Figure CabriII vers. MacOS 1.1.8

Used macro(s):

Tangente en M, no name
Icon:
0FFFFFFFFFFFFFF0
0FF0000FF0000FF0
0F00000FF00000F0
0F00000FF00000F0
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
0000000FF0000000
00000FFFFFF00000
Mth: 0
CN:3, ON:21, FN:1, PO:20
CT:
point, CS 0, R, W, t, DS:1 1, GT:1, V, nSt
point, CS 0, R, W, t, DS:1 1, GT:1, V, nSt
point, CS 0, R, W, t, DS:1 1, GT:1, V, nSt
Const:
Line, Mth:1, 0, 0, CN:2, VN:2, Const: 1 2
Perp, Mth:0, 0, 0, CN:2, VN:2, Const: 1 4
Cir, Mth:1, 0, 0, CN:2, VN:2, Const: 1 2
Int, Mth:1, 0, 256, CN:2, VN:1, Const: 5 6
PBiss, Mth:0, 0, 0, CN:2, VN:2, Const: 7 2
Perp, Mth:0, 0, 0, CN:2, VN:2, Const: 1 8
Perp, Mth:0, 0, 0, CN:2, VN:2, Const: 3 4
Perp, Mth:0, 0, 0, CN:2, VN:2, Const: 3 10
Int, Mth:0, 0, 0, CN:2, VN:1, Const: 10 4
Int, Mth:0, 0, 0, CN:2, VN:1, Const: 11 5
Seg, Mth:0, 0, 0, CN:2, VN:2, Const: 2 13
Refl, Mth:0, 0, 0, CN:2, VN:1, Const: 13 8
Par, Mth:0, 0, 0, CN:2, VN:2, Const: 15 14
Int, Mth:0, 0, 0, CN:2, VN:1, Const: 5 16
Refl, Mth:0, 0, 0, CN:2, VN:1, Const: 12 9
Vec, Mth:0, 0, 0, CN:2, VN:2, Const: 1 18
Tran, Mth:0, 0, 0, CN:2, VN:1, Const: 17 19
Vec, Mth:0, 0, 0, CN:2, VN:2, Const: 1 20
Tran, Mth:0, 0, 0, CN:2, VN:1, Const: 2 21
Vec, Mth:0, 0, 0, CN:2, VN:2, Const: 1 22
Par, Mth:0, 1, 0, CN:2, VN:2, Const: 3 23, Br, W, t, DS:1 1, GT:0, V, nSt

Figure description:

Window center x: 3.8 y: -1.73_

1: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Val: 0 0

2: Axes, 1, CN:1, VN:3
dGr, W, t, DS:1 1, GT:0, I, nSt
Const: 1, Val: 1 0, 0 1

"O", NP: -81, 110, NS: 14, 12
3: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: -2.4 -3.43_
p: 0, Times, S: 12 C: 6 Fa: 0

4: Line, 0, CN:1, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 3, Val: 1 0

"I", NP: -18, 108, NS: 10, 12
5: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, V, nSt
Const: 4, Val: -0.6 -3.43_
p: 0, Times, S: 12 C: 6 Fa: 0

6: Line, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, I, nSt
Const: 3 5

7: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, V, nSt
Const: 6, Val: -19.95 -3.43_

8: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, V, nSt
Const: 6, Val: 20.1 -3.43_

9: Seg, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, I, nSt
Const: 7 8

"x1", NP: 81, 109, NS: 17, 17
10: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, V, nSt
Const: 9, Val: 2.6 -3.43_
p: 0, Times, S: 12 C: 5 Fa: 0, p: 1, Times, S: 10 C: 5 Fa: 512

11: Perp, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 3 6

12: Cir, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 3 5

"J", NP: -84, 46, NS: 11, 12
13: Int, 256, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 11 12
p: 0, Times, S: 12 C: 6 Fa: 0

