| 1 | PROGRAM WURZEL2EXAKT(INPUT,OUTPUT); |
| 2 | |
| 3 | |
| 4 | (*) |
| 5 | ( QUADRATWURZEL AUS 2 AUF BIS ZU 1000 STELLEN GENAU |
| 6 | ( BERECHNET MITTELS DER |
| 7 | ( |
| 8 | ( BINOMIALREIHE SQRT(1+X) = |
| 9 | ( |
| 10 | ( (1/2) (1/2) (1/2) |
| 11 | ( = 1 + ( ) * X + ( ) * X^2 + ... + ( ) * X^K + ... |
| 12 | ( ( 1 ) ( 2 ) ( K ) |
| 13 | ( |
| 14 | ( |
| 15 | ( KONVERGENZBESCHLEUNIGUNG WIRD DURCH |
| 16 | ( |
| 17 | ( SQRT(2) = 10/7 * SQRT(2 * 49/100) = 10/7 * SQRT(1 - 2/100) |
| 18 | ( |
| 19 | ( ERREICHT, DAMIT IST X = -0.02 UND ALLE GLIEDER |
| 20 | ( DER REIHE (BIS AUF 1) SIND NEGATIV. |
| 21 | ( |
| 22 | ( SOMIT ERGIBT SICH: |
| 23 | ( |
| 24 | ( S Q R T ( 2 ) = ( 1 - REIHENSUMME ) * 10 / 7 |
| 25 | ( |
| 26 | (*) |
| 27 | |
| 28 | CONST BASE (*OF ARITHMETIC*) = 100000 (*DON'T CHANGE THIS!*); |
| 29 | |
| 30 | TYPE LONGNUMBER = ARRAY[0..205] OF INTEGER; |
| 31 | |
| 32 | VAR NUMBEROFDIGITS, FIRST, LAST, I, K, M, D, |
| 33 | DIGIT, CARRY, POWER: INTEGER; |
| 34 | A (*GLIED*), |
| 35 | S (*SUMME*): LONGNUMBER; |
| 36 | |
| 37 | |
| 38 | BEGIN |
| 39 | READ(NUMBEROFDIGITS); |
| 40 | FIRST := 1; LAST := NUMBEROFDIGITS DIV 5 + 4; |
| 41 | |
| 42 | (*) |
| 43 | ( AUFSTELLEN DES ANFANGSGLIEDES DER REIHE ( = 0.01 ) |
| 44 | ( UND DES ANFANGSWERTES DER SUMME |
| 45 | (*) |
| 46 | |
| 47 | K := 1; |
| 48 | FOR I := 0 TO LAST DO |
| 49 | BEGIN A[I] := 0; S[I] := 0 END; |
| 50 | A[1] := 10000; S[1] := 10000; |
| 51 | A[LAST+1] := BASE (* GUARD *); |
| 52 | |
| 53 | |
| 54 | REPEAT K := K + 1; |
| 55 | |
| 56 | (*) |
| 57 | ( REKURSIVE BERECHNUNG DES NAECHSTEN GLIEDES DER REIHE: |
| 58 | ( |
| 59 | ( A <=== A * (2*K - 3) / 100 / K |
| 60 | ( |
| 61 | (*) |
| 62 | |
| 63 | M := K+K - 3; |
| 64 | I := LAST+1; CARRY := 0; |
| 65 | WHILE (FIRST<I) OR (CARRY<>0) DO |
| 66 | BEGIN |
| 67 | I := I - 1; |
| 68 | DIGIT := M*A[I] + CARRY; |
| 69 | A[I] := DIGIT MOD BASE; |
| 70 | CARRY := DIGIT DIV BASE; |
| 71 | END; |
| 72 | FIRST := I; |
| 73 | |
| 74 | D := 100*K; |
| 75 | CARRY := 0; |
| 76 | FOR I := FIRST TO LAST DO |
| 77 | BEGIN |
| 78 | DIGIT := CARRY*BASE + A[I]; |
| 79 | A[I] := DIGIT DIV D; |
| 80 | CARRY := DIGIT MOD D |
| 81 | END; |
| 82 | |
| 83 | WHILE A[FIRST]=0 DO FIRST := FIRST + 1; |
| 84 | |
| 85 | (*) |
| 86 | ( AUFSUMMIERUNG SUMME <=== SUMME + GLIED |
| 87 | (*) |
| 88 | |
| 89 | I := LAST; CARRY := 0; |
| 90 | WHILE (FIRST<=I) OR (CARRY<>0) DO |
| 91 | BEGIN |
| 92 | DIGIT := S[I] + A[I] + CARRY; CARRY := 0; |
| 93 | IF DIGIT >= BASE THEN BEGIN |
| 94 | DIGIT := DIGIT - BASE; |
| 95 | CARRY := 1 |
| 96 | END; |
| 97 | S[I] := DIGIT; |
| 98 | I := I - 1 |
| 99 | END |
| 100 | |
| 101 | UNTIL FIRST>LAST; |
| 102 | |
| 103 | (*) |
| 104 | ( SUBTRAHIERE SUMME DER REIHE VON 1 |
| 105 | (*) |
| 106 | |
| 107 | I := LAST; |
| 108 | WHILE S[I]=0 DO I := I-1; |
| 109 | S[I] := BASE - S[I]; |
| 110 | FOR I := I-1 DOWNTO 1 DO S[I] := BASE-1 - S[I]; |
| 111 | S[0] := 9; |
| 112 | |
| 113 | (*) |
| 114 | ( DIVISION DURCH 7 |
| 115 | (*) |
| 116 | |
| 117 | CARRY := 0; |
| 118 | FOR I := 0 TO LAST DO |
| 119 | BEGIN |
| 120 | DIGIT := CARRY*BASE + S[I]; |
| 121 | S[I] := DIGIT DIV 7; |
| 122 | CARRY := DIGIT MOD 7 |
| 123 | END; |
| 124 | |
| 125 | (*) |
| 126 | ( A U S G A B E : |
| 127 | (*) |
| 128 | |
| 129 | WRITELN; WRITE("SQRT(2) = 1."); |
| 130 | FOR I := 1 TO LAST-4 DO |
| 131 | BEGIN |
| 132 | DIGIT := S[I]; POWER := BASE; |
| 133 | REPEAT POWER := POWER DIV 10; |
| 134 | WRITE(DIGIT DIV POWER:1); |
| 135 | DIGIT := DIGIT MOD POWER |
| 136 | UNTIL POWER=1; |
| 137 | IF I MOD 10 = 0 THEN BEGIN WRITELN; WRITE(" ":12) END |
| 138 | END; |
| 139 | WRITELN |
| 140 | END. |