\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Helvetica;}{\f4\fswiss\fprq2 Arial;}{\f5\fmodern\fprq1 Courier New;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red255\green0\blue0;\red0\green128\blue0;} \deflang1031\pard\ri4\plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f5\fs20\cf0 Inhalt....: Darstellung iterierter Funktionen \par Kategorie.: Arbeitsblatt \par Mathematik: Analysis, Numerik \par MuPAD.....: 3.0.0 \par Datum.....: 2003-06-23 \par Autoren...: Kai Gehrs \par Funktionen: plot, plot::Function2d, plot::line \par \plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs36\cf0\b \par \plain\f4\fs40\cf0\b Darstellung iterierter Funktionen \plain\f4\fs24\cf3 \par \par Iterationsverfahren spielen nicht nur in der Numerik, sondern auch in der Analysis eine wichtige \par Rolle. In diesem Arbeitsblatt wird eine Prozedur vorgestellt, die es erlaubt, die Iteration von \par Funktionen zu visualisieren. Dank geb\'fchrt an dieser Stelle \plain\f4\fs24\cf3\b OStR Stephan Neuhann\plain\f4\fs24\cf3 vom \par \plain\f4\fs24\cf3\b st\'e4dtischen Gymnasium in Wermelskirchen\plain\f4\fs24\cf3 , der eine Vorlage f\'fcr dieses Arbeitsblatt \par lieferte. \par \par \plain\f4\fs28\cf0 In diesem Arbeitsblatt lernen wir eine Prozedur kennen, die es erlaubt iterierte \par Funktionen darzustellen: Wir w\'e4hlen als Startpunkt eine reelle Zahl \plain\f5\fs28\cf2 x0\plain\f4\fs28\cf0 und eine \par Funktionsgleichung \plain\f5\fs28\cf2 f(x)\plain\f4\fs28\cf0 . \par \par Dann betrachten wir die Folge der reellen Zahlen \plain\f5\fs28\cf2 f(x0)\plain\f4\fs28\cf0 , \plain\f5\fs28\cf2 f(f(x0))\plain\f4\fs28\cf0 , \plain\f5\fs28\cf2 f(f(f(x0)))\plain\f4\fs28\cf0 \par etc. und versuchen \'fcber grafische Visualisierung des Verfahrens etwas \'fcber ein \par m\'f6gliches Konvergenzverhalten der Iteration herauszufinden. \par \par Ein solches Iterationsverfahren, bei dem man eine Funktion wieder und wieder \par auf "sich selbst" anwendet, kann man grafisch recht gut wie folgt visualisieren: \par \par Man zeichnet die Winkelhalbierende \plain\f5\fs28\cf2 y = x\plain\f4\fs28\cf0 zusammen mit der Funktion \plain\f5\fs28\cf2 f\plain\f4\fs28\cf0 in \par ein gemeinsames Koordinatensystem. Die Iteration startet an der Stelle \plain\f5\fs28\cf2 x0\plain\f4\fs28\cf0 . \par Der erste Wert der Folge, die wir berechnen, ist dann \plain\f5\fs28\cf2 f(x0)\plain\f4\fs28\cf0 . Um diesen Schritt \par zu visualisieren, zeichnen wir eine senkrechte Linie vom Startpunkt \plain\f5\fs28\cf2 (x0|x0)\plain\f4\fs28\cf0 zum \par Punkt \plain\f5\fs28\cf2 (x0|f(x0))\plain\f4\fs28\cf0 . Im n\'e4chsten Schritt berechnen wir \plain\f5\fs28\cf2 f(f(x0))\plain\f4\fs28\cf0 und interpre- \par tieren damit den zuletzt berechneten Funktionswert \plain\f5\fs28\cf2 f(x0)\plain\f4\fs28\cf0 wieder als Argument, \par auf das wir \plain\f5\fs28\cf2 f\plain\f4\fs28\cf0 erneut anwenden. Zur Visualisierung zeichnen wir daher zun\'e4chst \par eine waagerechte Linie vom Punkt \plain\f5\fs28\cf2 (x0|f(x0))\plain\f4\fs28\cf0 hin zum Punkt \plain\f5\fs28\cf2 (f(x0)|f(x0))\plain\f4\fs28\cf0 . \par Letzterer Punkt liegt dann genau auf der Winkelhalbierenden \plain\f5\fs28\cf2 y = x\plain\f4\fs28\cf0 . \par Von dort aus k\'f6nnen wir dann wieder eine waagerechte Linie hin zum Punkt \par \plain\f5\fs28\cf2 (f(x0)|f(f(x0)))\plain\f4\fs28\cf0 zeichnen. 