\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fmodern\fprq1 Courier New;}{\f4\fswiss\fprq2 Arial;}{\f5\fswiss\fprq2 Helvetica;}{\f6\froman\fcharset1 Times New Roman;}{\f7\fswiss\fprq2\fcharset1 Arial;}{\f8\fmodern\fprq1\fcharset1 Courier New;}} {\colortbl\red0\green0\blue0;\red0\green128\blue0;\red255\green0\blue0;\red0\green0\blue255;\red0\green0\blue128;} \deflang1031\pard\ri4\plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f3\fs20\cf0 Inhalt....: Das Newton-Verfahren \par Kategorie.: Unterrichtsmaterial \par Mathematik: Analysis \par MuPAD.....: 3.0.0 \par Datum.....: 2002-01-17 \par Autoren...: Vera Verspohl \par Funktionen: solve, plot, plot::Function2d, plot::Scene2d, plot::Line2d, diff, \par Funktionen: ViewingBoxYRange, plot::Point2d \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs36\cf0\b \par \plain\f4\fs40\cf0\b Das Newton-Verfahren\plain\f4\fs36\cf0\b \par \plain\f4\fs22\cf1 \par Dieses Beispiel soll das \plain\f4\fs22\cf1\i Newton-Verfahren\plain\f4\fs22\cf1 mit Hilfe von MuPAD veranschaulichen. \par \par \plain\f4\fs28\cf0 Wir m\'f6chten die Nullstelle des Polynoms \par \tab \tab \tab \tab \tab {\pict\wmetafile8\picw3722\pich1204\picscalex99\picscaley99\picwgoal2125\pichgoal686 010009000003B604000009001C0000000000050000000B0200000000050000000C02B4048A0E03 0000001E00050000000C02BB04A60E050000000B0200000000030000001E00050000000C02C204 B10E050000000B0200000000050000000B0200000000030000001E00050000000C02C904CE0E05 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C02D004D80E050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C02D704F60E050000000B0200000000050000000B02 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}\plain\f4\fs28\cf0 \par bestimmen. Mit schriftlichen Verfahren (der Analysis) kommen \par wir hier nicht weiter. Auch der Rechner ist mit der Funktion \plain\f3\fs28\cf2 solve\plain\f4\fs28\cf2 \plain\f4\fs28\cf0 keine \par Hilfe: \par \plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}f:= x^3 - x^2 - 5; \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 solve(f=0, x)\plain\f4\fs28\cf2 \par \par \pard\ri4\plain\f7\fs28\cf0 Betrachten wir also einmal den Graphen der Funktion: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}p:= plot::Function2d(x^3 - x^2 - 5, x = -1..3, \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 ViewingBoxYRange = -7..7): \par plot(p); \par \pard\ri4\plain\f4\fs28\cf0 \par Am Graphen kann man ablesen, dass sich die Nullstelle im Intervall [2; 3] \par befinden muss. \par \tab \tab \tab \tab \tab {\pict\wmetafile8\picw2124\pich1176\picscalex99\picscaley99\picwgoal1208\pichgoal672 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2D0106001C000000FB021000070000000000BC02000000000102022253797374656D0000D50B0A 7538E91200D89FF177E19FF1772020F377490C668B040000002D010700040000002701FFFF0400 0000F001000004000000F001010004000000F001020004000000F001030004000000F001040004 000000F0010500040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF 040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FF FF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701 FFFF030000000000 }\plain\f4\fs28\cf0 \par ist scheinbar ein recht guter N\'e4herungswert f\'fcr die gesuchte Nullstelle. \par \par Im Punkt \par \tab \tab \tab \tab \tab {\pict\wmetafile8\picw3512\pich1281\picscalex99\picscaley99\picwgoal1999\pichgoal733 0100090000038104000008001C0000000000050000000B0200000000050000000C020105B80D03 0000001E00050000000C020D05C70D050000000B0200000000030000001E00050000000C021005 