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{SECT 0 {EXCHG {PARA 205 "" 0 "" {TEXT 206 15 "M\351thode d'Euler" }
{TEXT 200 0 "" }}{PARA 206 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 207 0 "" }}
{PARA 207 "> " 0 "" {TEXT 208 7 "suite_a" }{TEXT 209 3 "et " }{TEXT 
208 7 "suite_t" }{TEXT 209 10 " sont des " }{TEXT 210 21 "variables de
 stockage" }{TEXT 208 0 "" }{TEXT 208 3 "\nan" }{TEXT 209 4 " et " }
{TEXT 208 2 "tn" }{TEXT 209 9 " sont des" }{TEXT 210 20 " variables de
 calcul" }{TEXT 209 45 " (toute nouvelle valeur \351crase la pr\351c\3
51dente)" }{TEXT 209 0 "" }{TEXT 209 26 "\nA chaque tour de Boucle (" 
}{TEXT 210 1 "i" }{TEXT 209 36 "-i\350me tour), on calcule la nouvelle
 " }{TEXT 210 48 "valeur approch\351e an (correspondant \340 l'indice \+
i)" }{TEXT 209 13 " idem pour tn" }{TEXT 209 0 "" }{TEXT 209 66 "\nPui
s on empile cette nouvelle valeur dans la variable de stockage" }
{TEXT 209 0 "" }{TEXT 209 107 "\nle R\351sultat de la procedure est un
e liste contenant la liste des instants et celles des valeurs approch
\351es" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 1 "\n" }}}{EXCHG {PARA 
208 "> " 0 "" {MPLTEXT 1 211 30 "Euler:=proc(f,t_0,y_0,N,delta)" }
{MPLTEXT 1 211 0 "" }{MPLTEXT 1 211 34 "\n local i,z,tn,an,suite_a,sui
te_t;" }{MPLTEXT 1 211 0 "" }{MPLTEXT 1 211 57 "\n suite_a:=y_0;suite_
t:=t_0;an:=y_0;tn:=t_0;             " }{MPLTEXT 1 211 0 "" }{MPLTEXT 
1 211 62 "\n for i from 1 to N do                                     \+
   " }{MPLTEXT 1 211 0 "" }{MPLTEXT 1 211 57 "\n    an:=an+delta*f(tn,
an); tn:=tn+delta;                " }{MPLTEXT 1 211 0 "" }{MPLTEXT 1 
211 46 "\n   suite_a:=suite_a,an; suite_t:=suite_t,tn; " }{MPLTEXT 1 
211 0 "" }{MPLTEXT 1 211 13 "\n od:        " }{MPLTEXT 1 211 0 "" }
{MPLTEXT 1 211 62 "\n [suite_t],[suite_a];                            \+
            " }{MPLTEXT 1 211 0 "" }{MPLTEXT 1 211 5 "\nend:" }
{MPLTEXT 1 211 0 "" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 207 0 "" 
}}}{EXCHG {PARA 209 "" 0 "" {TEXT 212 20 "Testons la proc\351dure" }
{TEXT 213 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 51 "N:=10: d
elta:=1:t_0:=0;y_0:=1;f:=(t,y)->1:         " }{MPLTEXT 1 0 0 "" }}
{PARA 210 "" 0 "" {TEXT 214 1 "f" }{TEXT 200 55 " est l'expression d\3
51finisant l'equation diff\351rentielle " }{TEXT 215 9 "y'=f(t,y)" }
{TEXT 200 32 "... la condition initiale \351tant " }{TEXT 215 10 "y(t_
0)=y_0" }{TEXT 200 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I$t_0G6\"
\"\"!" }{TEXT 20 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I$y_0G6\"\"
\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 53 "ED
O:=diff(y(t),t)=f(t,y(t));                         " }{MPLTEXT 1 0 0 "
" }}{PARA 210 "" 0 "" {TEXT 200 8 "Voici l'" }{TEXT 215 36 "Equation D
iff\351rentielle de r\351f\351rence" }{TEXT 200 0 "" }}{PARA 211 "" 1 
"" {XPPMATH 20 "6#>I$EDOG6\"/-I%diffGI*protectedGF)6$-I\"yGF%6#I\"tGF%
F.\"\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 
30 "Eul:=Euler(f,t_0,y_0,N,delta):" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 210 "" 0 "" {TEXT 200 10 "Voici les " }{TEXT 215 6 "listes" }
{TEXT 200 20 " correspondants aux " }{TEXT 215 10 "intstants " }{TEXT 
200 15 "cacul\351s et aux " }{TEXT 215 18 "valeurs approch\351es" }
{TEXT 200 27 " obtenues pour ces instants" }{TEXT 200 0 "" }}{PARA 
206 "> " 0 "" {MPLTEXT 1 0 61 "suite_des_temps:=Eul[1];suite_des_valeu
rs_approch\351es:=Eul[2];" }{MPLTEXT 1 0 0 "" }}{PARA 211 "" 1 "" 
{XPPMATH 20 "6#>I0suite_des_tempsG6\"7-\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&
\"\"'\"\"(\"\")\"\"*\"#5" }{TEXT 20 0 "" }}{PARA 211 "" 1 "" {XPPMATH 
20 "6#>I=suite_des_valeurs_approch|dyesG6\"7-\"\"\"\"\"#\"\"$\"\"%\"\"
&\"\"'\"\"(\"\")\"\"*\"#5\"#6" }{TEXT 20 0 "" }}}{EXCHG {PARA 205 "" 0
 "" {TEXT 200 0 "" }}{PARA 205 "" 0 "" {TEXT 200 0 "" }}{PARA 205 "" 0
 "" {TEXT 212 20 "Testons la proc\351dure" }{TEXT 200 0 "" }}}{EXCHG 
{PARA 206 "> " 0 "" {MPLTEXT 1 0 51 "N:=10: delta:=1:t_0:=0;y_0:=1;f:=
