{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 19 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 19 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 19 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 20 "Newton-Verfahren zur" }{TEXT 268 1 " " }{TEXT 267 21 "Nu llstellenbestimmung" }}{PARA 258 "" 0 "" {TEXT 266 43 "(Malte Hof, Thi lo Schuler, Bastian Trauter)" }}{PARA 259 "" 0 "" {TEXT -1 29 "Theodor -Heuss-Gymnasium Aalen" }}{PARA 260 "" 0 "" {TEXT -1 33 "Betreuender L ehrer: Hans Bergmann" }}{PARA 261 "" 0 "" {TEXT -1 48 "Email-Adresse: \+ hans.bergmann@thg.aa.bw.schule.de" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 256 10 "Herleitung" }}{SECT 1 {PARA 4 " " 0 "" {TEXT 259 4 "Idee" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Diese s nach dem englischen Physiker und Mathematiker Isaac Newton (1643-172 7) benannte Verfahren berechnet " }{TEXT 262 4 "eine" }{TEXT -1 203 " \+ Nullstellenn\344herung einer Funktion. Das Newton-Verfahren basiert au f dem wiederholten Anlegen einer Tangente an das Schaubild der gegeben en Funktion und wird deshalb auch oft Tangentenverfahren genannt." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Die Funkt ion f, welche untersucht wird mu\337 folgende Bedingungen erf\374llen: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 " \+ - die Funktion f mu\337 im Intervall ]a;b[ differenzierbar sein" }} {PARA 0 "" 0 "" {TEXT -1 76 " - f ' (x) <> 0, d.h. Funktion f is t streng monoton im Intervall [a;b]" }}{PARA 0 "" 0 "" {TEXT -1 86 " \+ - die Funktionswerte f(a) und f(b) m\374ssen von unterschiedliche m Vorzeichen sein" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 656 "Nun legt man die Tangente an den Startpunkt P(b / f(b)) \+ (oder P(a/f(a))) und berechnet die Abszisse (x-Koordinate) des Schnitt punkts dieser Tangente und der x-Achse. Im n\344chsten Schritt berech net man den Funktionswert dieses x-Werts und erh\344lt einen neuen Pun kt P* mit den Koordinaten b* und f(b*). Mit diesem Punkt P* verf\344h rt man wie mit dem Punkt P und man erh\344lt einen neuen Punkt P**. M eistens schon nach wenigen Wiederholungen (Iterationen) dieser Prozedu r erh\344lt man eine akzeptable N\344herung f\374r die gesuchte Nullst elle. Diese hohe \"Geschwindigkeit\" ist einer der Vorteile des Newto n- Verfahrens gegen\374ber z.B. dem Intervallhalbierungsverfahren." }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 8 "Beispiel" }}{PARA 0 "" 0 "" {TEXT -1 239 "Das folgende Diagramm zeigt das Schaubild der Funktion x -> 0.1 x^2-1 und drei nach obigem Verfah ren erhaltenen Tangenten wobei der Startpunkt P (6/2.6) war. Schon na ch drei Iterationen hat man eine recht gute N\344herung f\374r die Nul lstelle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "with(plots):\nf:=x->0.1*x^2-1:\nf1:=D(f):\nx1:=6:\nt 1:=x->f(x1)+f1(x1)*(x-x1):\nx2:=solve(f(x1)+f1(x1)*(x-x1)=0,x):\nt2:=x ->f(x2)+f1(x2)*(x-x2):\nx3:=solve(f(x2)+f1(x2)*(x-x2)=0,x):\nt3:=x->f( x3)+f1(x3)*(x-x3):\nb1:=plot(f(x),x=-2..10,color=black):\nb2:=plot(\{t 1(x),t2(x),t3(x)\},x=-2..10,color=red):\ndisplay(\{b1,b2\});" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7S7$$!