszeichen kommt daher, daß die Kraft der Auslenkung entgegengesetzt wirkt.) - - Wenn das Pendel die Länge $l$ hat, können wir dieselbe tangentiale Beschleunigung - mit Hilfe der zweiten Ableitung des Auslenkungswinkels $\phi(t)$ berechnen: - $a = l \cdot \phi''(t)$ (Winkel in Bogenmaß). - Durch Gleichsetzen erhalten wir eine Gleichung, - die nur noch eine Unbekannte enthält, nämlich die Funktion $\phi(t)$. - - Um $\phi(t)$ zu berechnen, müssen wir also die Differentialgleichung - \begin{displaymath} - \phi''(t) = -\frac{g}{l} \cdot \sin\phi(t) - \end{displaymath} - lösen. - \end{minipage}\hfill - \begin{minipage}{5.5cm} - \begin{center} - \begin{pdfpic} - \psset{unit=1.5cm}% - \begin{pspicture}(0,0)(0,0) - \SpecialCoor - \psline[linewidth=1pt](3;-110)(0,0) - \rput(3.25;-110){\psline[linewidth=0.5pt,linecolor=lightgray](0.34;-20)(1.00;-90)} - \rput(3.25;-110){\psline[linewidth=0.5pt,linecolor=lightgray](0.94;-110)(1.00;-90)} - \rput(3.25;-110){\psline[linewidth=0.5pt,arrows=->](0,0)(0.34;-20)} - \rput(3.25;-110){\psline[linewidth=0.5pt,arrows=->](0,0)(0.94;-110)} - \rput(3.25;-110){\psline[linewidth=0.5pt,arrows=->](0,0)(1.00;-90)} - \rput(3.25;-110){\rput(0.64;-20){\makebox(0,0){$F$}}} - \rput(3.25;-110){\rput(1.30;-90){\makebox(0,0){$m\cdot g$}}} - \pscircle*(3;-110){0.07} - \psline[linewidth=0.5pt,linestyle=dashed](0,0)(0,-1.9) - \psarc[linewidth=0.5pt,arrows=<-](0,0){1.5}{-110}{-90} - \rput(1.3;-100){\makebox(0,0){$\phi$}} - \psarc[linewidth=0.5pt,linestyle=dashed,arrows=<->,arcsepA=0.12](0,0){3}{-110}{-70} - \rput(0,-4.5){\psstrut} - \end{pspicture} - \end{pdfpic} - \end{center} - \end{minipage} - - \breath - - \begin{experts} - Diese Differentialgleichung läßt sich mit "`normalen"' Mitteln nicht lösen, - daher verwendet man in der Praxis meistens die Kleinwinkelnäherung - $\sin\phi \approx \phi$ (für $\phi \ll 1$) - und löst stattdessen die Differentialgleichung: - \begin{displaymath} - \phi''(t) = -\frac{g}{l} \cdot \phi(t) - \end{displaymath} - Für ein mit der Anfangsauslenkung $\phi(0)$ losgelassenes Pendel - lautet dann das Ergebnis: - \begin{displaymath} - \phi(t) = \phi(0)\cdot\cos(\omega t)\qquad\mbox{mit}\qquad\omega=\sqrt{\frac{g}{l}} - \end{displaymath} - Das Beispielprogramm \gitfile{hp}{script}{pendulum-1.c} illustriert, - welche Bewegung sich aus diesem $\phi(t)$ ergibt. - \end{experts} - - \subsubsection{Das explizite Euler-Verfahren} - - Um eine Differentialgleichung mit Hilfe eines Computers näherungsweise \newterm{numerisch\/} zu lösen, - stehen zahlreiche Lösungsverfahren zur Verfügung. - Im folgenden soll das einfachste dieser Verfahren, das \newterm{explizite Euler-Verfahren\/} - (auch \newterm{Eulersches Polygonzugverfahren\/} genannt) vorgestellt werden. - - Wir betrachten das System während eines kleinen Zeitintervalls $\Delta t$. - Während dieses Zeitintervalls sind alle von der Zeit $t$ abhängigen Funktionen - -- z.\,B.\ Ort, Geschwindigkeit, Beschleunigung, Kraft -- näherungsweise konstant. - - Bei konstanter Geschwindigkeit $v$ ist es einfach, - aus dem Ort $x(t)$ zu Beginn des Zeitintervalls - den Ort $x(t + \Delta t)$ am Ende des Zeitintervalls zu berechnen: - \begin{displaymath} - x(t + \Delta t) = x(t) + \Delta t \cdot v - \end{displaymath} - - Bei konstanter Kraft $F = m \cdot a$ - und somit konstanter Beschleunigung $a$ ist es ebenso einfach, - aus der Geschwindigkeit $v(t)$ zu Beginn des Zeitintervalls - die Geschwindigkeit $v(t + \Delta t)$ am Ende des Zeitintervalls zu berechnen: - \begin{displaymath} - v(t + \Delta t) = v(t) + \Delta t \cdot a - \end{displaymath} - - Wenn wir dies in einer Schleife durchführen und jedesmal $t$ um $\Delta t$ erhöhen, - erhalten wir Näherungen für die Funktionen $x(t)$ und $v(t)$. - - \breath - - Für das oben betrachtete Beispiel (Fadenpendel) - müssen wir in jedem Zeitintervall $\Delta t$ - zunächst die tangentiale Beschleunigung $a(t)$ aus der tangentialen Kraft berechnen, - die sich wiederum aus der momentanen Auslenkung $\phi(t)$ ergibt: - \begin{displaymath} - a = -g \cdot \sin\phi - \end{displaymath} - Mit Hilfe dieser -- innerhalb des Zeitintervalls näherungsweise konstanten -- Beschleunigung - berechnen wir die neue tangentiale Geschwindigkeit: - \begin{displaymath} - v(t + \Delta t) = v(t) + \Delta t \cdot a - \end{displaymath} - Mit Hilfe dieser -- innerhalb des Zeitintervalls näherungsweise konstanten -- Geschwindigkeit - berechnen wir schließlich die neue Winkelauslenkung $\phi$, - wobei wir einen kleinen Umweg über den Kreisbogen $x = l\cdot\phi$ machen: - \begin{displaymath} - \phi(t + \Delta t) = \frac{x(t + \Delta t)}{l} - = \frac{x(t) + \Delta t \cdot v}{l} - = \phi(t) + \frac{\Delta t \cdot v}{l} - \end{displaymath} - - Ein C-Programm, das diese Berechnungen durchführt (Datei: \gitfile{hp}{script}{pendulum-2.c}), enthält als Kernstück: - \goodbreak - \begin{lstlisting} - #define g 9.81 - #define l 1.0 - #define dt 0.05 - #define phi0 30.0 /* degrees */ - - float t = 0.0; - float phi = phi0 * M_PI / 180.0; - float v = 0.0; + \section{Hardwarenahe Programmierung} - void calc (void) - { - float a = -g * sin (phi); - v += dt * a; - phi += dt * v / l; - } - \end{lstlisting} - \goodbreak - Jeder Aufruf der Funktion \lstinline{calc()} - versetzt das Pendel um das Zeitintervall \lstinline{dt} in die Zukunft. - - Es ist vom Verfahren her nicht notwendig, mit der Kleinwinkelnäherung $\sin\phi\approx\phi$ zu arbeiten. - Das Beispielprogramm \gitfile{hp}{script}{pendulum-3.c} illustriert, - welchen Unterschied die Kleinwinkelnäherung ausmacht. + \subsection{Bit-Operationen} - Wie gut arbeitet das explizite Euler-Verfahren? - Um dies zu untersuchen, lösen wir eine Differentialgleichung, - deren exakte Lösung aus der Literatur bekannt ist, - nämlich die Differentialgleichung mit Kleinwinkelnäherung. - Das Beispielprogramm \gitfile{hp}{script}{pendulum-4.c} vergleicht beide Lösungen miteinander. - Für das betrachtete Beispiel ist die Übereinstimmung recht gut; - für Präzisionsberechnungen