14: PBiss, 0, CN:2, VN:2
lBl, W, t, DS:5 8, GT:0, V, nSt
Const: 13 5

15: Perp, 0, CN:2, VN:2
lBl, W, t, DS:5 8, GT:0, V, nSt
Const: 3 14

"(x0,y0)", NP: 27, 24, NS: 40, 17
16: Pt, 0, CN:0, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Val: 0.83_ -0.73_
p: 0, Times, S: 12 C: 3 Fa: 0, p: 2, Times, S: 10 C: 3 Fa: 512, p: 3, Times, S: 12 C: 3 Fa: 0, p: 5, Times, S: 10 C: 3 Fa: 512, p: 6, Times, S: 12 C: 3 Fa: 0

17: Perp, 0, CN:2, VN:2
O, W, t, DS:1 1, GT:0, V, nSt
Const: 16 6

18: Perp, 0, CN:2, VN:2
O, W, t, DS:1 1, GT:0, I, nSt
Const: 16 17

"x0", NP: 27, 105, NS: 17, 17
19: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 17 6
p: 0, Times, S: 12 C: 2 Fa: 0, p: 1, Times, S: 10 C: 2 Fa: 512

"y0", NP: -88, 7, NS: 17, 17
20: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 18 11
p: 0, Times, S: 12 C: 2 Fa: 0, p: 1, Times, S: 10 C: 2 Fa: 512

21: Seg, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, V, nSt
Const: 5 20

"y0", NP: 1, 108, NS: 17, 17
22: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 20 14
p: 0, Times, S: 12 C: 7 Fa: 0, p: 1, Times, S: 10 C: 7 Fa: 512

23: Par, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, V, nSt
Const: 22 21

"y02", NP: -95, -24, NS: 22, 21
24: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 11 23
p: 0, Times, S: 12 C: 9 Fa: 0, p: 1, Times, S: 10 C: 9 Fa: 512, p: 2, Times, S: 10 C: 9 Fa: 256

"-x0", NP: -67, 195, NS: 21, 17
25: Refl, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 19 15
p: 0, Times, S: 12 C: 7 Fa: 0, p: 2, Times, S: 10 C: 7 Fa: 512

26: Vec, 0, CN:2, VN:2
lBl, W, t, DS:1 1, GT:0, V, nSt
Const: 3 25

"y02-x0", NP: -95, 60, NS: 37, 21
27: Tran, 0, CN:2, VN:1
lGr, W, t, DS:1 1, GT:1, V, nSt
Const: 24 26
p: 0, Times, S: 12 C: 5 Fa: 0, p: 1, Times, S: 10 C: 5 Fa: 512, p: 2, Times, S: 10 C: 5 Fa: 256, p: 3, Times, S: 12 C: 5 Fa: 0, p: 5, Times, S: 10 C: 5 Fa: 512

28: Vec, 0, CN:2, VN:2
V, W, t, DS:1 1, GT:0, V, nSt
Const: 3 27

"Pente y'0 de la m�thode d'Euler", NP: -33, 65, NS: 156, 17
29: Tran, 0, CN:2, VN:1
P, W, t, DS:1 1, GT:0, V, nSt
Const: 5 28
p: 0, Times, S: 12 C: 4 Fa: 0, p: 8, Times, S: 10 C: 4 Fa: 512, p: 9, Times, S: 12 C: 4 Fa: 0

30: Vec, 0, CN:2, VN:2
P, W, t, DS:1 1, GT:0, V, nSt
Const: 3 29

31: Par, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, I, nSt
Const: 16 30

32: Perp, 0, CN:2, VN:2
G, W, t, DS:1 1, GT:0, V, nSt
Const: 10 6

"(x1, y'1)", NP: 78, -2, NS: 47, 17
33: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 31 32
p: 0, Times, S: 12 C: 9 Fa: 0, p: 2, Times, S: 10 C: 9 Fa: 512, p: 3, Times, S: 12 C: 9 Fa: 0, p: 7, Times, S: 10 C: 9 Fa: 512, p: 8, Times, S: 12 C: 9 Fa: 0