\"#\"\"!$!1+++++++g!#;7$$!1++ +]TVQzr(**)F-7$$\"1+++]o-h7F1$!1x!zNG6)4%)F-7$$\"1+++ 5cZ6:F1$!1F^P![Tar(F-7$$\"1++]xq!*QKF17$$\"1,++5zj_nF1$\"1=pNu=\")fNF17$$ \"1****\\<3;%*pF1$\"1E_5a&G=*QF17$$\"1++]Z=iYsF1$\"1Kk1?GN^UF17$$\"1** ****\\'[M\\(F1$\"1#o=qEx^h%F17$$\"1****\\PM&=v(F1$\"1c![;uqG`[)F17$$\"#5F*$\"\"*F*-%' COLOURG6&%$RGBGF*F*F*-F$6$7S7$F($!\"(F*7$F/$!1+++!)47'o'F17$F5$!1+++Bl ,8kF17$F:$!1*****z*>)e5'F17$F?$!1+++'e6nz&F17$FD$!1+++21,*[&F17$FI$!1+ ++6Ct._F17$FN$!1+++jLM3\\F17$FS$!1+++(e^Gg%F17$FY$!1+++F&R$)H%F17$Fhn$ !1*****RW6^)RF17$F]o$!1+++y*>#4PF17$Fbo$!1+++K:j)R$F17$Fgo$!1+++!ynn3$ F17$F\\p$!1+++o#Hiy#F17$Fap$!1+++2:J8DF17$Ffp$!1+++#*ey)=#F17$F[q$!1++ +G,(Q\">F17$F`q$!1+++Rx2%f\"F17$Feq$!1+++)G25J\"F17$Fjq$!1******QTV+5F 17$F_r$!1&*****p0&p/(F-7$Fer$!1(******\\B7'RF-7$Fjr$!1#******yiv7\"F-7 $F_s$\"1/+++J#*G>F-7$Fds$\"1,++IHv.^F-7$Fis$\"1+++I;XnyF-7$F^t$\"1+++% >K_3\"F17$Fct$\"1+++/&)f$R\"F17$Fht$\"1+++yZF&p\"F17$F]u$\"1+++&3ir)>F 17$Fbu$\"1+++uJD6BF17$Fgu$\"1*****>VjCg#F17$F\\v$\"1+++SDR8HF17$Fav$\" 1*****\\_S^>$F17$Ffv$\"1,++#\\lJ]$F17$F[w$\"1*****4)H*Hz$F17$F`w$\"1++ +prwAF17$Fhn$!1F3qx*)fw?F17$F]o$!1)[\\b)\\L+>F 17$Fbo$!1w+SyM!>q\"F17$Fgo$!1-uT#>dE]\"F17$F\\p$!1)4MkZY1J\"F17$Fap$!1 FfO^BGO6F17$Ffp$!1Hi+=aY*G*F-7$F[q$!1g*e\"[\"fI`(F-7$F`q$!1\"H.v5T**[& F-7$Feq$!1')Q\\[aV\"o$F-7$Fjq$!1*3Ly5=sp\"F-7$F_r$\"1Lh-q-EA>Fcr7$Fer$ \"1^y*fwiO;#F-7$Fjr$\"1B/M3r0uRF-7$F_s$\"1$GPB)>\"o#fF-7$Fds$\"1,4m%[v ^&zF-7$Fis$\"1JSF17$Fgu$\"1'>L\\T H@8#F17$F\\v$\"1c#\\scy2L#F17$Fav$\"1OpJCUy5DF17$Ffv$\"1#\\;K&zd2FF17$ F[w$\"1cqh:bu#*GF17$F`w$\"1fN$>(*)H'3$F17$Few$\"1qP$Q&G`vKF17$Fjw$\"1w )**3>VOZ$F17$F_x$\"1>mY\"*pWkOF17$Fdx$\"1^'\\fVr&fQF17$Fix$\"1dh^^,3`S F17$F^y$\"1HS3kX*3B%F17$Fcy$\"1vHBZ**oMWF17$Fhy$\"1\"z$Qnp'ph%F17$F]z$ \"1eNy)G78\"[F17$Fbz$\"1$Qn#fbK(*\\F17$Fgz$\"1,++AAA(>&F1Fedl-F$6$7S7$ F($!1++!GB**eK$F17$F/$!1Sv5axRdJF17$F5$!1U(p%=by5IF17$F:$!1')**zDZ!f%G F17$F?$!1%Q3i*4$*zEF17$FD$!1`9y5hu9DF17$FI$!1O^]nIuF-7$F[q$!1E)*[^*G[&fF-7$F`q$!1eE#e3p!Q UF-7$Feq$!1)pU4rX%=FF-7$Fjq$!1.4N@!y60\"F-7$F_r$\"1\\s.O4bk`Fcr7$Fer$ \"1t\"3ap$)H>#F-7$Fjr$\"1W$GT1%>9PF-7$F_s$\"1\"fv^&\\-b`F-7$Fds$\"13H& Q5(QfqF-7$Fis$\"1-lbk$RIa)F-7$F^t$\"1E)4_WUX,\"F17$Fct$\"1`A5/X3!