ist das explizite Euler-Verfahren jedoch nicht genau (und stabil) genug. - Hierfür sei auf die Lehrveranstaltungen zur numerischen Mathematik verwiesen. + \setlength{\unitlength}{12pt} - \subsubsection*{Bemerkung} + \subsubsection{Zahlensysteme} - Das Beispielprogramm \gitfile{hp}{script}{pendulum-4.c} berechnet mit überzeugender Übereinstimmung - dasselbe Ergebnis für die Auslenkung des Pendels auf zwei verschiedene Weisen: - \begin{enumerate} + \subsubsection*{Dezimalsystem} + \begin{itemize} \item - über eine Formel, die einen Cosinus enthält, + Basis: 10 \item - mit Hilfe der Funktion \lstinline{calc()}, die nur Grundrechenarten verwendet. - \end{enumerate} - Dies läßt die Natur der Verfahren erahnen, mit deren Hilfe es möglich ist, - Sinus, Cosinus und andere kompliziertere Funktionen - nur unter Verwendung der Grundrechenarten zu berechnen. + Gültige Ziffern: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 + \end{itemize} + \begin{verbatim} + 137 137 + + 42 + ---- + 179 + \end{verbatim} + \begin{picture}(0,0)(0,-0.25) + \color{red} + \put(5.0,4.25){\vector(-1,1){1.41}} + \put(5.5,4){\mbox{Einer: $7 \cdot 10^0$}} + \put(5.5,3.25){\vector(-1,1){2.41}} + \put(6.0,3){\mbox{Zehner: $3 \cdot 10^1$}} + \put(6.0,2.25){\vector(-1,1){3.41}} + \put(6.5,2){\mbox{Hunderter: $1 \cdot 10^2$}} + \put(2.5,0.5){\mbox{$137_{10} = 1 \cdot 10^2 + 3 \cdot 10^1 + 7 \cdot 10^0 + = 100 + 30 + 7 = 137$}} + \end{picture} \goodbreak - - \subsection{Rekursion} - - Aus der Mathematik ist das Beweisprinzip der \newterm{vollständigen Induktion\/} bekannt: - \begin{displaymath} - \hspace*{4cm} - \left. - \begin{array}{r} - \mbox{Aussage gilt für $n = 1$}\\[2pt] - \mbox{Schluß von $n - 1$ auf $n$} - \end{array} - \right\} - \mbox{Aussage gilt für alle $n\in\mathbb{N}$} - \end{displaymath} - Wenn auf diese Weise die Lösbarkeit eines Problems bewiesen wurde, - ist es direkt möglich, das Problem im Computer \emph{tatsächlich\/} zu lösen, - nämlich durch einen \newterm{rekursiven Algorithmus}. - - \breath - - Ein klassisches Beispiel für ein rekursiv lösbares Problem sind die Türme von Hanoi: + \subsubsection*{Hexadezimalsystem} \begin{itemize} \item - 64 Scheiben, 3 Plätze, immer 1 Scheibe verschieben - \item - Ziel: Turm verschieben + Basis: 16 \item - Es dürfen nur kleinere Scheiben auf größeren liegen. - \end{itemize} - - \goodbreak - \begin{center} - \includegraphics[width=12.2cm]{Tower_of_Hanoi.jpeg} - - \small - - Bildquelle: \url{http://commons.wikimedia.org/wiki/File:Tower\_of\_Hanoi.jpeg}\\ - Urheber: \url{http://en.wikipedia.org/wiki/User:Evanherk}\\ - Lizenz: GNU FDL (Version 1.2 oder später) oder\\ - Creative Commons Attribution-Share Alike (Version 3.0 Unported) - - \end{center} - - \goodbreak - Die rekursive Lösung des Problems lautet: - \begin{itemize} - \item - Wenn $n = 1$ ist, also nur eine Scheibe vorliegt, - läßt sich diese "`einfach so"' an den Zielplatz verschieben. - In diesem Fall sind wir direkt fertig. - \item - Wenn $n - 1$ Scheiben als verschiebbar vorausgesetzt werden, - lautet die Vorgehensweise:\\ - verschiebe die oberen $n - 1$ Scheiben auf einen Hilfsplatz,\\ - verschiebe die darunterliegende einzelne Scheibe auf den Zielplatz,\\ - verschiebe die $n - 1$ Scheiben vom Hilfsplatz auf den Zielplatz. - \end{itemize} - - \goodbreak - Dieser Algorithmus läßt sich unmittelbar in eine Programmiersprache übersetzen: - \begin{lstlisting} - void verschiebe (int n, int start, int ziel) - { - if (n == 1) - verschiebe_1_scheibe (start, ziel); - else - { - verschiebe (1, start, hilfsplatz); - verschiebe (n - 1, start, ziel); - verschiebe (1, hilfsplatz, ziel); - } - } - \end{lstlisting} - -\iffalse - - \subsection{Floodfill} - - Siehe die Vortragsfolien \file{ainf-20121220.pdf}\\ - sowie die Beispielprogramme \file{fill-\lstinline{*}.c} - - \subsection{Stack und FIFO} - - Siehe Vortragsfolien und Beispielprogramme - - \subsection{Wegfindungsalgorithmus für Roboterfahrzeug} - - Siehe die Vortragsnotizen \file{ainf-20130117.txt},\\ - die Beispielprogramme \file{text-parcour-\lstinline{*}.c}, - \file{robotext.c} und \file{robotext.h}\\ - sowie die E-Mails "`Wegfindung"' und "`Weg-Finde-Algorithmus"' - -\fi - - \subsection{Aufwandsabschätzungen} - - \subsubsection{Sortieralgorithmen} - - Am Beispiel von Sortieralgorithmen soll hier aufgezeigt werden, - wie man die Lösung eines Problems schrittweise effizienter gestalten kann. - - Als Problem wählen wir das Sortieren eines Arrays (z.\,B.\ von Namen). - - \begin{itemize} - \item - Minimum/Maximum ermitteln: \gitfile{hp}{script}{sort-0.c}, - \gitfile{hp}{script}{sort-1.c} und \gitfile{hp}{script}{sort-2.c} - \item - Selectionsort: \gitfile{hp}{script}{sort-3.c}, \gitfile{hp}{script}{sort-4.c} - \item - Bubblesort: \gitfile{hp}{script}{sort-5.c} - \item - Bubblesort mit Abbruch in äußerer Schleife: \gitfile{hp}{script}{sort-6.c} - \item - Bubblesort mit Abbruch in innerer Schleife: \gitfile{hp}{script}{sort-7.c}, - \gitfile{hp}{script}{sort-7a.c}, \gitfile{hp}{script}{sort-7b.c} - \item - Vergleich mit Quicksort: \gitfile{hp}{script}{sort-8.c}, \gitfile{hp}{script}{sort-8a.c}, \gitfile{hp}{script}{sort-8b.c} - \end{itemize} - - Bei "`zufällig"' sortierten Ausgangsdaten arbeitet Quicksort schneller als Bubblesort. - Wenn die Ausgangsdaten bereits nahezu sortiert sind, ist es umgekehrt. - Im jeweils ungünstigsten Fall arbeiten beide Algorithmen gleich langsam. - - \subsubsection{Landau-Symbole} - - Das Landau-Symbol $\mathcal{O}(g)$ mit einer Funktion $g(n)$ - steht für die \newterm{Ordnung\/} eines Algorithmus', - also die "`Geschwindigkeit"', mit der er arbeitet. - Die Variable $n$ bezeichnet die Menge der Eingabedaten, - hier also z.\,B.