Ma: Tangente en M, Const: 3 i: 0 5 i: 0 33 i: 0
54: Ma R, F No1, VN:2
Br, W, t, DS:1 1, GT:0, I, nSt

55: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:0, V, nSt
Const: 18 32

56: Vec, 0, CN:2, VN:2
V, W, t, DS:1 1, GT:0, I, nSt
Const: 16 55

57: Tran, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, I, nSt
Const: 33 56

58: Perp, 0, CN:2, VN:2
O, W, t, DS:1 1, GT:0, V, nSt
Const: 57 6

"(x2, y'2)", NP: 135, -59, NS: 47, 17
59: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 54 58
p: 0, Times, S: 12 C: 9 Fa: 0, p: 2, Times, S: 10 C: 9 Fa: 512, p: 3, Times, S: 12 C: 9 Fa: 0, p: 7, Times, S: 10 C: 9 Fa: 512, p: 8, Times, S: 12 C: 9 Fa: 0

"y1 par la m�thode des trap�zes", NP: 52, -29, NS: 149, 17
60: Mid, 0, CN:2, VN:1
V, W, t, DS:1 1, GT:1, V, nSt
Const: 16 59
p: 0, Times, S: 12 C: 5 Fa: 0, p: 1, Times, S: 10 C: 5 Fa: 512, p: 2, Times, S: 12 C: 5 Fa: 0

61: Seg, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 27 29

62: Seg, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 29 5

63: Seg, 0, CN:2, VN:2
P, W, t, DS:1 1, GT:0, V, nSt
Const: 16 33

64: Seg, 0, CN:2, VN:2
P, W, t, DS:1 1, GT:0, V, nSt
Const: 33 59

65: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 31, Val: -4.63_ -3.21358024691358

66: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 31, Val: 5.6 1.42932098765432

67: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 54, Val: 5.41372250725775 2.9

68: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 54, Val: -3.16452773503258 -5.73_

69: Seg, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 65 16

70: Seg, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 33 66

71: Seg, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 59 67

72: Seg, 0, CN:2, VN:2
Br, W, t, DS:1 1, GT:0, V, nSt
Const: 33 68

"x2 = x1 + (x1 - x0)", NP: 106, 117, NS: 94, 17
73: Int, 0, CN:2, VN:1
R, W, t, DS:1 1, GT:1, V, nSt
Const: 6 58
p: 0, Times, S: 12 C: 9 Fa: 0, p: 1, Times, S: 10 C: 9 Fa: 512, p: 2, Times, S: 12 C: 9 Fa: 0, p: 6, Times, S: 10 C: 9 Fa: 512, p: 7, Times, S: 12 C: 9 Fa: 0, p: 12, Times, S: 10 C: 9 Fa: 512, p: 13, Times, S: 12 C: 9 Fa: 0, p: 17, Times, S: 10 C: 9 Fa: 512, p: 18, Times, S: 12 C: 9 Fa: 0

74: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: 225 -65 0, A, nP, TP: 7.5, 2.16_, TS: 6.46_, -8.93_
"Dans cette m�thode de Runge, pour �valuer l'ordonn�e y1 d'un point d'abscisse x1, � partir d'une condition initiale (x0, y0),  on applique la m�thode d'Euler une premi�re fois - ce qui donne le point (x1, y'1), puis on l'applique � nouveau avec le m�me pas, et la condition initiale (x1, y'1), ce qui donne le point (x2, y'2).


On choisit alors pour pente la moynne des pentes, c'est-�-dire qu'en pratique y1 est le milieu du segment d'extr�mit�s (x0, y0) et (x2, y'2).