(t,y)->y:         " }{MPLTEXT 1 0 0 "" }}{PARA 211 "" 1 "" {XPPMATH 
20 "6#>I$t_0G6\"\"\"!" }{TEXT 20 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 
"6#>I$y_0G6\"\"\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 206 "> " 0 "" 
{MPLTEXT 1 0 51 "EDO:=diff(y(t),t)=f(t,y(t));                       " 
}{MPLTEXT 1 0 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I$EDOG6\"/-I%di
ffGI*protectedGF)6$-I\"yGF%6#I\"tGF%F.F+" }{TEXT 20 0 "" }}}{EXCHG 
{PARA 206 "> " 0 "" {MPLTEXT 1 0 30 "Eul:=Euler(f,t_0,y_0,N,delta):" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 61 "suite
_des_temps:=Eul[1];suite_des_valeurs_approch\351es:=Eul[2];" }
{MPLTEXT 1 0 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I0suite_des_temp
sG6\"7-\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5" }
{TEXT 20 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I=suite_des_valeurs_
approch|dyesG6\"7-\"\"\"\"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%
C5" }{TEXT 20 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 200 34 "On consta
te qu'il s'agit l\340 d'une " }{TEXT 215 29 "suite g\351om\351trique d
e raison 2" }{TEXT 200 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 212 28 "
Calcul approch\351 de exp(-1/2)" }{TEXT 200 0 "" }}}{EXCHG {PARA 206 "
> " 0 "" {MPLTEXT 1 0 53 "N:=10: delta:=1/N:t_0:=0:y_0:=1:f:=(t,y)->-t
*y:      " }{MPLTEXT 1 0 0 "" }}{PARA 210 "" 0 "" {TEXT 200 0 "" }}}
{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 51 "EDO:=diff(y(t),t)=f(t,y(t)
);                       " }{MPLTEXT 1 0 0 "" }}{PARA 210 "" 0 "" 
{TEXT 200 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I$EDOG6\"/-I%diffGI
*protectedGF)6$-I\"yGF%6#I\"tGF%F.,$*&F.\"\"\"F+F1!\"\"" }{TEXT 20 0 "
" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 30 "Eul:=Euler(f,t_0,y_0,
N,delta):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 
1 0 61 "suite_des_temps:=Eul[1];suite_des_valeurs_approch\351es:=Eul[2
];" }{MPLTEXT 1 0 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I0suite_des
_tempsG6\"7-\"\"!#\"\"\"\"#5#F)\"\"&#\"\"$F*#\"\"#F,#F)F0#F.F,#\"\"(F*
#\"\"%F,#\"\"*F*F)" }{TEXT 20 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#
>I=suite_des_valeurs_approch|dyesG6\"7-\"\"\"F'#\"#**\"$+\"#\"%^[\"%+]
#\"'Z0Z\"'++]#\"(T;T\"\"(+Dc\"#\")z6#o#\")++DJ#\"+8afg7\"+++]i:#\"-4MP
Ns6\"-+++]i:#\".2%)e8kp#\".+++]i!R#\"0P]aOOPX#\"0++++]i!R" }{TEXT 20 
0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 215 42 "Augmentons le nombre de
 valeurs approch\351es" }{TEXT 200 54 " calcul\351es sur [0,1] ce qui \+
affine la pr\351cision de la " }{TEXT 215 34 "valeur approch\351e de y
(1)=exp(-1/2)" }{TEXT 200 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 
1 0 21 "N:=1000: delta:=1/N: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 
"> " 0 "" {MPLTEXT 1 0 30 "Eul:=Euler(f,t_0,y_0,N,delta):" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 29 "evalf(Eul[2][N
+1]-exp(-1/2));" }{MPLTEXT 1 0 0 "" }}{PARA 211 "" 1 "" {XPPMATH 20 "6
#$\"(<J-#!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 200 68 
"En calculant la m\351thode d'Euler avec un pas de 1/1000 on obtient u
ne" }{TEXT 215 55 " pr\351cision de l'ordre de 10^(-3) pour la valeur \+
de y(1)" }{TEXT 200 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 18 "point_suite:=NULL:" 
}{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 31 "\nfor N from 10 to 1000 by 50 do"
 }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 45 "\n if N<>0 then delta:=1/N: else
 delta:=0; fi;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 32 "\n Eul:=Euler(f,t
_0,y_0,N,delta):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 30 "\n point_suite:
=point_suite,[N," }{MPLTEXT 1 0 20 "evalf(Eul[2][N+1])];" }{MPLTEXT 1 
0 0 "" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 210 
"" 0 "" {TEXT 200 99 "On trace ici les valeurs de la suite de valeurs \+
approch\351es calcul\351es \340 partir de la m\351thode d'Euler" }}}
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