DD0D050000000B0200000000050000000B0200000000030000001E00050000000C021C05EC0D05 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}\plain\f4\fs28\cf0 \par legen wir die Tangente an den Graphen von \plain\f3\fs28\cf2 f\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}xN:=2.5: \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 f1:=diff(f, x): \par tangente1:= subs(f1, x=xN) * (x-xN) + subs(f, x=xN); \par t1:=plot::Function2d(tangente1, x=-1..3, \par ViewingBoxYRange=-7..13, \par Color=RGB::Red): \par P:=plot::Point2d(xN, subs(f, x=xN), \par PointSize = 2.5*unit::mm): \par plot(p, t1, P)\plain\f3\fs22\cf2 \par \pard\ri4\plain\f4\fs28\cf0 \par Der Schnittpunkt der Tangente mit der x-Achse ist sicher ein besserer \par N\'e4herungswert f\'fcr die Nullstelle. 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}\plain\f4\fs28\cf0 \par berechnen. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}xN1:= xN -(subs(f, x=xN))/(subs(f1, x=xN)); \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}tangente2:= subs(f1, x=xN1) * (x-xN1) + subs(f, x=xN1); \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 t2:=plot::Function2d(tangente2, x=-1..3, \par ViewingBoxYRange=-7..13, \par Color=RGB::Red): \par P2:=plot::Point2d(xN1, subs(f, x=xN1), \par PointSize = 2.5*unit::mm): \par plot(p, t2, P2):\plain\f4\fs28\cf2 \par \plain\f3\fs22\cf2 \par \pard\ri4\plain\f4\fs28\cf0 Wieder betrachten wir den Schnittpunkt der Tangente mit der x-Achse. Die \par N\'e4herung ist wieder deutlich besser geworden. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}xN2:= xN1 -(subs(f, x=xN1))/(subs(f1, x=xN1)); \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}tangente3:= subs(f1, x=xN2) * (x-xN2) + subs(f, x=xN2); \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 t3:=plot::Function2d(tangente3, x=-1..3, \par ViewingBoxYRange=-7..13, \par Color=RGB::Red): \par P3:=plot::Point2d(xN2, subs(f, x=xN2), \par PointSize = 2.5*unit::mm): \par plot(p, t3, P3):\plain\f4\fs28\cf2 \par \plain\f3\fs22\cf2 \par \pard\ri4\plain\f7\fs28\cf0 Beschreiben wir also das Verfahren einmal allgemein: \par \par Gegeben sei eine stetig differenzierbare Funktion \plain\f8\fs28\cf2 f\plain\f7\fs28\cf0 . Wir w\'e4hlen einen \par geeigneten Startpunkt \par \tab \tab \tab \tab \plain\f7\fs28\cf2 {\pict\wmetafile8\picw3624\pich1281\picscalex99\picscaley99\picwgoal2068\pichgoal733 010009000003CA04000009001C0000000000050000000B0200000000050000000C020105280E03 0000001E00050000000C020D05410E050000000B0200000000030000001E00050000000C021005 4F0E050000000B0200000000050000000B0200000000030000001E00050000000C021C05670E05 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C021F05750E050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C022C058F0E050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000030000001E 00050000000C023B059C0E050000000B0200000000050000000B0200000000050000000B020000 0000050000000B0200000000050000000B0200000000050000000B0200000000030000001E0005 0000000C024A05B70E050000000B0200000000050000000B0200000000050000000B0200000000 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040000002D010800040000002701FFFF04000000F001000004000000F001010004000000F00102 0004000000F001030004000000F001040004000000F001050004000000F0010600040000002701 FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF0400000027 01FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF04000000 2701FFFF040000002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f7\fs28\cf2 \par \plain\f7\fs28\cf0 und legen durch diesen eine \par Tangente an den Graphen von \plain\f8\fs28\cf2 f\plain\f7\fs28\cf0 . Ihr Schnittpunkt mit der x-Achse \par bildet nun unseren Wert \plain\f8\fs28\cf2 x2\plain\f7\fs28\cf0 . \par Dementsprechend berechnen wir nun \par \tab \tab \tab \tab {\pict\wmetafile8\picw3491\pich1251\picscalex99\picscaley99\picwgoal1986\pichgoal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}\plain\f7\fs28\cf0 \par und damit den Schnittpunkt mit der x-Achse und einer neuen Tangente \par an den Graphen von \plain\f8\fs28\cf2 f\plain\f7\fs28\cf0 , usw. \par \par Dieses Verfahren konvergiert in den meisten F\'e4llen schnell gegen eine \par gesuchte Nullstelle \plain\f8\fs28\cf2 xN\plain\f7\fs28\cf0 . \par \par Berechnen wir ein weiteres Beispiel: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}f:= sqrt(x+1) - x; \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 f1:= diff(f, x); \par It:= x-f/f1\plain\f4\fs28\cf2 \par \pard\ri4\plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}Newton:= fp::unapply(It, x)\plain\f4\fs28\cf2 \par \pard\ri4\plain\f4\fs28\cf0 \par Damit berechnen wir die Iteration mit dem Startwert \plain\f3\fs28\cf2 x=5\plain\f4\fs28\cf0 : \par \plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}Newton(5)\plain\f4\fs28\cf2 \par \pard\ri4\plain\f4\fs28\cf0 \par Diesen Wert lassen wir uns nochmals als Dezimalbruch von MuPAD \par ausgeben: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}float(Newton(5)) \par \pard\ri4\plain\f4\fs28\cf0 \par Das Ergebnis m\'fcssen wir wieder einsetzen. Das ist folgenderma\'dfen \par m\'f6glich: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}Newton(%) \par \pard\ri4\plain\f4\fs28\cf0 \par Wir k\'f6nnen aber auch mit Hilfe des Operators \plain\f3\fs28\cf2 @@ \plain\f4\fs28\cf0 die Iteration mehrfach \par hintereinander ausf\'fchren; dabei bezeichnet \plain\f3\fs28\cf2 Newton@@i\plain\f4\fs28\cf0 die \plain\f3\fs28\cf2 i\plain\f4\fs28\cf0 -malige, \par iterative Anwendung des Verfahrens. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}float((Newton@@2)(5)); \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 float((Newton@@3)(5));\plain\f3\fs22\cf2 \par \pard\ri4\plain\f4\fs28\cf0 \par Alternativ lassen wir uns mehrere Iterationsschritte hintereinander ausgeben, \par z.B. so: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}n:=6: \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 for i from 1 to n do \par x[i]:= float((Newton@@i)(5)); \par print(Unquoted, "x".expr2text(i)." = ".expr2text(x[i]) ); \par end_for;\plain\f4\fs28\cf2 \par \plain\f3\fs22\cf2 \par \pard\ri4\plain\f4\fs28\cf0 Weitere M\'f6glichkeiten bietet die \plain\f3\fs28\cf2 analysis\plain\f4\fs28\cf0 -Bibliothek. Sie stellt die \par Prozeduren \plain\f3\fs28\cf2 nsNewton\plain\f4\fs28\cf2 \plain\f4\fs28\cf0 zur numerischen Berechnung von Nullstellen, und \par \plain\f3\fs28\cf2 graphNewton\plain\f4\fs28\cf0 , zur grafischen Veranschaulichung des Newton-Verfahrens, \par zur Verf\'fcgung. \par \par Das Paket muss zuerst geladen werden: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}package("analysis"): \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 export(analysis): \par \pard\ri4\plain\f3\fs28\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}delete x: \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 nsNewton(x^3 - x^2 -5, 2.5, 10^-10); \par \pard\ri4\plain\f4\fs28\cf0 \par Hier wurde die Nullstelle der Funktion \par \tab \tab \tab \tab \tab {\pict\wmetafile8\picw3722\pich1176\picscalex99\picscaley99\picwgoal2125\pichgoal672 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}\plain\f4\fs28\cf0 \tab \par mit einer Genauigkeit von 10^(-10) bestimmt. \par \plain\f4\fs20\cf0\b \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}graphIt:= graphNewton(x^3 - x^2 -5, 2.5, -1..4, 0.001): \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 plot(graphIt);\plain\f3\fs22\cf2 \par \pard\ri4\plain\f4\fs28\cf0 \par Es besteht aber auch die M\'f6glichkeit, die einzelnen Tangenten \par nacheinander in verschiedene Schaubilder zu zeichnen und zwar ohne das \par Analysis-Paket: \par \plain\f4\fs20\cf0\b \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}newt:= proc(xStart, f, xBereich, yBereich, n) \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf2 local p1, p2, p3, p4, pSzene, i, f1, tang, phi, fx, w; \par begin \par w:= xStart; \par for i from 1 to n do \par f1:= diff(f, x); \par phi:= fp::unapply(x- f/f1, x): \par fx:= subs(f, x=w): \par tang:= (subs(f1, x=w))*(x-w)+fx: \par p1:= plot::Function2d(f, x=xBereich, \par ViewingBoxYRange=yBereich, \par Color=RGB::Blue): \par p2:= plot::Point2d([w, fx], Color=RGB::Red): \par p3:= plot::Function2d(tang, x=xBereich, \par ViewingBoxYRange=yBereich, \par Color=RGB::Violet): \par p4:= plot::Line2d([w, fx],[w, 0], Color=RGB::Green): \par pSzene:= plot::Scene2d(p1, p2, p3, p4): \par plot(pSzene); \par w:= float((phi@@i)(xStart)): \par end_for; \par end;\plain\f3\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf2 {\pntext\f1\'b7\tab}newt(2.5, x^3-x^2-5, -1..3, -7..13, 3): \par \pard\ri4\plain\f3\fs22\cf3 \par \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf3\b \par \plain\f4\fs22\cf4\b \'dcbungen: \par \plain\f4\fs20\cf4\b 1.\plain\f4\fs20\cf4 Bestimmen Sie N\'e4herungswerte f\'fcr die Nullstellen von f mit Hilfe des Newton Verfahrens. 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6D00006F0C0A6B38E91200D89FF177E19FF1772020F377960C6690040000002D01070004000000 2701FFFF04000000F001000004000000F001010004000000F001020004000000F0010300040000 00F001040004000000F0010500040000002701FFFF040000002701FFFF040000002701FFFF0400 00002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF04 0000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF 040000002701FFFF040000002701FFFF030000000000 }\plain\f4\fs20\cf4 \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf0 \par \plain\f4\fs22\cf1\b Anmerkungen:\plain\f4\fs22\cf1 \par \plain\f4\fs20\cf1\b 1. \plain\f4\fs20\cf1 Der obige Prozess funktionert nat\'fcrlich nicht immer so gut - startet man mit dem Newton-Verfahren an \par einer Stelle, an der die Funktion, deren Nullstelle zu berechnen ist, ein lokales Maximum oder Minimum \par hat, so scheitert es. An diesen Stellen ist die Steigung des Graphen Null und die Tangente ist parallel zur \par x-Achse. Au\'dferdem ben\'f6tigen wir Funktionen, die auf dem jeweiligen Intervall stetig sind. \par \par \plain\f4\fs20\cf1\b 2\plain\f4\fs20\cf1 . Ein \'e4hnliches Notebook steht zur Verf\'fcgung, in dem die \plain\f4\fs20\cf2 Approximation einer Funktion durch ihre \par Taylorreihe\plain\f4\fs20\cf1 graphisch visualisiert wird. \plain\f4\fs20\cf2 \par \plain\f4\fs20\cf1 \par \plain\f4\fs20\cf1\b 3\plain\f4\fs20\cf1 . Weitere Anregungen finden Sie in der Buchreihe \plain\f4\fs20\cf2 Mathematik 1 x anders\plain\f4\fs20\cf1 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f5\fs20\cf3 www.schule.mupad.de/literatur\plain\f4\fs20\cf1 kostenfrei kopiert werden. \par \plain\f4\fs20\cf3 \par \plain\f3\fs20\cf0\b _______________________________________________________________________________ \par \par \par }