{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Times New Roman" -1 256 "" 0 0 0 255 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 14 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" 256 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE " " -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "Quelle" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 330 "Dateiname: heron.mws\nDateigr\366\337e: 7 KB\nNam e: Christoph Schill\nSchule: Isolde-Kurz-Gymnasium\nKlasse: Mathe-Lk 1 2\nDatum: 12.06.97\nFach: Mathematik\nThema: Verfahren von Heron\nStic hw\366rter: N\344herungswerte f\374rWurzel\nKurzbeschreibung: Durch st \344ndiges Durchf\374hren eines des Heron-Algorithmuses kommt man zu e inem Naeherungswert der Wurzel\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Update auf Maple 8 am 7.6.2004 | Sebastia n Saur" }}}}{EXCHG {PARA 256 "" 0 "" {TEXT 262 16 "Mathe-Lk Kl.12/1" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 13 "Refer at Nr.1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 258 6 "Thema:" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 259 23 "Das Verfahren von Heron" }{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Heron von Alexandrien" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 163 "Heron von Alexandrien(ca. 100 n.Chr.) war ein gri echischer Verfasser von Werken zur praktischen Mathematik, Mechanik, P neumatik, Vermessungskunde und Gesch\374tzbau. " }}{PARA 0 "" 0 "" {TEXT -1 153 "Ferner beschrieb er den Heronsball, der ein Gef\344\337 \+ mit einer R\366hre darstellt, in der durch eingeblasene, zusammengepre \337te Luft Wasser hochgetrieben wird. " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Prinzip des H eron-Algorithmus" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Es sei " } {XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 34 " ein N\344herungs wert f\374r die Wurzel " }{XPPEDIT 18 0 "sqrt(a) " "6#-%%sqrtG6#%\"aG " }{TEXT -1 8 " (wobei " }{XPPEDIT 18 0 "x[n] " "6#&%\"xG6#%\"nG" } {TEXT -1 10 " ungleich " }{XPPEDIT 18 0 "sqrt(a) " "6#-%%sqrtG6#%\"aG " }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 13 "Dann ist 1/2(" } {XPPEDIT 18 0 "x[n] " "6#&%\"xG6#%\"nG" }{TEXT -1 6 " + a/(" } {XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x[n+1]" "6#&%\"xG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 38 " ein erhebli ch besserer N\344herungswert." }}{PARA 0 "" 0 "" {TEXT -1 20 "In jedem Fall gilt: " }{XPPEDIT 18 0 "x[n+1]" "6#&%\"xG6#,&%\"nG\"\"\"F(F(" } {TEXT -1 3 " > " }{XPPEDIT 18 0 "sqrt(a)" "6#-%%sqrtG6#%\"aG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 33 "Beispiel: N\344herungsverfahren f\374r " }{XPPEDIT 256 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 125 "N\344herung \+ | zugeh\366rige N\344herung durch Division | \+ Neue N\344herung durch Mittelwert " }{TEXT 256 2 " " }} {PARA 0 "" 0 "" {TEXT 256 108 " | \+ | " }}{PARA 0 "" 0 "" {TEXT 256 127 " 1 | \+ 2/1 = 2 | \+ (1 + 2)/2 = 1.5" }}{PARA 0 "" 0 "" {TEXT 256 108 " \+ | \+ |" }}{PARA 0 "" 0 "" {TEXT 256 132 " 1.5 \+ | 2/1.5 = 1.33333 \+ | (1.5 + 1.33333)/2 = 1,41666" }}{PARA 0 "" 0 " " {TEXT 256 109 " | \+ |\031" }}{PARA 0 "" 0 "" {TEXT 256 128 " 1.41666 | 1. 41666/2 = 1.41176 | (1.41666 + 1.41176 )/2 = 1.41421" }}{PARA 0 "" 0 "" {TEXT 256 135 " . \+ . \+ ." }}{PARA 0 "" 0 "" {TEXT 256 130 " u .s.w u.s.w \+ u.s.w " }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 256 128 " . \+ . \+ ." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Automatisierung dieses Algorithmuses" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`Die Wurzel aus`:=144;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Hier kann man nun die Zahl eingeben aus der die Wurzel ge zogen werden soll!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "`Erster Naeherungswert`:=1; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Erste N\344herung kann man st ets bei diesem Wert lassen!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Algorithmus:=(`Erst er Naeherungswert`)->1/2*((`Erster Naeherungswert`)+(`Die Wurzel aus`) /(`Erster Naeherungswert`));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "D as ist nun der Heron-Algorithmus als Funktion geschrieben" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for i to 5 do\nx[i]:=Algorithmus(`Erster Naeherungsw ert`): \n`Erster Naeherungswert`:=Algorithmus(`Erster Naeherungswert`) :\nod;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Diese For-Schleif f \374hrt nun das Prinzip dieses Algorithmuses durch" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "`Die Wurzel lautet`:=evalf(\",3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Das ist der Wert, der mi t dem Heron-Verfahren berechnet wurde" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 10 "Die Probe:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Probe:=evalf(sqrt(`Die Wu rzel aus`),3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Die Probe ist mit dem gew\366hnlic hen Maplebefehl sqrt(n) gel\366st worden" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Graphik" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "Pkte:=plot(\{seq([p,x[p]],p =1..6)\},style=point,symbol=box,color=green,thickness=5,title=`Heron-V erfahren`):\nWurzel:=plot(`Die Wurzel lautet`,1..6,thickness=3):\ndisp lay(\{Pkte,Wurzel\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Dieser P lot soll nochmals die Ann\344herung an " }{XPPEDIT 18 0 "sqrt(a" "6#-% %sqrtG6#%\"aG" }{TEXT -1 17 " veranschaulichen" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 263 18 "Rekursionsrelation" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rr:=f(n+1)=(f(n)+(`Die Wurze l aus`)/f(n))/2;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rsolve(rr,n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Maple kennt \+ diese Folge auch nicht." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 227 "Also \+ Maple kann diese Rekursionen nicht l\366sen, weil es bei dem Verfahren von Heron nicht um Differenzen, sondern um Quotienten handelt (vgl.ge ometrische Folge). Erst richtig definiert kann Maple etwas mit dieser \+ Folge anfangen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f:=proc(n::\{integer\})\nop tion remember;\nif n=1 then 1\nelse (1/2*(f(n-1)+(`Die Wurzel aus`)/f( n-1))) fi\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Wu:=seq(f(i),i=1..10):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "evalf(Wu);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 " Und siehe da, es stimmt!" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK " 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }