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0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 330 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 331 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT 263 6 "Quelle" }}{PARA 331 "" 0 "" {TEXT -1 313 "Dateiname: newton.mws\nDateigr\366\337e: 183 KB\nName : Martin Kaufmann\nSchule: Quenstedt Gymnasium\nKlasse: 12\nDatum: 16. 7.97\nFach: Mathematik\nThema: Newton-Verfahren\nStichw\366rter: Newto n-Verfahren, Iterationsvorschrift, Animation\nKurzbeschreibung: Vorste llung des Newton-Verfahrens mit praktischen Beispielen und Animation" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 21 "Das Newton-Verfahren " }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 258 10 "Einf\374hrung" }}{EXCHG {PARA 285 "" 0 "" {TEXT -1 251 "Wenn wir Funktionen untersuchen, so m\374sse n wir Gleichungen l\366sen wie zum Beispiel f(x)=0 oder f ' (x)=0. Bei einfacheren ganzrationalen Funktionen niederen Grades haben wir hierf \374r L\366sungsformeln, welche uns allgemein das gew\374nschte Ergebn is liefern. " }}{PARA 286 "" 0 "" {TEXT -1 195 "In vielen F\344llen st eht uns jedoch kein solches L\366sungsverfahren zur Verf\374gung. Zun \344chst m\374ssen wir feststellen, ob \374berhaupt eine L\366sung exi stiert. Dabei hilft uns der sogenannte Nullstellensatz:" }}{PARA 288 " " 0 "" {TEXT -1 0 "" }}{PARA 289 "" 0 "" {TEXT 256 16 "Nullstellensatz :" }}{PARA 290 "" 0 "" {TEXT -1 190 "Ist eine Funktion auf einem abges chlossenen Intervall [a;b] stetig und haben f(a) und f(b) verschiedene Vorzeichen, so hat f in [a;b] mindestens eine Nullstelle, d.h. es gib t eine Stelle x* " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 20 " [a;b] mit f(x*)=0. " }}{PARA 291 "" 0 "" {TEXT -1 0 "" }}{PARA 292 "" 0 "" {TEXT -1 250 "Wenn wir wissen, da\337 sich auf einem besti mmten Intervall eine Nullstelle befindet, k\366nnten wir diese durch e infaches \"Ausprobieren\" mit dem Taschenrechner ermitteln. Dies kann \+ jedoch mitunter sehr umst\344ndlich und arbeitsintensiv sein. Hier f \374hrt das " }{TEXT 257 15 "Newtonverfahren" }{TEXT -1 25 " viel schn eller zum Ziel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 297 "" 0 "" {TEXT -1 35 "Herleitung der Iterationsvorschrift" }} {EXCHG {PARA 293 "" 0 "" {TEXT -1 64 "Gegeben sei eine differenzierbar e (also auch stetige Funktion) " }{XPPEDIT 18 0 "f" "I\"fG6\"" } {TEXT -1 31 " . Wir w\344hlen einen Startpunkt " }{XPPEDIT 18 0 "P[1] " "&%\"PG6#\"\"\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "x[1]" "&%\"xG6#\" \"\"" }{TEXT -1 5 " | f(" }{XPPEDIT 18 0 "x[1]" "&%\"xG6#\"\"\"" } {TEXT -1 72 ") ) und legen durch diesen die Tangente. Wir berechnen nu n die Abszisse " }{XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT -1 39 " \+ ihres Schnittpunktes mit der x-Achse. " }}{PARA 295 "" 0 "" {TEXT -1 35 "Nun verfahren wir entsprechend bei " }{XPPEDIT 18 0 "P[2]" "&%\"PG 6#\"\"#" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" } {TEXT -1 5 " | f(" }{XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT -1 31 ") ) und erhalten eine Abszisse " }{XPPEDIT 18 0 "x[3]" "&%\"xG6#\"\"$ " }{TEXT -1 45 " usw. Auf diese Weise ergibt sich eine Folge " } {XPPEDIT 18 0 "x[1]" "&%\"xG6#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x[3]" "&%\" xG6#\"\"$" }{TEXT -1 81 " , ... , die in der Regel schnell gegen die \+ gesuchte Nullstelle x* konvergiert. 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::5:" }}}{EXCHG {PARA 295 "" 0 "" {TEXT -1 140 "Wir wollen nun die allgemeine Iterationsvorschrift (von lat. iter are: wiederholen) herleiten, welche uns jeweils den n\344chsten N\344h erungswert " }{XPPEDIT 18 0 "x[n+1]" "&%\"xG6#,&%\"nG\"\"\"F'F'" } {TEXT -1 33 " zu einem fr\374heren N\344herungswert " }{XPPEDIT 18 0 " x[n]" "&%\"xG6#%\"nG" }{TEXT -1 10 " liefert. " }}{PARA 298 "" 0 "" {TEXT -1 81 "Aus der allgemeinen Tangentenfunktion erhalten wir die Gl eichung der Tangente in " }{XPPEDIT 18 0 "P[n]" "&%\"PG6#%\"nG" } {TEXT -1 3 " ( " }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT -1 6 " | f( " }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT -1 3 "). " }}{PARA 298 "" 0 "" {TEXT -1 11 "Sie lautet:" }}{PARA 294 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(x [n])" "-%\"fG6#&%\"xG6#%\"nG" }{TEXT -1 8 " + f ' (" }{XPPEDIT 18 0 "x [n]" "&%\"xG6#%\"nG" }{TEXT -1 4 " )( " }{XPPEDIT 18 0 "(x-x[n])" ",&% \"xG\"\"\"&F#6#%\"nG!\"\"" }{TEXT -1 3 " ) " }}{PARA 299 "" 0 "" {TEXT -1 158 "Die Abszisse des Schnittpunktes der Tangente mit der x-A chse berechnet man, indem man den obigen Ausdruck gleich Null setzt un d die Gleichung nach x aufl\366st. " }}{PARA 300 "" 0 "" {TEXT -1 0 " " }}{PARA 301 "" 0 "" {TEXT -1 62 "Die Iterationsvorschrift f\374r das Newton-Verfahren lautet also:" }}{PARA 302 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x[n+1]=x[n]-f(x[n]" "/&%\"xG6#,&%\"nG\"\"\"F(F(,&&F$6#F 'F(-%\"fG6#&F$6#F'!\"\"" }{TEXT -1 8 " / f ' (" }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT -1 1 ")" }}{PARA 303 "" 0 "" {TEXT -1 0 "" }} {PARA 304 "" 0 "" {TEXT 261 6 "Achtun" }{TEXT 19 1 "g" }{TEXT 262 1 ": " }{TEXT 19 19 " Der N\344herungswert " }{XPPEDIT 19 0 "x[n+1]" "&%\"x G6#,&%\"nG\"\"\"F'F'" }{TEXT 19 38 " kann nur berechnet werden, wenn f ' (" }{XPPEDIT 19 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT 19 21 ") ungleich N ull ist !" }}{PARA 287 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 9 "Beispiele" }}{EXCHG {PARA 275 "" 0 "" {TEXT -1 25 "Geg eben sei die Funktion " }{XPPEDIT 18 0 "f(x)=sqrt(x+1)-x" "/-%\"fG6#% \"xG,&-%%sqrtG6#,&F&\"\"\"F,F,F,F&!