\ die Anzahl der Namen. - - \begin{center} - \begin{pdfpic} - \psset{unit=1pt} - \begin{pspicture}(-20,-20)(250,200) - \psline[arrows=->](-10,0)(200,0) - \psline[arrows=->](0,-10)(0,200) - \psplot[plotpoints=200]{1}{125}{2 x 0.06 mul exp} - \put(100,190){\mbox{$g(n) \sim 2^n$}} - \psplot[plotpoints=200]{0}{190}{x x mul 0.005 mul} - \put(190,190){\mbox{$g(n) \sim n^2$}} - \psplot[plotpoints=200]{1}{190}{x ln x mul 0.1 mul} - \put(195,100){\mbox{$g(n) \sim n \log n$}} - \psplot[plotpoints=200]{0}{190}{x 0.4 mul} - \put(195,75){\mbox{$g(n) \sim n$}} - \psplot[plotpoints=200]{1}{190}{x ln 10 mul} - \put(195,50){\mbox{$g(n) \sim \log n$}} - \put(210,0){\makebox(0,0)[l]{$n$}} - \put(0,210){\makebox(0,0)[l]{$g(n)$}} - \end{pspicture} - \end{pdfpic} - \end{center} - - \begin{itemize} - \item - $\mathcal{O}(n)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten linear wächst. - Um doppelt so viele Namen zu sortieren, benötigt das Programm doppelt so lange. - \item - $\mathcal{O}(n^2)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten quadratisch wächst. - Um doppelt so viele Namen zu sortieren, benötigt das Programm viermal so lange. - \item - $\mathcal{O}(2^n)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten exponentiell wächst. - Für jeden Namen, der dazukommt, benötigt das Programm doppelt so lange. - - Ein derartiges Programm gilt normalerweise als inakzeptabel langsam. - \item - $\mathcal{O}(\log n)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten logarithmisch wächst. - Für jede Verdopplung der Namen benötigt das Programm nur einen Rechenschritt mehr. - - Ein derartiges Programm gilt als "`traumhaft schnell"'. - Dies wird jedoch nur selten erreicht. - \item - $\mathcal{O}(1)$ bedeutet, daß die Rechenzeit von der Menge der Eingabedaten unabhängig ist: - 1\,000\,000 Namen werden genau so schnell sortiert wie 10. - - Dies ist nur in Ausnahmefällen erreichbar. - \item - $\mathcal{O}(n \log n)$ liegt zwischen $\mathcal{O}(n)$ und $\mathcal{O}(n^2)$. - - Ein derartiges Programm gilt als schnell. - Viele Sortieralgorithmen erreichen dieses Verhalten. - \end{itemize} - - Wie sieht man einem Programm an, wie schnell es arbeitet? - - \begin{itemize} - \item - Vorfaktoren interessieren nicht. - - Wenn ein Code immer -- also unabhängig von den Eingabedaten -- zweimal ausgeführt wird, - beeinflußt das die Ordnung des Algorithmus nicht. - - Wenn ein Code immer -- also unabhängig von den Eingabedaten -- 1\,000\,000mal ausgeführt wird, - mag das Programm für kleine Datenmengen langsam erscheinen. - Für die Ordnung interessiert jedoch nur das Verhalten für große Datenmengen, - und dort kann dasselbe Programm durchaus schnell sein. - \item - Jede Schleife, die von $0$ bis $n$ geht, - multipliziert die Rechenzeit des innerhalb der Schleife befindlichen Codes mit $n$. - - Eine Doppelschleife (Schleife innerhalb einer Schleife) hat demnach $\mathcal{O}(n^2)$. - \goodbreak - \item - Wenn sich die Grenzen einer Schleife ständig ändern, nimmt man den Durschschnitt. - - Beispiel: - \begin{lstlisting}[gobble=8] - for (int i = 0; i < n; i++) - for (int j = 0; j < i; j++) - ... - \end{lstlisting} - Die äußere Schleife wird immer $n$-mal ausgeführt, - die innere \emph{im Durchschnitt\/} $\frac{n}{2}$-mal, was proportional zu $n$ ist. - - Zusammen ergibt sich $\mathcal{O}(n^2)$. - \item - Bei Rekursionen muß man mitzählen, wie viele Schleifen hinzukommen. - - Bei Quicksort wird z.\,B.\ in jeder Rekursion - eine Schleife von $0$ bis $n$ (aufgeteilt) ausgeführt. - Bei jeder Rekursion wird das Array "`normalerweise"' halbiert, - d.\,h.\ die Rekursionstiefe ist proportional zum Logarithmus von $n$ (zur Basis 2). - Daraus ergibt sich die Ordnung $\mathcal{O}(n\log n)$ für den "`Normalfall"' des Quicksort. - (Im ungünstigsten Fall kann sich auch $\mathcal{O}(n^2)$ ergeben.) - \end{itemize} - - Für eine präzise Definition der Landau-Symbole siehe z.\,B.: - \url{http://de.wikipedia.org/wiki/Landau-Symbole} - - \section{Hardwarenahe Programmierung} - - \subsection{Bit-Operationen} - - \setlength{\unitlength}{12pt} - - \subsubsection{Zahlensysteme} - - \subsubsection*{Dezimalsystem} - \begin{itemize} - \item - Basis: 10 - \item - Gültige Ziffern: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 - \end{itemize} - \begin{verbatim} - 137 137 - + 42 - ---- - 179 - \end{verbatim} - \begin{picture}(0,0)(0,-0.25) - \color{red} - \put(5.0,4.25){\vector(-1,1){1.41}} - \put(5.5,4){\mbox{Einer: $7 \cdot 10^0$}} - \put(5.5,3.25){\vector(-1,1){2.41}} - \put(6.0,3){\mbox{Zehner: $3 \cdot 10^1$}} - \put(6.0,2.25){\vector(-1,1){3.41}} - \put(6.5,2){\mbox{Hunderter: $1 \cdot 10^2$}} - \put(2.5,0.5){\mbox{$137_{10} = 1 \cdot 10^2 + 3 \cdot 10^1 + 7 \cdot 10^0 - = 100 + 30 + 7 = 137$}} - \end{picture} - - \goodbreak - \subsubsection*{Hexadezimalsystem} - \begin{itemize} - \item - Basis: 16 - \item - Gültige Ziffern: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F + Gültige Ziffern: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F \end{itemize} \begin{verbatim} 137 A380 @@ -4279,6 +3856,63 @@ \put(14.5,3){\mbox{IP-Adresse des Sub-Netzes}} \end{picture}\vspace{-6ex} + \subsection{Programmierung von Mikrocontrollern} + + Ein Mikrocontroller ist ein elektronischer Baustein, + der einen kompletten Computer mit eingeschränkter Funktionalität enthält. + + Mit "`eingeschränkter Funktionalität"' ist gemeint, + daß auf einem Mikrocontroller kein Universal-Betriebssystem läuft, + sondern daß darauf nur ein einziges Programm läuft, + nämlich die Anwendungs-Software. + + Wenn ein Baustein einen kompletten Computer enthält, + der leistungsfähig genug ist, + daß darauf ein Universal-Betriebssystem laufen kann, + spricht man normalerweise nicht mehr von einem Mikrocontroller, + sondern von einem Ein-Chip-Computer. + Der Übergang ist fließend. + + \breath + + Da ein Mikrocontroller nicht über die Leistung verfügt, + einen Compiler laufen zu lassen, + erfolgt die Entwicklung von Software für Mikrocontroller + auf einem "`normalen"' Computer. + Diese Art der Software-Entwicklung, + bei der ein Computer Software für einen ganz anderen Computer erzeugt, + bezeichnet man als \newterm{Cross-Entwicklung}. + Die einzelnen Werkzeuge heißen entsprechend + \newterm{Cross-Compiler}, \newterm{Cross-Assembler\/} und \newterm{Cross-Linker}. + + Beispiel: Erzeugen einer ausführbaren Datei \file{blink.elf} + aus einem C-Quelltext \file{blink.c} + für einen Mikrocontroller vom Typ ATmega328P + \begin{lstlisting}[style=cmd] + avr-gcc -Wall -Os -mmcu=atmega328p blink.c -o blink.elf + \end{lstlisting} + + Damit der Mikrocontroller die ausführbare Datei ausführen kann, + muß man sie mit einem speziellen Werkzeug auf den Mikrocontroller + \newterm{herunterladen}. + Hierfür ist es oft notwendig, die Datei vorher in ein anderes Dateiformat + zu konvertieren. + + Beispiel: Konvertierung der ausführbaren Datei \file{blink.elf} + aus dem ELF-Dateiformat in das Intel-HEX-Format: + \begin{lstlisting}[style=cmd] + avr-objcopy -O ihex blink.elf blink.hex + \end{lstlisting} + + Anschließend kann die Datei auf den Mikrocontroller heruntergeladen werden. + Beispiel: Herunterladen der Datei \file{blink.hex} + in den Flash-Speicher eines ATmega328P-Mikrocontrollers + auf einem Mikrocontroller-Board vom Typ Arduino Uno + über die Schnittstelle \file{/dev/ttyACM0} + \begin{lstlisting}[style=cmd] + avrdude -P /dev/ttyACM0 -c arduino -p m328p -U flash:w:blink.hex + \end{lstlisting} + \subsection{I/O-Ports} Es gibt drei grundlegende Mechanismen für die Kommunikation zwischen dem Prozessor @@ -4293,38 +3927,7 @@ \end{itemize} \begin{center} - \begin{pdfpic} - \psset{unit=1cm}% - \begin{pspicture}(0,0)(10,6) - \rput(0,0){\psframe[framearc=0.25](2,5)} - \rput(1,4.5){\makebox(0,0)[t]{Prozessor}} - \rput(2.0,3.7){\pnode{p0}} - \rput(2.0,3.3){\pnode{q0}} - \rput(2.0,1.0){\pnode{i0}} - \rput(2.12,1.2){\pnode{j0}} - \rput(2.02,1.3){\pnode{k0}} - \rput(2.12,1.4){\pnode{l0}} - \rput(3.5,1.4){\pnode{m0}} - \rput(8,0){\psframe[framearc=0.25](2,5)} - \rput(9,4.5){\makebox(0,0)[t]{\shortstack{externes\\Gerät}}} - \rput(8.0,3.7){\pnode{p1}} - \rput(7.88,3.3){\pnode{q1}} - \rput(7.98,3.2){\pnode{r1}} - \rput(7.88,3.1){\pnode{s1}} - \rput(6.5,3.1){\pnode{t1}} - \rput(8.0,1.0){\pnode{i1}} - \rput(8.0,1.2){\pnode{j1}} - \ncline{->}{p0}{p1} - \ncline{q0}{q1} - \nccurve[angleB=90]{q1}{r1} - \nccurve[angleA=-90]{r1}{s1} - \ncline{->}{s1}{t1} - \rput(2.2,3.8){\makebox(0,0)[lb]{Prozessor schreibt in Output-Port}} - \rput(2.2,3.1){\makebox(0,0)[lt]{Prozessor liest Input-Port}} - \ncline{->}{i1}{i0} - \rput(7.8,1.1){\makebox(0,0)[rb]{externes Gerät ruft Interrupt auf}} - \end{pspicture} - \end{pdfpic} + \includegraphics{io-ports-and-interrupts.pdf} \end{center} \setlength{\unitlength}{1cm} @@ -4582,120 +4185,6 @@ \fi - \let\savesubsection\thesubsection - \gdef\thesubsection{\thesection.\arabic{subsection}\protect\half} - \addtocounter{subsection}{-1} - \subsection[\quad Binärdarstellung von Zahlen]{Binärdarstellung von Zahlen} - - Es gibt unendlich viele verschiedene ganze Zahlen. - Da Speicherplatz begrenzt ist, können Rechner nur begrenzt viele Zahlen darstellen. - In der Praxis legt man sich auf eine Anzahl von Bits fest, - die eine Zahl maximal belegen darf. - Typischerweise handelt es sich bei dieser Anzahl von Bits - um eine Zweierpotenz: 8, 16, 32, 64. - - Eine 8-Bit-Zahl kann per definitionem Binärzahlen von 0000\,0000 bis 1111\,1111 - darstellen. Dezimal sind dies die Zahlen von 0 bis 255. - Wenn man die 8-Bit-Zahl 255 inkrementiert (= um 1 erhöht), - gibt es einen Überlauf, und das Rechenergebnis ist 0: - Die Binärzahl 1111\,1111 + 1 = 1\,0000\,0000 hat 9 Bits. - Wenn man das oberste Bit abschneidet, bleibt der Wert 0 übrig. - - Auf Computern macht man sich dieses Verhalten - für die Binärdarstellung negativer Zahlen zunutze: - Wenn 255 + 1 (dezimal geschrieben, mit 8-Bit-Zahlen berechnet) den Wert 0 ergibt, - dann ist 255 dasselbe wie $-1$. - - Allgemein gilt: Diejenige Zahl $y$, die ich auf eine Zahl $x$ addieren muß, - um auf einem $n$-Bit-Rechner einen Überlauf und das Rechenergebnis $0$ zu erhalten, - ist die \emph{$n$-Bit-Binärdarstellung von $-x$\/}). - - Um diese Zahl direkt auszurechnen, geht man folgendermaßen vor: - \begin{enumerate} - \item - Man invertiert alle Bits der Zahl.\\ - Aus 0010\,1100 (= Binärdarstellung von 44) wird so zum Beispiel 1101\,0011. - \item - Man addiert 1 zu der erhaltenen Zahl:\\ - 1101\,0011 + 1 = 1101\,0100 (= 8-Bit-Binärdarstellung von $-44$). - \end{enumerate} - Diese Darstellung negativer Zahlen heißt \newterm{Zweierkomplement}. - - \goodbreak - - Theoretisch kann man jede Zahl bei gegebener Rechengenauigkeit - sowohl als postive als auch als negative Zahl interpretieren. - Die folgende Konvention hat sich als sinnvoll heraugestellt: - \begin{itemize} - \item - Entweder betrachtet man alle Zahlen als positiv, - \item - oder man betrachtet Zahlen, deren oberstes Bit gesetzt ist, als negativ\\ - und die anderen als positiv. - \end{itemize} - - Für normale Anwendungen ist die genaue Anzahl der Bits einer Ganzzahl-Variablen - unerheblich, sofern der Wertebereich groß genug für die durchzuführenden - Rechnungen ist. In diesen Fällen verwendet man in C den Datentyp \lstinline{int} - für vorzeichenbehaftete ganze Zahlen und \lstinline{unsigned int} (oder kurz: - \lstinline{unsigned}) für vorzeichenlose. Für die Ausgabe mit \lstinline{printf()} - verwendet man \lstinline{%d} oder \lstinline{%i} für vorzeichenbehaftete und - \lstinline{%u} für vorzeichenlose ganze Zahlen. - - Für spezielle Situationen, in denen die genaue Anzahl der Bits eine Rolle spielt, - stellt C (unter Verwendung von \lstinline{#include <stdint.h>}) spezielle Datentypen bereit: - - \begin{center} - \renewcommand{\arraystretch}{1.2} - \newcommand{\xdots}{\hspace*{-0.7em}\dots,\hspace*{-0.7em}} - \begin{tabular}{|c|c|c|rcl|}\hline - C-Datentyp & Bits & Vorzeichen & \multicolumn{3}{c|}{Wertebereich} \\\hline\hline - \lstinline,int8_t, & 8 & ja & $-128$, & \xdots & $127$ \\\hline - \lstinline,uint8_t, & 8 & nein & $0$, & \xdots & $255$ \\\hline - \lstinline,int16_t, & 16 & ja & $-32\,768$, & \xdots & $32\,767$ \\\hline - \lstinline,uint16_t, & 16 & nein & $0$, & \xdots & $65\,535$ \\\hline - \lstinline,int32_t, & 32 & ja & $-2\,147\,483\,648$, & \xdots & $2\,147\,483\,647$ \\\hline - \lstinline,uint32_t, & 32 & nein & $0$, & \xdots & $4\,294\,967\,295$ \\\hline - \lstinline,int64_t, & 64 & ja & $9\,223\,372\,036\,854\,775\,808$, & \xdots & $9\,223\,372\,036\,854\,775\,807$ \\\hline - \lstinline,uint64_t, & 64 & nein & $0$, & \xdots & $18\,446\,744\,073\,709\,551\,615$ \\\hline - \end{tabular} - \end{center} - - Man beachte, daß es keine "`allein richtige"' Binärdarstellung einer negativen Zahl gibt; - diese hängt vielmehr von der Genauigkeit $n$ des $n$-Bit-Rechenwerks ab. - Auf einem 8-Bit-Rechner ist $255$ dasselbe wie $-1$, - auf einem 16-Bit-Rechner ist $255$ eine "`völlig normale"' Zahl; - stattdessen ist $65535$ dasselbe wie $-1$. - - Ebensowenig gibt es einen "`allein richtigen"' Zahlenwert eines Bitmusters. - Dieser Zahlenwert hängt von der Genauigkeit $n$ des $n$-Bit-Rechenwerks ab - und davon, ob man überhaupt negative Zahlen zuläßt - oder vielleicht alle Zahlen als positive Zahlen interpretiert. - - \breath - - Beispiel: Für welche Zahl steht der Speicherinhalt 1001\,0000\,1100\,0011 - (binär) = 90a3 (hexadezimal)? - - Die richtige Antwort auf diese Frage hängt vom Datentyp ab, - also von der Bitgenauigkeit des Rechenwerks - und davon, ob wir überhaupt mit Vorzeichen rechnen: - - \begin{tabular}{lrl} - als \lstinline,int8_t,: & $-93$ & (nur unteres Byte, Little-Endian)\\ - als \lstinline,uint8_t,: & $163$ & (nur unteres Byte, Little-Endian)\\ - als \lstinline,int16_t,: & $-28509$\\ - als \lstinline,uint16_t,: & $37027$\\ - \lstinline,int32_t, oder größer: & $37027$ - & (zusätzliche Bytes mit Nullen aufgefüllt) - \end{tabular} - - \breath - - Siehe auch: \url{http://xkcd.com/571/} - - \let\thesubsection\savesubsection - \subsection{Byte-Reihenfolge -- Endianness} \subsubsection{Konzept} @@ -4810,6 +4299,115 @@ daß die vier Bytes von links nach rechts (Big-Endian) übertragen werden, die acht Bits innerhalb jedes Bytes hingegen von rechts nach links (LSB first). + \subsection{Binärdarstellung von Zahlen} + + Es gibt unendlich viele verschiedene ganze Zahlen. + Da Speicherplatz begrenzt ist, können Rechner nur begrenzt viele Zahlen darstellen. + In der Praxis legt man sich auf eine Anzahl von Bits fest, + die eine Zahl maximal belegen darf. + Typischerweise handelt es sich bei dieser Anzahl von Bits + um eine Zweierpotenz: 8, 16, 32, 64. + + Eine 8-Bit-Zahl kann per definitionem Binärzahlen von 0000\,0000 bis 1111\,1111 + darstellen. Dezimal sind dies die Zahlen von 0 bis 255. + Wenn man die 8-Bit-Zahl 255 inkrementiert (= um 1 erhöht), + gibt es einen Überlauf, und das Rechenergebnis ist 0: + Die Binärzahl 1111\,1111 + 1 = 1\,0000\,0000 hat 9 Bits. + Wenn man das oberste Bit abschneidet, bleibt der Wert 0 übrig. + + Auf Computern macht man sich dieses Verhalten + für die Binärdarstellung negativer Zahlen zunutze: + Wenn 255 + 1 (dezimal geschrieben, mit 8-Bit-Zahlen berechnet) den Wert 0 ergibt, + dann ist 255 dasselbe wie $-1$. + + Allgemein gilt: Diejenige Zahl $y$, die ich auf eine Zahl $x$ addieren muß, + um auf einem $n$-Bit-Rechner einen Überlauf und das Rechenergebnis $0$ zu erhalten, + ist die \emph{$n$-Bit-Binärdarstellung von $-x$\/}). + + Um diese Zahl direkt auszurechnen, geht man folgendermaßen vor: + \begin{enumerate} + \item + Man invertiert alle Bits der Zahl.\\ + Aus 0010\,1100 (= Binärdarstellung von 44) wird so zum Beispiel 1101\,0011. + \item + Man addiert 1 zu der erhaltenen Zahl:\\ + 1101\,0011 + 1 = 1101\,0100 (= 8-Bit-Binärdarstellung von $-44$). + \end{enumerate} + Diese Darstellung negativer Zahlen heißt \newterm{Zweierkomplement}. + + \goodbreak + + Theoretisch kann man jede Zahl bei gegebener Rechengenauigkeit + sowohl als postive als auch als negative Zahl interpretieren. + Die folgende Konvention hat sich als sinnvoll heraugestellt: + \begin{itemize} + \item + Entweder betrachtet man alle Zahlen als positiv, + \item + oder man betrachtet Zahlen, deren oberstes Bit gesetzt ist, als negativ\\ + und die anderen als positiv. + \end{itemize} + + Für normale Anwendungen ist die genaue Anzahl der Bits einer Ganzzahl-Variablen + unerheblich, sofern der Wertebereich groß genug für die durchzuführenden + Rechnungen ist. In diesen Fällen verwendet man in C den Datentyp \lstinline{int} + für vorzeichenbehaftete ganze Zahlen und \lstinline{unsigned int} (oder kurz: + \lstinline{unsigned}) für vorzeichenlose. Für die Ausgabe mit \lstinline{printf()} + verwendet man \lstinline{%d} oder \lstinline{%i} für vorzeichenbehaftete und + \lstinline{%u} für vorzeichenlose ganze Zahlen. + + Für spezielle Situationen, in denen die genaue Anzahl der Bits eine Rolle spielt, + stellt C (unter Verwendung von \lstinline{#include <stdint.h>}) spezielle Datentypen bereit: + + \begin{center} + \renewcommand{\arraystretch}{1.2} + \newcommand{\xdots}{\hspace*{-0.7em}\dots,\hspace*{-0.7em}} + \begin{tabular}{|c|c|c|rcl|}\hline + C-Datentyp & Bits & Vorzeichen & \multicolumn{3}{c|}{Wertebereich} \\\hline\hline + \lstinline,int8_t, & 8 & ja & $-128$, & \xdots & $127$ \\\hline + \lstinline,uint8_t, & 8 & nein & $0$, & \xdots & $255$ \\\hline + \lstinline,int16_t, & 16 & ja & $-32\,768$, & \xdots & $32\,767$ \\\hline + \lstinline,uint16_t, & 16 & nein & $0$, & \xdots & $65\,535$ \\\hline + \lstinline,int32_t, & 32 & ja & $-2\,147\,483\,648$, & \xdots & $2\,147\,483\,647$ \\\hline + \lstinline,uint32_t, & 32 & nein & $0$, & \xdots & $4\,294\,967\,295$ \\\hline + \lstinline,int64_t, & 64 & ja & $9\,223\,372\,036\,854\,775\,808$, & \xdots & $9\,223\,372\,036\,854\,775\,807$ \\\hline + \lstinline,uint64_t, & 64 & nein & $0$, & \xdots & $18\,446\,744\,073\,709\,551\,615$ \\\hline + \end{tabular} + \end{center} + + Man beachte, daß es keine "`allein richtige"' Binärdarstellung einer negativen Zahl gibt; + diese hängt vielmehr von der Genauigkeit $n$ des $n$-Bit-Rechenwerks ab. + Auf einem 8-Bit-Rechner ist $255$ dasselbe wie $-1$, + auf einem 