{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 "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "Times New Roman" -1 256 "" 0 0 0 255 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" 256 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -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 "Headi ng 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 Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 }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 }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 }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 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 1 {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" }}}}{EXCHG {PARA 256 "" 0 "" {TEXT 262 16 "Mathe-Lk Kl.12/1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 13 "Referat 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 " Heron von Alexandrien" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "Heron vo n Alexandrien(ca. 100 n.Chr.) war ein griechischer Verfasser von Werke n zur praktischen Mathematik, Mechanik, Pneumatik, Vermessungskunde un d Gesch\374tzbau. " }}{PARA 0 "" 0 "" {TEXT -1 153 "Ferner beschrieb e r 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 Heron-Algorithmus" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Es sei " }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT -1 34 " ein N\344herungswert f\374r die Wurzel " }{XPPEDIT 18 0 "sqrt(a) " "-%%sqrtG6#%\"aG" }{TEXT -1 8 " (wobei " }{XPPEDIT 18 0 " x[n] " "&%\"xG6#%\"nG" }{TEXT -1 10 " ungleich " }{XPPEDIT 18 0 "sqrt( a) " "-%%sqrtG6#%\"aG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 13 "Dann ist 1/2(" }{XPPEDIT 18 0 "x[n] " "&%\"xG6#%\"nG" }{TEXT -1 6 " + a/(" }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "x[n+1]" "&%\"xG6#,&%\"nG\"\"\"\"\"\"F'" }{TEXT -1 38 " \+ ein erheblich besserer N\344herungswert." }}{PARA 0 "" 0 "" {TEXT -1 20 "In jedem Fall gilt: " }{XPPEDIT 18 0 "x[n+1]" "&%\"xG6#,&%\"nG\"\" \"\"\"\"F'" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "sqrt(a)" "-%%sqrtG6#%\"a G" }}}{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)" "-%%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);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Das ist der Wert, der mit dem Heron-Verfahren berechnet w urde" }}}{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 Wurzel aus`),3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Die Probe ist mit dem gew\366hnlichen Maplebefehl sqrt(n) gel \366st worden" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {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,symb ol=box,color=green,thickness=5,title=`Heron-Verfahren`):\nWurzel:=plot (`Die Wurzel lautet`,1..6,thickness=3):\ndisplay(\{Pkte,Wurzel\});" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Dieser Plot soll nochmals die Ann \344herung an " }{XPPEDIT 18 0 "sqrt(a" "-%%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 Wurzel 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 Rekur sionen nicht l\366sen, weil es bei dem Verfahren von Heron nicht um Di fferenzen, sondern um Quotienten handelt (vgl.geometrische Folge). Ers t 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\})\noption 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! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }