■正多角形の近似作図問題(その4)

  x=(3n(n−4)+(n−4)√(n^2+16n−32))/2(2n^2−4n+8)

では数値計算誤差が入り込みやすい.

 そこで,

  x=(3(1−4/n)+(1−4/n)√(1+16/n−32/n^2))/2(2−4/n+8/n^2)

としてみよう.

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 n=12が最も精度が悪く,その後,相対誤差は0に近づくようである.すなわち,かなり精確な近似作図法であるという結論は揺らがない.

n   x/cos(2π/n)

3 1

4 1

5 1.0025

6 1

7 .998036

8 .996724

9 .995919

10 .995468

11 .995254

12 .995195

13 .995236

14 .995338

15 .99548

16 .995643

17 .995817

18 .995994

19 .996171

20 .996344

21 .996511

22 .996672

23 .996825

24 .99697

25 .997107

26 .997238

27 .99736

28 .997476

29 .997585

30 .997689

31 .997786

32 .997877

33 .997964

34 .998046

35 .998123

36 .998196

37 .998265

38 .99833

39 .998392

40 .99845

41 .998506

42 .998558

43 .998609

44 .998656

45 .998701

46 .998744

47 .998785

48 .998825

49 .998862

50 .998897

51 .998931

52 .998964

53 .998995

54 .999024

55 .999053

56 .99908

57 .999106

58 .999131

59 .999155

60 .999178

61 .9992

62 .999221

63 .999242

64 .999261

65 .99928

66 .999298

67 .999316

68 .999333

69 .999349

70 .999364

71 .99938

72 .999394

73 .999408

74 .999422

75 .999435

76 .999448

77 .99946

78 .999472

79 .999483

80 .999494

81 .999505

82 .999516

83 .999526

84 .999535

85 .999545

86 .999554

87 .999563

88 .999572

89 .99958

90 .999588

91 .999596

92 .999604

93 .999611

94 .999619

95 .999626

96 .999633

97 .999639

98 .999646

99 .999652

100 .999658

101 .999664

102 .99967

103 .999676

104 .999681

105 .999687

106 .999692

107 .999697

108 .999702

109 .999707

110 .999712

111 .999717

112 .999721

113 .999726

114 .99973

115 .999734

116 .999738

117 .999742

118 .999746

119 .99975

120 .999754

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