16-Bit-Rechner ist $255$ eine "`völlig normale"' Zahl; + stattdessen ist $65535$ dasselbe wie $-1$. + + Ebensowenig gibt es einen "`allein richtigen"' Zahlenwert eines Bitmusters. + Dieser Zahlenwert hängt von der Genauigkeit $n$ des $n$-Bit-Rechenwerks ab + und davon, ob man überhaupt negative Zahlen zuläßt + oder vielleicht alle Zahlen als positive Zahlen interpretiert. + + \breath + + Beispiel: Für welche Zahl steht der Speicherinhalt 1001\,0000\,1100\,0011 + (binär) = 90a3 (hexadezimal)? + + Die richtige Antwort auf diese Frage hängt vom Datentyp ab, + also von der Bitgenauigkeit des Rechenwerks + und davon, ob wir überhaupt mit Vorzeichen rechnen: + + \begin{tabular}{lrl} + als \lstinline,int8_t,: & $-93$ & (nur unteres Byte, Little-Endian)\\ + als \lstinline,uint8_t,: & $163$ & (nur unteres Byte, Little-Endian)\\ + als \lstinline,int16_t,: & $-28509$\\ + als \lstinline,uint16_t,: & $37027$\\ + \lstinline,int32_t, oder größer: & $37027$ + & (zusätzliche Bytes mit Nullen aufgefüllt) + \end{tabular} + + \breath + + Siehe auch: \url{http://xkcd.com/571/} + \subsection{Speicherausrichtung -- Alignment} Ein 32-Bit-Prozessor kann auf eine 32-Bit-Variable effizienter zugreifen, @@ -4992,6 +4590,435 @@ \fi + \section{Algorithmen} + + \subsection{Differentialgleichungen} + + Eine mathematische Gleichung mit einer gesuchten Zahl $x$, z.\,B. + \begin{displaymath} + x + 2 = -x + \end{displaymath} + läßt sich leicht nach der Unbekannten $x$ auflösen. + (In diesem Fall lautet die Lösung: $x = -1$.) + + Wesentlich schwieriger ist es, + eine mathematische Gleichung mit einer gesuchten Funktion $x(t)$ zu lösen, z.\,B.: + \begin{displaymath} + x'(t) = -x(t)\qquad\mbox{mit}\qquad x(0) = 1 + \end{displaymath} + Um hier auf die Lösung $x(t) = e^{-t}$ zu kommen, + sind bereits weitreichende mathematische Kenntnisse erforderlich. + + Eine derartige Gleichung, die einen Zusammenhang zwischen der gesuchten Funktion + und ihren Ableitungen vorgibt, heißt \newterm{Differentialgleichung}. + Viele physikalisch-technische Probleme werden durch Differentialgleichungen beschrieben. + + \goodbreak + + \subsubsection{Beispiel: Pendelschwingung} + + \begin{minipage}{9.4cm} + \setlength{\parskip}{\smallskipamount} + Das Verhalten eines Fadenpendels (mathematisches Pendel) + wird durch seine Auslenkung $\phi$ als Funktion der Zeit $t$ beschrieben. + Wie kann man $\phi(t)$ berechnen? + + Wie aus anderen Veranstaltungen (Grundlagen der Physik, Mechanik) her bekannt sein sollte, + wirkt auf ein Fadenpendel, das um den Winkel $\phi(t)$ ausgelenkt ist, + die tangentiale Kraft $F = -m \cdot g \cdot \sin\phi(t)$. + Gemäß der Formel $F = m \cdot a$ bewirkt diese Kraft + eine tangentiale Beschleunigung $a = -g \cdot \sin\phi(t)$. + (Das Minuszeichen kommt daher, daß die Kraft der Auslenkung entgegengesetzt wirkt.) + + Wenn das Pendel die Länge $l$ hat, können wir dieselbe tangentiale Beschleunigung + mit Hilfe der zweiten Ableitung des Auslenkungswinkels $\phi(t)$ berechnen: + $a = l \cdot \phi''(t)$ (Winkel in Bogenmaß). + Durch Gleichsetzen erhalten wir eine Gleichung, + die nur noch eine Unbekannte enthält, nämlich die Funktion $\phi(t)$. + + Um $\phi(t)$ zu berechnen, müssen wir also die Differentialgleichung + \begin{displaymath} + \phi''(t) = -\frac{g}{l} \cdot \sin\phi(t) + \end{displaymath} + lösen. + \end{minipage}\hfill + \begin{minipage}{5.5cm} + \begin{center} + \begin{pdfpic} + \psset{unit=1.5cm}% + \begin{pspicture}(0,0)(0,0) + \SpecialCoor + \psline[linewidth=1pt](3;-110)(0,0) + \rput(3.25;-110){\psline[linewidth=0.5pt,linecolor=lightgray](0.34;-20)(1.00;-90)} + \rput(3.25;-110){\psline[linewidth=0.5pt,linecolor=lightgray](0.94;-110)(1.00;-90)} + \rput(3.25;-110){\psline[linewidth=0.5pt,arrows=->](0,0)(0.34;-20)} + \rput(3.25;-110){\psline[linewidth=0.5pt,arrows=->](0,0)(0.94;-110)} + \rput(3.25;-110){\psline[linewidth=0.5pt,arrows=->](0,0)(1.00;-90)} + \rput(3.25;-110){\rput(0.64;-20){\makebox(0,0){$F$}}} + \rput(3.25;-110){\rput(1.30;-90){\makebox(0,0){$m\cdot g$}}} + \pscircle*(3;-110){0.07} + \psline[linewidth=0.5pt,linestyle=dashed](0,0)(0,-1.9) + \psarc[linewidth=0.5pt,arrows=<-](0,0){1.5}{-110}{-90} + \rput(1.3;-100){\makebox(0,0){$\phi$}} + \psarc[linewidth=0.5pt,linestyle=dashed,arrows=<->,arcsepA=0.12](0,0){3}{-110}{-70} + \rput(0,-4.5){\psstrut} + \end{pspicture} + \end{pdfpic} + \end{center} + \end{minipage} + + \breath + + \begin{experts} + Diese Differentialgleichung läßt sich mit "`normalen"' Mitteln nicht lösen, + daher verwendet man in der Praxis meistens die Kleinwinkelnäherung + $\sin\phi \approx \phi$ (für $\phi \ll 1$) + und löst stattdessen die Differentialgleichung: + \begin{displaymath} + \phi''(t) = -\frac{g}{l} \cdot \phi(t) + \end{displaymath} + Für ein mit der Anfangsauslenkung $\phi(0)$ losgelassenes Pendel + lautet dann das Ergebnis: + \begin{displaymath} + \phi(t) = \phi(0)\cdot\cos(\omega t)\qquad\mbox{mit}\qquad\omega=\sqrt{\frac{g}{l}} + \end{displaymath} + Das Beispielprogramm \gitfile{hp}{script}{pendulum-1.c} illustriert, + welche Bewegung sich aus diesem $\phi(t)$ ergibt. + \end{experts} + + \subsubsection{Das explizite Euler-Verfahren} + + Um eine Differentialgleichung mit Hilfe eines Computers näherungsweise \newterm{numerisch\/} zu lösen, + stehen zahlreiche Lösungsverfahren zur Verfügung. + Im folgenden soll das einfachste dieser Verfahren, das \newterm{explizite Euler-Verfahren\/} + (auch \newterm{Eulersches Polygonzugverfahren\/} genannt) vorgestellt werden. + + Wir betrachten das System während eines kleinen Zeitintervalls $\Delta t$. + Während dieses Zeitintervalls sind alle von der Zeit $t$ abhängigen Funktionen + -- z.\,B.