\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\froman\fprq2 Times New Roman;}{\f4\fmodern\fprq1 Courier New;}{\f5\fswiss\fprq2 Arial;}{\f6\froman\fcharset1 Times New Roman;}{\f7\fswiss\fprq2\fcharset1 Arial;}{\f8\fswiss\fprq2 Helvetica;}{\f9\fswiss\fprq2 Verdana;}{\f10\froman\fprq2 Symbol;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red255\green0\blue0;\red0\green128\blue0;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f4\fs20\cf0 Inhalt....: Einige N\'e4herungsverfahren zur Berechnung von PI \par Kategorie.: Arbeitsblatt \par Mathematik: Zahlentheorie, Analysis, Numerik \par MuPAD.....: 3.0.0 \par Datum.....: 2002-12-23 \par Autoren...: Roman Seidel \par Funktionen: expr2text, read, write, stringlib::contains, plot::Polygon2d, \par Funktionen: plot::Circle2d\plain\f4\fs20\cf0\b \par ________________________________________________________________________________ \par \plain\f5\fs36\cf0\b \par \plain\f5\fs40\cf0\b Einige N\'e4herungsverfahren zur Berechnung \par von \plain\f10\fs56\cf0 p \par \plain\f5\fs22\cf0 \par \plain\f5\fs24\cf3 Wir wollen MuPAD benutzen, um einige historische Formeln zur Berechnung von PI selbst \par zu programmieren, d.h. wir geben die entsprechende Formel zur Berechnung von PI an und \par definieren sie dann mit MuPAD. Anschlie\'dfend f\'fchren wir einen Vergleich der verschiedenen \par N\'e4herungsverfahren durch. \par \par \plain\f7\fs28\cf0 \par Alle im folgenden vorgestellten Prozeduren zur n\'e4herungsweisen Berechnung \par von PI erhalten stets eine Zahl n - die Anzahl der durchgef\'fchrten Berechnungs- \par schritte nach der entsprechenden Methode - als Eingabe. Vergr\'f6\'dfern dieser \par Zahl erm\'f6glicht stets eine genauere Approximation von PI. \par \par \par \plain\f7\fs40\cf0\b Archimedes \par \plain\f3\fs20 \par \plain\f7\fs28\cf0 Archimedes lebte von ca. 287 bis 212 v.Chr. in Syrakus, wo er geboren wurde \par und auch starb (get\'f6tet wurde). \plain\f5\fs28 Er studierte in Alexandria und war sp\'e4ter \par Mathematiker, Ingenieur und technischer Berater der K\'f6nige. Er entdeckte \par einige mathematische und physikalische Gesetze und berechnete Kreisfl\'e4chen, \par Kreisumfang, Kegelschnitte u.v.m. \par \par Das von ihm verwandte N\'e4herungsverfahren ist gegeben durch die rekursiven \par Zahlenfolgen: \par \par \plain\f4\fs22\cf1 {\pict\wmetafile8\picw2188\pich841\picscalex99\picscaley99\picwgoal1242\pichgoal478 0100090000034004000009001C0000000000050000000B0200000000050000000C0249038C0803 0000001E00050000000C024C039008050000000B0200000000030000001E00050000000C025303 A508050000000B0200000000050000000B0200000000030000001E00050000000C025703A90805 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C025E03BE08050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C026103C108050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000030000001E 00050000000C026803D708050000000B0200000000050000000B0200000000050000000B020000 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0000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f4\fs22\cf1 \par \plain\f5\fs28 \par Sowohl die Folge der a_n als auch die Folge der b_n konvergiert gegen PI. \par \par Geometrisch l\'e4\'dft sich das Approximationsverfahren wie folgt interpretieren: \par Man betrachtet einen Einheitskreis (Radius r = 1), dessen Fl\'e4cheninhalt \par gerade die Zahl PI ist (A = r^2 * PI). In diesen Kreis schreibt man regelm\'e4\'dfige \par n-Ecke ein, deren Fl\'e4cheninhalt explizit berechnet werden kann. Erh\'f6ht man \par sukzessive die Anzahl der Ecken dieser n-Ecke, so n\'e4hert ihr Fl\'e4cheninhalt \par die Zahl PI immer besser