\"\"" }{TEXT -1 95 " . Wir untersuc hen die Funktion mit Hilfe von Maple \374ber das Newton-Verfahren auf \+ Nullstellen. " }}{PARA 275 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->sqrt(x+1)- x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%& arrowGF(,&-%%sqrtG6#,&9$\"\"\"F2F2F2F1!\"\"F(F(" }}}{EXCHG {PARA 276 " > " 0 "" {MPLTEXT 1 0 28 "f1:=unapply(diff(f(x),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$,&9$\" \"\"F0F0#!\"\"\"\"##F0F3F2F0F(F(" }}}{EXCHG {PARA 305 "" 0 "" {TEXT -1 0 "" }}{PARA 306 "" 0 "" {TEXT -1 117 "Nachdem wir die Funktion def iniert und die Ableitung berechnet haben, k\366nnen wir die Iterations vorschrift aufstellen:" }}}{EXCHG {PARA 277 "> " 0 "" {MPLTEXT 1 0 35 "Iteration:=unapply(x-f(x)/f1(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*IterationG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*&,&*$,&F-F .F.F.#F.\"\"#F.F-!\"\"F.,&*$F2#F5F4F3F5F.F5F5F(F(" }}}{EXCHG {PARA 307 "" 0 "" {TEXT -1 0 "" }}{PARA 307 "" 0 "" {TEXT -1 25 "Als Startwe rt w\344hlen wir " }{XPPEDIT 18 0 "x[1]=5" "/&%\"xG6#\"\"\"\"\"&" } {TEXT -1 42 " und berechnen den n\344chsten N\344herungswert:" }}} {EXCHG {PARA 308 "> " 0 "" {MPLTEXT 1 0 13 "Iteration(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"&\"\"\"*&,&*$\"\"'#F%\"\"#F%!\"&F%F%,&F(# F%\"#7!\"\"F%F0F0" }}}{EXCHG {PARA 309 "" 0 "" {TEXT -1 28 "Das geht n och anschaulicher:" }}}{EXCHG {PARA 310 "> " 0 "" {MPLTEXT 1 0 9 "eval f(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$eT`z\"!\"*" }}}{EXCHG {PARA 311 "" 0 "" {TEXT -1 0 "" }}{PARA 311 "" 0 "" {TEXT -1 117 "Den \+ n\344chsten N\344herungswert erh\344lt man, indem man das vorherige Er gebnis einfach in die Iterationsvorschrift einsetzt:" }}}{EXCHG {PARA 312 "> " 0 "" {MPLTEXT 1 0 13 "Iteration(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\">t#>;!\"*" }}}{EXCHG {PARA 313 "" 0 "" {TEXT -1 0 "" }}{PARA 313 "" 0 "" {TEXT -1 368 "Wir k\366nnen Maple auch einwei sen, eine bestimmte Anzahl von Iterationsschritten durchzuf\374hren un d direkt das Ergebnis auszugeben. Im folgenden Beispiel lassen wir den N\344herungswert mit 3 Iterationsschritten berechnen. Man kann deutli ch sehen, da\337 sich bereits mit wenigen Schritten das Ergebnis kaum \+ noch \344ndert. Wir haben also bereits einen recht guten N\344herungsw ert. " }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 0 27 "(Iteration@@3)(5 ):evalf(\");" }}{PARA 314 "" 1 "" {XPPMATH 20 "6#$\"+bS.=;!