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01050008000000FA0200000000000000000000040000002D01050004000000F001060007000000 FC020000FFFFFF000000040000002D01060004000000F00101001C000000FB0210000700000000 00BC02000000000102022253797374656D0000070A0A3838E91200D89FF177E19FF1772020F377 0A0A66C5040000002D01010004000000F001020004000000F001030004000000F0010400040000 002701FFFF030000000000 }}}\plain\f4\fs28\cf0 \par \pard\ri4\plain\f4\fs28\cf0 Im obigen Bild sieht man sch\'f6n: Der "Fixpunkt", auf den die dargestellte \par Iteration hinauszulaufen scheint, ist genau der Schnittpunkt zwischen der \par in \plain\f4\fs28\cf1 blauer Farbe\plain\f4\fs28\cf0 dargestellten Funktion und der in \plain\f4\fs28\cf3 gr\'fcner Farbe\plain\f4\fs28\cf0 darge- \par stellten Winkelhalbierenden \plain\f5\fs28\cf2 y = x\plain\f4\fs28\cf0 . \par \par Nun zu der Prozedur, die solche Darstellungen f\'fcr uns anfertigen wird: \par \plain\f5\fs28\cf2 \par \plain\f4\fs28\cf0 Sie ist in der MuPAD \plain\f5\fs28\cf2 plot\plain\f4\fs28\cf0 -Bibliothek enthalten und hei\'dft \plain\f5\fs28\cf2 plot::Iteration\plain\f4\fs28\cf0 . \par Sie erh\'e4lt standardm\'e4\'dfig \plain\f4\fs28\cf0\i vier Argumente\plain\f4\fs28\cf0 : \par \par \pard\li500\ri4\plain\f5\fs28\cf2\b f\plain\f4\fs28\cf0 : Die zu \plain\f4\fs28\cf0\i iterierende Funktion\plain\f4\fs28\cf0 \plain\f5\fs28\cf2 f\plain\f4\fs28\cf0 . \par \par \plain\f5\fs28\cf2\b Startpunkt\plain\f4\fs28\cf0 : Der \plain\f4\fs28\cf0\i Startpunkt \plain\f4\fs28\cf0 f\'fcr die Iteration. \par \par \plain\f5\fs28\cf2\b n\plain\f4\fs28\cf0 : Die \plain\f4\fs28\cf0\i Anzahl von Iterationsschritten\plain\f4\fs28\cf0 . \par \par \plain\f5\fs28\cf2\b Intervall\plain\f4\fs28\cf0 : Das \plain\f4\fs28\cf0\i Darstellungsintervall\plain\f4\fs28\cf0 f\'fcr die Funktion \plain\f5\fs28\cf2 f\plain\f4\fs28\cf0 . \par \pard\ri4\plain\f4\fs28\cf0 \par Mit Hilfe des folgenden Aufrufs k\'f6nnen wir dann das Bild erzeugen, das \par wir schon oben gesehen haben: \plain\f5\fs28\cf2 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Winkelhalbierende:= plot::Function2d(x, x = 0..1, Color = RGB::DarkGreen): \par {\pntext\f1\'b7\tab}Funktion:= plot::Function2d(3 * x * (1 - x), x = 0..1): \par {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * x * (1 - x), 0.2, 20, x = 0..1): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f4\fs28\cf0 \par Wir erh\'f6hen die Anzahl der Iterationsschritte auf \plain\f5\fs28\cf2 50\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * x * (1 - x), 0.2, 50, x = 0..1): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f5\fs28\cf2 \par \plain\f4\fs28\cf0 Ebenso k\'f6nnen wir nat\'fcrlich auch den Startpunkt modifizieren: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * x * (1 - x), 0.8, 50, x = 0..1): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f4\fs28\cf0 \par Wie wir sehen, scheint eine Ver\'e4nderung des Startpunktes wenig Einfluss \par auf das endg\'fcltige Verhalten der Iteration zu haben. Auch wenn wir \plain\f5\fs28\cf2 0.8\plain\f4\fs28\cf0 als \par Startpunkt w\'e4hlen, bewegt sich die Iteration auf den gleichen Fixpunkt zu. \par \par Wir betrachten daher zum Abschluss noch eine andere Funktion, die Funktion \par f(x) = 2 * sin(x). Als Startpunkt f\'fcr unsere Iteration w\'e4hlen wir die Stelle \plain\f5\fs28\cf2 x0 = \par 0.5\plain\f4\fs28\cf0 und die Anzahl von Iterationen setzen wir auf den Wert \plain\f5\fs28\cf2 n = 30\plain\f4\fs28\cf0 . Als Dar- \par stellungsintervall w\'e4hlen wir \plain\f5\fs28\cf2 0..PI\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Winkelhalbierende:= plot::Function2d(x, x = 0..PI, Color = RGB::DarkGreen): \par {\pntext\f1\'b7\tab}Funktion:= plot::Function2d(2 * sin(x), x = 0..PI): \par {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(2 * sin(x), 0.5, 30, x = 0..PI): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f4\fs28\cf0 \par Wie auf einer unsichtbaren Treppe bewegen sich die Werte der Iteration \par auf den Schnittpunkt zwischen f(x) = 2 * sin(x) und der Ursprungsgerade \par y = x zu. Wir erh\'f6hen den Faktor vor \plain\f5\fs28\cf2 sin(x)\plain\f4\fs28\cf0 von \plain\f5\fs28\cf2 2\plain\f4\fs28\cf0 auf \plain\f5\fs28\cf2 3\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Funktion:= plot::Function2d(3 * sin(x), x = 0..PI): \par {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * sin(x), 0.5, 30, x = 0..PI): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f5\fs28\cf2 \par \plain\f4\fs28\cf0 In diesem Fall scheint sich unsere Iteration jedoch ganz und gar nicht um \par einen festen Wert stabilisieren zu wollen. Wir erh\'f6hen die Anzahl von \par Iterationsschritten von \plain\f5\fs28\cf2 30\plain\f4\fs28\cf0 auf \plain\f5\fs28\cf2 100\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * sin(x), 0.5, 100, x = 0..PI): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration)\plain\f4\fs28\cf2 \par \pard\ri4\plain\f4\fs28\cf0 und von \plain\f5\fs28\cf2 100\plain\f4\fs28\cf0 auf \plain\f5\fs28\cf2 250\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * sin(x), 0.5, 250, x = 0..PI): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f4\fs28\cf0 \par Mit zunehmender Anzahl von Iterationsschritten scheint sich doch allm\'e4hlig \par das "Spinnennetz der Iteration" enger um den Schnittpunkt zwischen der \par Funktion \plain\f5\fs28\cf2 f(x) =\plain\f4\fs28\cf0 \plain\f5\fs28\cf2 3 * sin(x)\plain\f4\fs28\cf0 und der Winkelhalbierenden zu wickeln. \par Der Grafik ist jedoch nicht wirklich anzusehen, ob die Iteration gegen einen \par festen Wert konvergiert. \par \par Eine M\'f6glichkeit, wie man genauer beurteilen kann, ob das Verfahren kon- \par vergiert, besteht einfach darin, den Startpunkt n\'e4her bei dem Punkt zu w\'e4hlen, \par von dem glauben, dass er ein Fixpunkt der Iteration sein k\'f6nnte. Wir vermuten, \par dass sich die berechneten Werte dem Schnittpunkt zwischen \plain\f5\fs28\cf2 3 * sin(x)\plain\f4\fs28\cf0 \par und der Winkelhalbierenden n\'e4hern. An der Grafik lesen wir ab, dass dieser \par Punkt eine \plain\f5\fs28\cf2 x\plain\f4\fs28\cf0 -Koordinate von ungef\'e4hr \plain\f5\fs28\cf2 2.5\plain\f4\fs28\cf0 hat. Wir w\'e4hlen also als Startpunkt \par \plain\f5\fs28\cf2 2.1\plain\f4\fs28\cf0 und verringern die Anzahl von Iterationsschritten zun\'e4chst wieder auf \plain\f5\fs28\cf2 30\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * sin(x), 2.1, 30, x = 0..PI): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f4\fs28\cf0 \par Verwenden wir wieder \plain\f5\fs28\cf2 100\plain\f4\fs28\cf0 Iterationsschritte, so erhalten wir praktisch die \par gleiche Grafik wie oben: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf2 {\pntext\f1\'b7\tab}Iteration:= plot::Iteration(3 * sin(x), 2.1, 100, x = 0..PI): \par {\pntext\f1\'b7\tab}plot(Winkelhalbierende, Funktion, Iteration) \par \pard\ri4\plain\f5\fs28\cf2 \par \plain\f4\fs28\cf0 Es scheint also so zu sein, dass sich die w\'e4hrend der Iteration berechneten \par Werte doch sehr stark vom gew\'e4hlten Startpunkt \plain\f5\fs28\cf2 x0 = 2.1\plain\f4\fs28\cf0 entfernen. Dies \par ist ein Indiz daf\'fcr, dass die Iteration nicht gegen einen Fixpunkt zu konvergieren \par scheint. \par \par \plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf0 \par \plain\f4\fs22\cf3\b Anmerkungen:\plain\f4\fs22\cf3 \par \plain\f4\fs20\cf3\b 1\plain\f4\fs20\cf3 . Wir danken \plain\f4\fs20\cf3\b OStR Stephan Neuhann\plain\f4\fs20\cf3 vom \plain\f4\fs20\cf3\b st\'e4dtischen Gymnasium in Wermelskirchen\plain\f4\fs20\cf3 \par f\'fcr die Anregungen zu diesem Notebook. \par \par \plain\f4\fs20\cf3\b 2\plain\f4\fs20\cf3 . Weitere Anregungen finden Sie in der Buchreihe \plain\f4\fs20\cf2 Mathematik 1 x anders\plain\f4\fs20\cf3 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f3\fs20\cf1 www.schule.mupad.de/literatur\plain\f4\fs20\cf3 kostenfrei kopiert werden. \par \plain\f4\fs20\cf1 \par \plain\f5\fs20\cf0\b _______________________________________________________________________________ \par \par \plain\f4\fs28\cf0 \par \par \plain\f5\fs28\cf2 \par }