\ Ort, Geschwindigkeit, Beschleunigung, Kraft -- näherungsweise konstant. + + Bei konstanter Geschwindigkeit $v$ ist es einfach, + aus dem Ort $x(t)$ zu Beginn des Zeitintervalls + den Ort $x(t + \Delta t)$ am Ende des Zeitintervalls zu berechnen: + \begin{displaymath} + x(t + \Delta t) = x(t) + \Delta t \cdot v + \end{displaymath} + + Bei konstanter Kraft $F = m \cdot a$ + und somit konstanter Beschleunigung $a$ ist es ebenso einfach, + aus der Geschwindigkeit $v(t)$ zu Beginn des Zeitintervalls + die Geschwindigkeit $v(t + \Delta t)$ am Ende des Zeitintervalls zu berechnen: + \begin{displaymath} + v(t + \Delta t) = v(t) + \Delta t \cdot a + \end{displaymath} + + Wenn wir dies in einer Schleife durchführen und jedesmal $t$ um $\Delta t$ erhöhen, + erhalten wir Näherungen für die Funktionen $x(t)$ und $v(t)$. + + \breath + + Für das oben betrachtete Beispiel (Fadenpendel) + müssen wir in jedem Zeitintervall $\Delta t$ + zunächst die tangentiale Beschleunigung $a(t)$ aus der tangentialen Kraft berechnen, + die sich wiederum aus der momentanen Auslenkung $\phi(t)$ ergibt: + \begin{displaymath} + a = -g \cdot \sin\phi + \end{displaymath} + Mit Hilfe dieser -- innerhalb des Zeitintervalls näherungsweise konstanten -- Beschleunigung + berechnen wir die neue tangentiale Geschwindigkeit: + \begin{displaymath} + v(t + \Delta t) = v(t) + \Delta t \cdot a + \end{displaymath} + Mit Hilfe dieser -- innerhalb des Zeitintervalls näherungsweise konstanten -- Geschwindigkeit + berechnen wir schließlich die neue Winkelauslenkung $\phi$, + wobei wir einen kleinen Umweg über den Kreisbogen $x = l\cdot\phi$ machen: + \begin{displaymath} + \phi(t + \Delta t) = \frac{x(t + \Delta t)}{l} + = \frac{x(t) + \Delta t \cdot v}{l} + = \phi(t) + \frac{\Delta t \cdot v}{l} + \end{displaymath} + + Ein C-Programm, das diese Berechnungen durchführt (Datei: \gitfile{hp}{script}{pendulum-2.c}), enthält als Kernstück: + \goodbreak + \begin{lstlisting} + #define g 9.81 + #define l 1.0 + #define dt 0.05 + #define phi0 30.0 /* degrees */ + + float t = 0.0; + float phi = phi0 * M_PI / 180.0; + float v = 0.0; + + void calc (void) + { + float a = -g * sin (phi); + v += dt * a; + phi += dt * v / l; + } + \end{lstlisting} + \goodbreak + Jeder Aufruf der Funktion \lstinline{calc()} + versetzt das Pendel um das Zeitintervall \lstinline{dt} in die Zukunft. + + Es ist vom Verfahren her nicht notwendig, mit der Kleinwinkelnäherung $\sin\phi\approx\phi$ zu arbeiten. + Das Beispielprogramm \gitfile{hp}{script}{pendulum-3.c} illustriert, + welchen Unterschied die Kleinwinkelnäherung ausmacht. + + Wie gut arbeitet das explizite Euler-Verfahren? + Um dies zu untersuchen, lösen wir eine Differentialgleichung, + deren exakte Lösung aus der Literatur bekannt ist, + nämlich die Differentialgleichung mit Kleinwinkelnäherung. + Das Beispielprogramm \gitfile{hp}{script}{pendulum-4.c} vergleicht beide Lösungen miteinander. + Für das betrachtete Beispiel ist die Übereinstimmung recht gut; + für Präzisionsberechnungen ist das explizite Euler-Verfahren jedoch nicht genau (und stabil) genug. + Hierfür sei auf die Lehrveranstaltungen zur numerischen Mathematik verwiesen. + + \subsubsection*{Bemerkung} + + Das Beispielprogramm \gitfile{hp}{script}{pendulum-4.c} berechnet mit überzeugender Übereinstimmung + dasselbe Ergebnis für die Auslenkung des Pendels auf zwei verschiedene Weisen: + \begin{enumerate} + \item + über eine Formel, die einen Cosinus enthält, + \item + mit Hilfe der Funktion \lstinline{calc()}, die nur Grundrechenarten verwendet. + \end{enumerate} + Dies läßt die Natur der Verfahren erahnen, mit deren Hilfe es möglich ist, + Sinus, Cosinus und andere kompliziertere Funktionen + nur unter Verwendung der Grundrechenarten zu berechnen. + + \goodbreak + + \subsection{Rekursion} + + Aus der Mathematik ist das Beweisprinzip der \newterm{vollständigen Induktion\/} bekannt: + \begin{displaymath} + \hspace*{4cm} + \left. + \begin{array}{r} + \mbox{Aussage gilt für $n = 1$}\\[2pt] + \mbox{Schluß von $n - 1$ auf $n$} + \end{array} + \right\} + \mbox{Aussage gilt für alle $n\in\mathbb{N}$} + \end{displaymath} + Wenn auf diese Weise die Lösbarkeit eines Problems bewiesen wurde, + ist es direkt möglich, das Problem im Computer \emph{tatsächlich\/} zu lösen, + nämlich durch einen \newterm{rekursiven Algorithmus}. + + \breath + + Ein klassisches Beispiel für ein rekursiv lösbares Problem sind die Türme von Hanoi: + \begin{itemize} + \item + 64 Scheiben, 3 Plätze, immer 1 Scheibe verschieben + \item + Ziel: Turm verschieben + \item + Es dürfen nur kleinere Scheiben auf größeren liegen. + \end{itemize} + + \goodbreak + \begin{center} + \includegraphics[width=12.2cm]{Tower_of_Hanoi.jpeg} + + \small + + Bildquelle: \url{http://commons.wikimedia.org/wiki/File:Tower\_of\_Hanoi.jpeg}\\ + Urheber: \url{http://en.wikipedia.org/wiki/User:Evanherk}\\ + Lizenz: GNU FDL (Version 1.2 oder später) oder\\ + Creative Commons Attribution-Share Alike (Version 3.0 Unported) + + \end{center} + + \goodbreak + Die rekursive Lösung des Problems lautet: + \begin{itemize} + \item + Wenn $n = 1$ ist, also nur eine Scheibe vorliegt, + läßt sich diese "`einfach so"' an den Zielplatz verschieben. + In diesem Fall sind wir direkt fertig. + \item + Wenn $n - 1$ Scheiben als verschiebbar vorausgesetzt werden, + lautet die Vorgehensweise:\\ + verschiebe die oberen $n - 1$ Scheiben auf einen Hilfsplatz,\\ + verschiebe die darunterliegende einzelne Scheibe auf den Zielplatz,\\ + verschiebe die $n - 1$ Scheiben vom Hilfsplatz auf den Zielplatz. + \end{itemize} + + \goodbreak + Dieser Algorithmus läßt sich unmittelbar in eine Programmiersprache übersetzen: + \begin{lstlisting} + void verschiebe (int n, int start, int ziel) + { + if (n == 1) + verschiebe_1_scheibe (start, ziel); + else + { + verschiebe (1, start, hilfsplatz); + verschiebe (n - 1, start, ziel); + verschiebe (1, hilfsplatz, ziel); + } + } + \end{lstlisting} + +\iffalse + + \subsection{Floodfill} + + Siehe die Vortragsfolien \file{ainf-20121220.pdf}\\ + sowie die Beispielprogramme \file{fill-\lstinline{*}.c} + + \subsection{Stack und FIFO} + + Siehe Vortragsfolien und Beispielprogramme + + \subsection{Wegfindungsalgorithmus für Roboterfahrzeug} + + Siehe die Vortragsnotizen \file{ainf-20130117.txt},\\ + die Beispielprogramme \file{text-parcour-\lstinline{*}.c}, + \file{robotext.c} und \file{robotext.h}\\ + sowie die E-Mails "`Wegfindung"' und "`Weg-Finde-Algorithmus"' + +\fi + + \subsection{Aufwandsabschätzungen} + + \subsubsection{Sortieralgorithmen} + + Am Beispiel von Sortieralgorithmen soll hier aufgezeigt werden, + wie man die Lösung eines Problems schrittweise effizienter gestalten kann. + + Als Problem wählen wir das Sortieren eines Arrays (z.