\"*" }}} {EXCHG {PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 316 "" 0 "" {TEXT -1 281 "Um dies noch zu veranschaulichen, schreiben wir ein kleines Progr amm, in dem Maple die Ergebnisse f\374r mehrere Iterationsschritte unt ereinander auflistet. Auf diese Weise kann man sehr sch\366n beobachte n, wie sich nach mehreren Schritten die einzelnen Kommastellen nicht m ehr \344ndern. " }}{PARA 317 "" 0 "" {TEXT -1 50 "Das folgende Program m geht wiederum vom Startwert " }{XPPEDIT 18 0 "x[1]=5" "/&%\"xG6#\"\" \"\"\"&" }{TEXT -1 60 " aus und listet die Ergebnisse f\374r 6 Iterati onsschritte auf:" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 0 5 "n:=6; " }}{PARA 280 "> " 0 "" {MPLTEXT 1 0 8 "x[1]:=3:" }}{PARA 281 "> " 0 " " {MPLTEXT 1 0 13 "for i to n do" }}{PARA 282 "> " 0 "" {MPLTEXT 1 0 31 "x[i]:=evalf((Iteration@@i)(5)):" }}{PARA 283 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 318 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"'" }}{PARA 319 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$\"+$eT`z\"!\"*" }}{PARA 320 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#$\"+\">t#>;!\"*" }}{PARA 321 "" 1 " " {XPPMATH 20 "6#>&%\"xG6#\"\"$$\"+bS.=;!\"*" }}{PARA 322 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%$\"+*)R.=;!\"*" }}{PARA 323 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&$\"+*)R.=;!\"*" }}{PARA 324 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'$\"+*)R.=;!\"*" }}}{EXCHG {PARA 325 "" 0 "" {TEXT -1 103 "Dieses Beispiel zeigt, da\337 f\374r ein ausreichend \+ genaues Ergebnis bereits wenige Rechenschritte gen\374gen. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 260 9 "Anima tion" }}{EXCHG {PARA 326 "" 0 "" {TEXT -1 111 "Im folgenden lassen wir Maple eine Animation zur Ann\344herung der Tangenten an die gesuchte \+ Nullstelle erzeugen. " }}{PARA 326 "" 0 "" {TEXT -1 0 "" }}{PARA 326 " " 0 "" {TEXT -1 36 "Das im Anschlu\337 definierte Kommando " }{TEXT 0 6 "newton" }{TEXT -1 30 " setzt folgende Syntax voraus:" }}{PARA 326 " " 0 "" {TEXT 0 34 "newton(xs,f,xbereich,ybereich,n) " }}{PARA 327 "" 0 "" {TEXT -1 14 "Hierbei steht " }{TEXT 0 2 "xs" }{TEXT -1 20 " f\374 r den Startwert, " }{TEXT 0 1 "f" }{TEXT -1 43 " stellt die zu untersu chende Funktion dar, " }{TEXT 0 8 "xbereich" }{TEXT -1 37 " steht f \374r den x-Achsenabschnitt und " }{TEXT 0 8 "ybereich" }{TEXT -1 53 " f\374r den y-Achsenabschnitt. Schlie\337lich kann man mit " }{TEXT 0 1 "n" }{TEXT -1 43 " die Anzahl der Iterationsschritte w\344hlen. " }} {PARA 270 "" 0 "" {TEXT -1 0 "" }}{PARA 328 "> " 0 "" {MPLTEXT 1 0 34 "restart:with(student):with(plots):" }}{PARA 256 "> " 0 "" {MPLTEXT 1 0 79 "newton:=proc(xs::numeric,f::anything,xbereich::range,ybereich::r ange,n::posint)" }}{PARA 257 "> " 0 "" {MPLTEXT 1 0 47 "local p1,p2,p3 ,p4,p5,p6,i,f1,tangente,phi,fx,w;" }}{PARA 258 "> " 0 "" {MPLTEXT 1 0 22 " for i from 0 to n do" }}{PARA 273 "> " 0 "" {MPLTEXT 1 0 13 " \+ w[0]:=xs;" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 18 " f1:=diff(f,x); " }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 34 " phi:=unapply(x-f(x)/f1(x) ,x); " }}{PARA 271 "> " 0 "" {MPLTEXT 1 0 32 " w[i+1]:=evalf((phi@@ i)(xs));" }}{PARA 272 "> " 0 "" {MPLTEXT 1 0 26 " fx[i]:=subs(x=w[i ],f);" }}{PARA 261 "> " 0 "" {MPLTEXT 1 0 49 " tangente:=((subs(x=w [i],f1))*(x-w[i])+fx[i]);" }}{PARA 262 "> " 0 "" {MPLTEXT 1 0 80 " \+ p1[i]:=plot(f,x=xbereich,y=ybereich,color=black,title=convert(w[i],str ing));" }}{PARA 263 "> " 0 "" {MPLTEXT 1 0 68 " p2[i]:=plot(\{[w[i] ,fx[i]]\},style=POINT,symbol=CIRCLE,color=RED);" }}{PARA 264 "> " 0 " " {MPLTEXT 1 0 85 " p3[i]:=plot(tangente,x=xbereich,y=ybereich,styl e=line,linestyle=2,color=magenta);" }}{PARA 274 "> " 0 "" {MPLTEXT 1 0 87 " p4[i]:=plot(\{[[w[i],fx[i]],[w[i],0]]\},y=ybereich,style=lin e,linestyle=2,color=red);" }}{PARA 284 "> " 0 "" {MPLTEXT 1 0 53 " \+ p5[i]:=textplot([w[i],fx[i],`P(n)`],align=ABOVE);" }}{PARA 265 "> " 0 "" {MPLTEXT 1 0 84 " p6[i]:=plots[display](\{p1[i],p2[i],p3[i],p4[i ],p5[i]\},view=[xbereich,ybereich]);" }}{PARA 266 "> " 0 "" {MPLTEXT 1 0 5 " od;" }}{PARA 267 "> " 0 "" {MPLTEXT 1 0 52 "plots[display]([s eq(p6[i],i=1..n)],insequence=true);" }}{PARA 268 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 330 "" 0 "" {TEXT -1 0 "" }}{PARA 329 "" 0 "" {TEXT -1 468 "Die Animation kann gestartet werden, indem man mit de r Maus auf das untenstehende Schaubild klickt wird und anschlie\337end den Wiedergabeknopf auf der Symbolleiste dr\374ckt. Iteressant ist be i diesem Beispiel auch die Einzelbildwiedergabe: Man aktiviert sie ein fach, indem man auf das Symbol rechts neben dem Wiedergabepfeil klickt . Man gelangt dann jeweils ein Bild weiter. In der Titelleiste kann ma n zus\344tzlich bei jedem Einzelbild den jeweiligen N\344herungswert a blesen." }}{PARA 329 "" 0 "" {TEXT -1 70 "Die vorgegebenen Werte der A nimation k\366nnen beliebig ver\344ndert werden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 0 33 "newton(2, (sin(x)),0..6,-3..3,10);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%(ANIMATEG6, 7*-%'CURVESG6&7S7$\"\"!$\"1m*>**4\"fTm ro\"F/7$$\"1++]([kdW#F3$\"1?(yWQ6)R;F/7$$\"1+++v;\\DPF3$\"1qlSUfb'e\"F /7$$\"1+++Dq0 ]\"F/$\"1)y`-cLr6\"F/7$$\"1+++DM^I;F/$\"1q&eh4eI1\"F/7$$\"1+++0ytb@F/$\"1))*zPwSof)F37$$\"1++v3wY_AF/$\"1wu8,$QB/) F37$$\"1+++IOTqBF/$\"1\"Qs[m4:b(F37$$\"1++v3\">)*\\#F/$\"1pbteG*H,(F37 $$\"1++DEP/BEF/$\"1/?;6v>+lF37$$\"1++](o:;v#F/$\"16Nth$\\^'fF37$$\"1++ v$)[opGF/$\"1)\\-m83QZ&F37$$\"1++]i%Qq*HF/$\"1,[SZ-$Q%\\F37$$\"1++vQIK HJF/$\"1L15o@L$R%F37$$\"1++D^rZWKF/$\"1Yzxr>79RF37$$\"1++]Zn%)oLF/$\"1 EF[v7$$\"1+++b*=jP%F/$!1nzX#*)=+'zF[v7$$\"1++v3/ 3(\\%F/$!1gZzeoa)H\"F37$$\"1++vB4JBYF/$!1GWs`6&Q#=F37$$\"1+++DVsYZF/$! 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