=\"F1 7$Fht$\"18+^7^.U8F17$F]u$\"1c'*[`1t)\\\"F17$Fbu$\"1m\"*)eL9Fn\"F17$Fgu $\"14?w7k/H=F17$F\\v$\"1\\uBtU'f*>F17$Fav$\"1[-qjm@Z@F17$Ffv$\"1>xYxad 7BF17$F[w$\"18#4XBl\"oCF17$F`w$\"1i#o90()f;JF17$Fdx$\"1BJAy[b!G$F17$Fix$\"1I:T6M :VMF17$F^y$\"1#G\"ewXc#f$F17$Fcy$\"1er!Hj1Qw$F17$Fhy$\"1&Rw%pv'p\"RF17 $F]z$\"13E(H2p-3%F17$Fbz$\"1@3i[%plB%F17$Fgz$\"1+++[b`/WF1Fedl-%+AXESL ABELSG6$%\"xG%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 2 474 474 474 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 142 312 0 0 0 0 0 0 }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 20 "Iterationsvorschrift" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Die " }{TEXT 263 31 "allge meine Iterationsvorschrift" }{TEXT -1 78 " (d.h. wie man von der n-ten N\344herung auf die n\344chste N\344herung kommt) f\374r das " } {TEXT 264 16 "Newton-Verfahren" }{TEXT -1 8 " lautet:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "x[n+1]=x[n]-f(x[n] )/`f '`(x[n])" "/&%\"xG6#,&%\"nG\"\"\"F(F(,&&F$6#F'F(*&-%\"fG6#&F$6#F' F(-%$f~'G6#&F$6#F'!\"\"F7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Man erh\344lt dies e Vorschrift indem man:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " 1. die Gleichung der Tangenten im Punkt Pn(xn/ f(xn)) aufstellt:" }}{PARA 0 "" 0 "" {TEXT -1 66 " y-y1 = m(x -x1) (Punkt-Steigungs-Form einer Geraden)" }}{PARA 0 "" 0 "" {TEXT -1 21 " m = f '(xn)" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ x1 = xn" }}{PARA 0 "" 0 "" {TEXT -1 20 " y1 = f(xn)" } }{PARA 0 "" 0 "" {TEXT -1 37 " => y = f(xn)+f '(xn) (x-xn)" } }{PARA 0 "" 0 "" {TEXT -1 99 " 2. diese Tangentengleichung Null g leichsetzt um den Schnittpunkt mit der x-Achse zu bekommen:" }}{PARA 0 "" 0 "" {TEXT -1 62 " f(xn)+f '(xn) (x-xn) = 0 <=> x = xn \+ - f(xn)/f '(xn)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT 260 9 "Probleme\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 241 "Aus der Iterationsvorschrift geht schon hervor, da\337 f ' (xn) <> O sein mu\337. Aber auch wenn alle Beding ungen (siehe auch unter Idee) erf\374llt sind kann ein Fall auftreten \+ bei dem durch das Newton-Verfahren keine N\344herung gefunden werden k ann." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Be ispiel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "restart;\nwith(plots): \nf:=x->arctan(x):\nf1:=D(f):\nx1:=2:\nt1:=x->f(x1)+f1(x1)*(x-x1):\nx2 :=solve(f(x1)+f1(x1)*(x-x1)=0,x):\nt2:=x->f(x2)+f1(x2)*(x-x2):\nx3:=so lve(f(x2)+f1(x2)*(x-x2)=0,x):\nt3:=x->f(x3)+f1(x3)*(x-x3):\nb1:=plot(f (x),x=-20..20,color=black):\nb2:=plot(\{t1(x),t2(x),t3(x)\},x=-20..20, color=red):\ndisplay(\{b1,b2\});" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CU RVESG6$7gn7$$!#?\"\"!$!1aH2Jz$3_\"!#:7$$!1LLL$Q6G\">!#9$!1Mo%p![c=:F-7 $$!1nm;M!\\p$=F1$!1Q*Gb\">T;:F-7$$!1LLL))Qj^qR/]\"F-7$$!1LL$3WDTL\"F1$!1qfu f2)f\\\"F-7$$!1++]d(Q&\\7F1$!1#zF&)*p$4\\\"F-7$$!1nmmc4`i6F1$!1wD\"fL) )\\[\"F-7$$!1LLLQW*e3\"F1$!1\"Q1DNl*y9F-7$$!1*******p)>'***F-$!1%f*\\B +4r9F-7$$!1*******\\5*H\"*F-$!15T(p5,#3 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