\,B.\ von Namen). + + \begin{itemize} + \item + Minimum/Maximum ermitteln: \gitfile{hp}{script}{sort-0.c}, + \gitfile{hp}{script}{sort-1.c} und \gitfile{hp}{script}{sort-2.c} + \item + Selectionsort: \gitfile{hp}{script}{sort-3.c}, \gitfile{hp}{script}{sort-4.c} + \item + Bubblesort: \gitfile{hp}{script}{sort-5.c} + \item + Bubblesort mit Abbruch in äußerer Schleife: \gitfile{hp}{script}{sort-6.c} + \item + Bubblesort mit Abbruch in innerer Schleife: \gitfile{hp}{script}{sort-7.c}, + \gitfile{hp}{script}{sort-7a.c}, \gitfile{hp}{script}{sort-7b.c} + \item + Vergleich mit Quicksort: \gitfile{hp}{script}{sort-8.c}, \gitfile{hp}{script}{sort-8a.c}, \gitfile{hp}{script}{sort-8b.c} + \end{itemize} + + Bei "`zufällig"' sortierten Ausgangsdaten arbeitet Quicksort schneller als Bubblesort. + Wenn die Ausgangsdaten bereits nahezu sortiert sind, ist es umgekehrt. + Im jeweils ungünstigsten Fall arbeiten beide Algorithmen gleich langsam. + + \subsubsection{Landau-Symbole} + + Das Landau-Symbol $\mathcal{O}(g)$ mit einer Funktion $g(n)$ + steht für die \newterm{Ordnung\/} eines Algorithmus', + also die "`Geschwindigkeit"', mit der er arbeitet. + Die Variable $n$ bezeichnet die Menge der Eingabedaten, + hier also z.\,B.\ die Anzahl der Namen. + + \begin{center} + \begin{pdfpic} + \psset{unit=1pt} + \begin{pspicture}(-20,-20)(250,200) + \psline[arrows=->](-10,0)(200,0) + \psline[arrows=->](0,-10)(0,200) + \psplot[plotpoints=200]{1}{125}{2 x 0.06 mul exp} + \put(100,190){\mbox{$g(n) \sim 2^n$}} + \psplot[plotpoints=200]{0}{190}{x x mul 0.005 mul} + \put(190,190){\mbox{$g(n) \sim n^2$}} + \psplot[plotpoints=200]{1}{190}{x ln x mul 0.1 mul} + \put(195,100){\mbox{$g(n) \sim n \log n$}} + \psplot[plotpoints=200]{0}{190}{x 0.4 mul} + \put(195,75){\mbox{$g(n) \sim n$}} + \psplot[plotpoints=200]{1}{190}{x ln 10 mul} + \put(195,50){\mbox{$g(n) \sim \log n$}} + \put(210,0){\makebox(0,0)[l]{$n$}} + \put(0,210){\makebox(0,0)[l]{$g(n)$}} + \end{pspicture} + \end{pdfpic} + \end{center} + + \begin{itemize} + \item + $\mathcal{O}(n)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten linear wächst. + Um doppelt so viele Namen zu sortieren, benötigt das Programm doppelt so lange. + \item + $\mathcal{O}(n^2)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten quadratisch wächst. + Um doppelt so viele Namen zu sortieren, benötigt das Programm viermal so lange. + \item + $\mathcal{O}(2^n)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten exponentiell wächst. + Für jeden Namen, der dazukommt, benötigt das Programm doppelt so lange. + + Ein derartiges Programm gilt normalerweise als inakzeptabel langsam. + \item + $\mathcal{O}(\log n)$ bedeutet, daß die Rechenzeit mit der Menge der Eingabedaten logarithmisch wächst. + Für jede Verdopplung der Namen benötigt das Programm nur einen Rechenschritt mehr. + + Ein derartiges Programm gilt als "`traumhaft schnell"'. + Dies wird jedoch nur selten erreicht. + \item + $\mathcal{O}(1)$ bedeutet, daß die Rechenzeit von der Menge der Eingabedaten unabhängig ist: + 1\,000\,000 Namen werden genau so schnell sortiert wie 10. + + Dies ist nur in Ausnahmefällen erreichbar. + \item + $\mathcal{O}(n \log n)$ liegt zwischen $\mathcal{O}(n)$ und $\mathcal{O}(n^2)$. + + Ein derartiges Programm gilt als schnell. + Viele Sortieralgorithmen erreichen dieses Verhalten. + \end{itemize} + + Wie sieht man einem Programm an, wie schnell es arbeitet? + + \begin{itemize} + \item + Vorfaktoren interessieren nicht. + + Wenn ein Code immer -- also unabhängig von den Eingabedaten -- zweimal ausgeführt wird, + beeinflußt das die Ordnung des Algorithmus nicht. + + Wenn ein Code immer -- also unabhängig von den Eingabedaten -- 1\,000\,000mal ausgeführt wird, + mag das Programm für kleine Datenmengen langsam erscheinen. + Für die Ordnung interessiert jedoch nur das Verhalten für große Datenmengen, + und dort kann dasselbe Programm durchaus schnell sein. + \item + Jede Schleife, die von $0$ bis $n$ geht, + multipliziert die Rechenzeit des innerhalb der Schleife befindlichen Codes mit $n$. + + Eine Doppelschleife (Schleife innerhalb einer Schleife) hat demnach $\mathcal{O}(n^2)$. + \goodbreak + \item + Wenn sich die Grenzen einer Schleife ständig ändern, nimmt man den Durschschnitt. + + Beispiel: + \begin{lstlisting}[gobble=8] + for (int i = 0; i < n; i++) + for (int j = 0; j < i; j++) + ... + \end{lstlisting} + Die äußere Schleife wird immer $n$-mal ausgeführt, + die innere \emph{im Durchschnitt\/} $\frac{n}{2}$-mal, was proportional zu $n$ ist. + + Zusammen ergibt sich $\mathcal{O}(n^2)$. + \item + Bei Rekursionen muß man mitzählen, wie viele Schleifen hinzukommen. + + Bei Quicksort wird z.\,B.\ in jeder Rekursion + eine Schleife von $0$ bis $n$ (aufgeteilt) ausgeführt. + Bei jeder Rekursion wird das Array "`normalerweise"' halbiert, + d.\,h.\ die Rekursionstiefe ist proportional zum Logarithmus von $n$ (zur Basis 2). + Daraus ergibt sich die Ordnung $\mathcal{O}(n\log n)$ für den "`Normalfall"' des Quicksort. + (Im ungünstigsten Fall kann sich auch $\mathcal{O}(n^2)$ ergeben.) + \end{itemize} + + Für eine präzise Definition der Landau-Symbole siehe z.\,B.: + \url{http://de.wikipedia.org/wiki/Landau-Symbole} + \section{Objektorientierte Programmierung} \iffalse diff --git a/script/io-ports-and-interrupts.pdf b/script/io-ports-and-interrupts.pdf new file mode 120000 index 0000000..bcd46f7 --- /dev/null +++ b/script/io-ports-and-interrupts.pdf @@ -0,0 +1 @@ +../common/io-ports-and-interrupts.pdf \ No newline at end of file -- GitLab