an. Ein grafische Veranschaulichung des Verfahrens \par findet sich weiter unten. \par \par \plain\f5\fs28\b\ul Quellen: \par \plain\f7\fs28\cf1 \par http://www.mathe.tu-freiberg.de/~hebisch/cafe/archimedes.html \par http://village.ch/~aeschbacher/archie.html \par \plain\f6\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}archimedes:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local temp1, temp2, i; \par begin \par temp1:= float(2*sqrt(3)); \par temp2:= 3; \par for i from 0 to n do \par temp1:= float( (2*temp1*temp2)/(temp1+temp2) ); \par temp2:= float( sqrt(temp1*temp2) ); \par end_for; \par return(temp1); \par end_proc: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Kreis:= plot::Circle2d(1, [0,0], Color = RGB::Black): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 pi:= float(PI): \par nEck:= (n,c) -> plot::Polygon2d( \par [[sin((2*PI*k)/n), cos((2*PI*k)/n)] $ k=0..n], \par Color = c, Closed = TRUE): \par plot(Kreis, nEck(24, RGB::Red ), nEck(12, RGB::Black), \par nEck( 6, RGB::Green), nEck( 3, RGB::Blue ), \par Scaling=Constrained): \par \par \pard\ri4\plain\f7\fs40\cf0\b Vi\'e9te \par \plain\f5\fs28\cf0 \par Fran\'e7ois Vi\'e9te wurde 1540 in Fontenay-le-Comte geboren. Er studierte Jura \par in Poitiers, doch ab 1564 arbeitete er als Privatlehrer im Dienste einer adligen \par Familie. Ab 1570 war er als Rechtsanwalt in Paris t\'e4tig und wurde 1573 zum \par Mitglied des Parlaments in Rennes ernannt. Ca. 1579 berechnete er mit der \par folgenden Formel PI. Er starb am 23.02.1603 in Paris. \par \par Fran\'e7ois Vi\'e9te n\'e4herte PI \'fcber die folgende Iterationsvorschrift an: \par \par {\pict\wmetafile8\picw10143\pich1566\picscalex99\picscaley99\picwgoal5805\pichgoal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000700000021050100B3012D01C211040000002D01020007000000210501003101E8016E120700 0000210501003201E1026E12040000002D0104000700000021050100C501050251120700000021 050100C50105027312040000002D01030007000000210501002B0156023F13040000002D010200 07000000210501003101E8011C1407000000210501003201E1021C14040000002D010400070000 0021050100C5010502FF130700000021050100C50105022114040000002D010200040000002D01 0400040000002D010200040000002D0104000700000021050100B4012D01B114040000002D0103 000700000021050100D70156027015040000002D010200040000002D010400040000002D010200 040000002D010400040000002D010200040000002D01040007000000210501007201D700DC1507 00000021050100C501D700CC160700000021050100C501D700EE16040000002D01020007000000 210501003101E801E91607000000210501003201E102E916040000002D01040007000000210501 00C5010502CC160700000021050100C5010502EE1608000000FA02000000000000000000000400 00002D0105001C000000FB021000070000000000BC02000000000102022253797374656D0000AC 0C0A5138E91200D89FF177E19FF1772020F3774A0C66C6040000002D010600040000002701FFFF 04000000F001000004000000F001010004000000F001020004000000F001030004000000F00104 00040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701 FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF0400000027 01FFFF040000002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f5\fs28\cf1 . . . \plain\f5\fs28\cf0 \par \par \plain\f5\fs28\b\ul Quellen: \par \plain\f5\fs28\cf0 \par \plain\f5\fs28\cf1 http://www.mathe.tu-freiberg.de/~hebisch/cafe/vieta.html \par \plain\f4\fs28\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}viete:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local ergebnis, i, temp; \par begin \par temp:= sqrt(1/2); \par ergebnis:= temp; \par for i from 0 to n do \par temp:= float( sqrt(1/2+1/2*temp)); \par ergebnis:= float(temp*ergebnis); \par end_for; \par return(2/ergebnis); \par end_proc: \par \par \pard\ri4\plain\f7\fs28\cf0 \par \plain\f7\fs40\cf0\b Madhava, James Gregory, Gottfried Wilhelm \par Leibniz \par \plain\f7\fs28\cf0 \par Um 1658 fanden Madhava, James Gregory und Gottfried Wilhelm Leibniz eine \par Formel zur Berechnung von PI \'fcber eine unendliche Summe der Form: \par \par {\pict\wmetafile8\picw8718\pich1056\picscalex99\picscaley99\picwgoal4986\pichgoal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}\plain\f7\fs28\cf1 . . . \plain\f7\fs28\cf0 \par \plain\f6\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}mgl:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local ergebnis, k; \par begin \par ergebnis:= float( _plus((-1)^k/(2*k+1) $ k=0..n) ); \par return(4*ergebnis); \par end_proc: \par \par \pard\ri4\plain\f7\fs40\cf0\b \par Leonard Euler \par \plain\f7\fs28\cf0 \par Leonard Euler wurde am \plain\f5\fs28 15.04.1707 in Basel geboren und studierte an der \par dortigen Universit\'e4t. Mit 16 Jahren bekam er seinen Magistertitel und wurde \par 1733 Mathematikprofessor. Von 1744 bis 1765 war er Direktor an der \par Universit\'e4t Berlin. In dieser Zeit, ca. 1748, fand er die unten stehende Formel, \par mit der sich auch PI berechnen l\'e4\'dft. Er starb am 15.09.1783. \par \plain\f7\fs28\cf0 \par {\pict\wmetafile8\picw5770\pich1300\picscalex99\picscaley99\picwgoal3301\pichgoal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04000000F0010500040000002701FFFF040000002701FFFF040000002701FFFF040000002701FF FF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701 FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f7\fs28\cf1 . . . \plain\f7\fs28\cf0 \par \par \plain\f5\fs28\b\ul Quellen: \par \plain\f7\fs28\cf0 \par \plain\f7\fs28\cf1 http://www.gm.nw.schule.de/~gymwiehl/prim/euler.htm \par \plain\f6\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}euler:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local ergebnis, k; \par begin \par ergebnis:= float( _plus((1/k^2) $ k=1..n) ); \par return( sqrt(ergebnis*6) ); \par end_proc: \par \par \pard\ri4\plain\f4\fs28\cf2 \par \plain\f5\fs40\b Bailey, Borwein, Plouffe \par \plain\f5\fs28 \par 1997 entdeckten Bailey, Borwein und Plouffe eine Formel, mit der man \par eine einzelne Ziffer der Bin\'e4rdarstellung von PI direkt berechnen kann: \par \par 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}\plain\f5\fs28 \par \par \plain\f5\fs28\b\ul Quellen: \par \plain\f5\fs28 \par \plain\f5\fs28\cf1 http://www.uni-paderborn.de/fachbereich/fachschaft/service/ \par VKOM/VKOM_SS2000/HTML_FILES/vonzurGathenS_Pi.html\plain\f6\fs28\cf1 \par \plain\f6\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}bbp:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local ergebnis, k; \par begin \par ergebnis:= float( _plus( (1/16^k) * ( ( 4/(8*k+1) - \par 2/(8*k+4) - \par 1/(8*k+5) - \par 1/(8*k+6) \par ) ) \par $ k = 0..n ) ); \par return(ergebnis); \par end_proc: \par \par \pard\ri4\plain\f9\fs28\cf0 \par \plain\f5\fs40\b Fabrice Bellard \par 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0000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF 040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FF FF040000002701FFFF030000000000 }\plain\f9\fs28\cf0 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}bellard:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local ergebnis, k; \par begin \par ergebnis:= float( 1/(2^6) * _plus( (-1)^k/(2^(10*k)) * ( \par - 2^5/(4*k+1) - \par 1/(4*k+3) + \par 2^8/(10*k+1) - \par 2^6/(10*k+3) - \par 2^2/(10*k+5) - \par 2^2/(10*k+7) + \par 1/(10*k+9) ) \par $ k = 0..n ) ); \par return(ergebnis); \par end_proc: \par \pard\ri4\plain\f4\fs28\cf2 \par \par \plain\f5\fs28\cf0 Wir schreiben uns nun eine kleine Prozedur, mit deren Hilfe wir die \par verschiedenen N\'e4herungsverfahren, die wir oben vorgestellt haben, \par vergleichen k\'f6nnen. Als Eingabe erh\'e4lt die Prozedur die Anzahl n von \par Iterationsschritten, mit dem jedes der obigen N\'e4herungsverfahren \par aufgerufen werden soll: \par \plain\f4\fs28\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}pi:= proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 local piEcht, List, Namen, temp; \par begin \par DIGITS:= n; \par piEcht:= float(PI); \par List:= [0$7]; \par Namen:= List: \par \par List[1]:= archimedes(n); \par List[2]:= viete(n); \par List[3]:= mgl(n); \par List[4]:= euler(n); \par List[5]:= bbp(n); \par List[6]:= bellard(n); \par \par Namen[1]:= "Archimedes"; \par Namen[2]:= "Viete"; \par Namen[3]:= "Madhava, Gregory, Leibniz"; \par Namen[4]:= "Euler"; \par Namen[5]:= "Bailey, Borwein, Plouffe"; \par Namen[6]:= "Bellard"; \par \par delete DIGITS: \par \par print(Unquoted, "===================================================="); \par print(Unquoted, "Approximation von PI im Vergleich"); \par print(Unquoted, "===================================================="); \par \par for i from 1 to 6 do \par \par print(Unquoted, Namen[i]); \par print(Unquoted, "N\'e4herungswert: ".expr2text(List[i]) ); \par print(Unquoted, "Differenz zu PI: ".expr2text(float(piEcht-List[i]))); \par \par print(Unquoted, "----------------------------------------------------"); \par end_for; \par \par return(); \par end_proc: \par \par \pard\ri4\plain\f5\fs28\cf0 \par Mit Aufrufen der Form \plain\f5\fs28\cf2 pi(3)\plain\f5\fs28\cf0 , \plain\f5\fs28\cf2 pi(4)\plain\f5\fs28\cf0 oder \plain\f5\fs28\cf2 pi(10)\plain\f5\fs28\cf0 k\'f6nnen wir nun schon experimentell \par entdecken, wie gut sich welche der Formeln wirklich zur Berechnung von PI eignet: \par \plain\f4\fs28\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}pi(3) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}pi(4) \par \pard\ri4\plain\f5\fs28\cf0 \par Dass das aktuellste Verfahren - die Methode von Fabrice Bellard - in der Tat ein \par sehr gutes Verfahren darstellt, sehen wir bereits f\'fcr den Fall n = 10: \par \plain\f4\fs28\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}pi(10) \par \pard\ri4\plain\f5\fs28\cf0 \par Die Abweichung zwischen der N\'e4herung, die wir z.B. mit der Formel von Bellard \par berechnet haben, und der Zahl PI (bis auf zehn Nachkommastellen genau) \par ist in diesem Fall bereits kleiner als -4 * 10^-19. Damit gen\'fcgen zehn Iterations- \par schritte nach Bellards Methode, um PI auf zehn Nachkommastellen genau \par bestimmen zu k\'f6nnen. \par \plain\f4\fs28\cf2 \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \plain\f5\fs22\cf0 \par \plain\f5\fs22\cf3\b Anmerkungen:\plain\f5\fs22\cf3 \par \plain\f5\fs20\cf3\b 1\plain\f5\fs20\cf3 . Weitere Anregungen finden Sie in der Buchreihe \plain\f5\fs20\cf2 Mathematik 1 x anders\plain\f5\fs20\cf3 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f8\fs20\cf1 www.schule.mupad.de/literatur\plain\f5\fs20\cf3 kostenfrei kopiert werden. \plain\f5\fs20\cf1 \par \par \plain\f5\fs20\cf3\b 2\plain\f5\fs20\cf3 . Eine genauere Diskussion des Verfahrens von Archimedes findet sich auch in dem Notebook \par "Berechnung_von_PI.mnb", das ebenfalls unter \plain\f5\fs20\cf1 www.schule.mupad.de/material\plain\f5\fs20\cf3 zum Download bereitsteht. \plain\f5\fs20\cf1 \par \plain\f4\fs20\cf0\b _______________________________________________________________________________\plain\f4\fs28\cf0\b \par \plain\f4\fs28\cf2 \par }