Runge a remarqu� le premier que cette m�thode donnait une approximation d'ordre 2."
p: 0, Times, S: 12 C: 11 Fa: 0, p: 53, Times, S: 12 C: 5 Fa: 0, p: 54, Times, S: 10 C: 5 Fa: 512, p: 55, Times, S: 12 C: 11 Fa: 0, p: 78, Times, S: 12 C: 5 Fa: 0, p: 79, Times, S: 10 C: 5 Fa: 512, p: 80, Times, S: 12 C: 11 Fa: 0, p: 115, Times, S: 12 C: 3 Fa: 0, p: 118, Times, S: 10 C: 3 Fa: 512, p: 119, Times, S: 12 C: 3 Fa: 0, p: 122, Times, S: 10 C: 3 Fa: 512, p: 123, Times, S: 12 C: 3 Fa: 0, p: 124, Times, S: 12 C: 11 Fa: 0, p: 199, Times, S: 12 C: 9 Fa: 0, p: 202, Times, S: 10 C: 9 Fa: 512, p: 203, Times, S: 12 C: 9 Fa: 0, p: 207, Times, S: 10 C: 9 Fa: 512, p: 208, Times, S: 12 C: 9 Fa: 0, p: 209, Times, S: 12 C: 11 Fa: 0, p: 282, Times, S: 12 C: 9 Fa: 0, p: 285, Times, S: 10 C: 9 Fa: 512, p: 286, Times, S: 12 C: 9 Fa: 0, p: 290, Times, S: 10 C: 9 Fa: 512, p: 291, Times, S: 12 C: 9 Fa: 0, p: 292, Times, S: 12 C: 11 Fa: 0, p: 315, Times, S: 12 C: 9 Fa: 0, p: 318, Times, S: 10 C: 9 Fa: 512, p: 319, Times, S: 12 C: 9 Fa: 0, p: 323, Times, S: 10 C: 9 Fa: 512, p: 324, Times, S: 12 C: 9 Fa: 0, p: 325, Times, S: 12 C: 11 Fa: 0, p: 407, Times, S: 12 C: 5 Fa: 0, p: 408, Times, S: 10 C: 5 Fa: 512, p: 409, Times, S: 12 C: 11 Fa: 0, p: 448, Times, S: 12 C: 3 Fa: 0, p: 450, Times, S: 10 C: 3 Fa: 512, p: 451, Times, S: 12 C: 3 Fa: 0, p: 454, Times, S: 10 C: 3 Fa: 512, p: 455, Times, S: 12 C: 3 Fa: 0, p: 456, Times, S: 12 C: 11 Fa: 0, p: 459, Times, S: 12 C: 9 Fa: 0, p: 462, Times, S: 10 C: 9 Fa: 512, p: 463, Times, S: 12 C: 9 Fa: 0, p: 467, Times, S: 10 C: 9 Fa: 512, p: 468, Times, S: 12 C: 9 Fa: 0, p: 469, Times, S: 12 C: 11 Fa: 0, p: 472, Times, S: 12 C: 4 Fa: 0

75: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 18, Val: -5.83_ -0.73_

76: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 18, Val: 5.23_ -0.73_

77: Seg, 0, CN:2, VN:2
O, W, t, DS:1 1, GT:0, V, nSt
Const: 75 76

78: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 6, Val: 6 -3.43_

79: Pt/, 0, CN:1, VN:3
R, W, t, DS:1 1, GT:1, I, nSt
Const: 6, Val: -6.3 -3.43_

80: Seg, 0, CN:2, VN:2
lGr, W, t, DS:1 1, GT:0, V, nSt
Const: 78 79

81: Text, 0, CN:0, VN:1
B, W, BTh:1, DS:1 1, GT:0, V, nSt
Val: -85 -159 0, A, nP, TP: -2.83_, 5.3, TS: 12.9, -1.13_
"         It�ration sur l'�quation de Riccati y' = y2 - x
M�thode des trap�zes (moyenne des pentes des tangentes)"
p: 0, Times, S: 14 C: 5 Fa: 1, p: 51, Times, S: 12 C: 5 Fa: 257, p: 52, Times, S: 14 C: 5 Fa: 1

82: Seg, 0, CN:2, VN:2
V, W, t, DS:1 1, GT:0, V, nSt
Const: 16 59