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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 202 10 "Exercice 1" }{TEXT 203 
0 "" }}{PARA 201 "> " 0 "" {TEXT 204 11 "Question a)" }{MPLTEXT 1 205 
0 "" }{MPLTEXT 1 205 2 "\n " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {TEXT 206 
17 "M\351thode it\351rative" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "
\n   " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
205 24 "iteration:=proc(a,b,n)  " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 23 "\n local s,an,bn,temp,k;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 15 "\n an:=a;bn:=b; " }{TEXT 207 73 "initialisation des valeurs de
 la suite an et bn repr\351sentent ici a0 et bo" }{MPLTEXT 1 205 0 "" 
}{MPLTEXT 1 205 24 "\n for k from 1 to n do  " }{TEXT 207 0 "" }{TEXT 
207 138 "\n    on utilise des variables temporaires atemp et btemp pou
r ne pas interferer dans le calcul du nouveau an et bn en fonction des
 anciens" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 36 "\n  temp:=sqrt(an*b
n);bn:=(an+bn)/2; " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 14 "\n  an:=t
emp;  " }{TEXT 207 62 "les anciennes valeurs de an et bn deviennent ic
i les nouvelles" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nod:" }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 10 "\n [an,bn];" }{MPLTEXT 1 205 
0 "" }{MPLTEXT 1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 21 "iteration(0.5,1,25); " }{TEXT 207 11 "Calcul de [" 
}{TEXT 206 11 "a(25),b(25)" }{TEXT 207 7 "] pour " }{TEXT 206 14 "a0=0
.5 et b0=1" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#7$
$\"+c^&RG(!#5F$" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {TEXT 206 17 "M\351t
hode r\351cursive" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 39 "iteration:=proc(a,b,n) option remember;" }{MPLTEXT 
1 205 0 "" }{MPLTEXT 1 205 12 "\n local s,u;" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 23 "\n if n=0 then s:=[a,b] " }{TEXT 207 80 "la recursi
on doit pouvoir s'arr\352ter pour donner une valeur effective ici [a0,
b0]" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 8 "\n  else " }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 29 "\n     u:=iteration(a,b,n-1); " }{TEXT 
207 82 "on fait appel \340 notre propre proc\351dure pour connaitre le
s valeurs a(n-1) et b(n-1)" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 42 "
\n     s:=[sqrt(u[1]*u[2]),(u[1]+u[2])/2]; " }{TEXT 207 22 " on en d\3
51duit an et bn" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nfi:" }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\n s;" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> \+
" 0 "" {MPLTEXT 1 205 1 " " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "
> " 0 "" {MPLTEXT 1 205 21 "iteration(0.5,1,25); " }{TEXT 207 11 "Calc
ul de [" }{TEXT 206 11 "a(25),b(25)" }{TEXT 207 7 "] pour " }{TEXT 
206 14 "a0=0.5 et b0=1" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#7$$\"+c^&RG(!#5F$" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 
1 205 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 209 10 "Question b" }
{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {TEXT 206 29 "Calcul d'un
e limite approch\351e" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0
 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 203 13 "M\35
1thode naive" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 
1 205 25 "limite:=proc(a,b,epsilon)" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 17 "\nlocal an,bn,u,n;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 18 "
\nan:=a;bn:=b;n:=0;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 33 "\nwhile \+
is(abs(an-bn)>epsilon) do " }{TEXT 207 92 "tant que la distance s\351p
arant an, bn (suites adjacentes) d\351passe epsilon, le calcul continu
e" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 47 "\n n:=n+1:u:=iteration(a,b
,n);an:=u[1];bn:=u[2];" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nod;"
 }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 17 "\n['an'=an,'n'=n];" }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}
{PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 25 "limite(0.5,3,1/1000000); " }{TEXT 207 19 "limite ap
proch\351e \340 " }{TEXT 206 7 "10^(-6)" }{TEXT 207 10 " pr\350s avec"
 }{TEXT 206 15 " a0=0.5 et b0=3" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 
1 "" {XPPMATH 20 "6#7$/I#anG6\"$\"+rVmv9!\"*/I\"nGF&\"\"%" }{TEXT 208 
0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG 
{PARA 203 "" 0 "" {TEXT 203 17 "M\351thode it\351rative" }{TEXT 203 0 
"" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 26 "limite:=proc(a,b,e
psilon) " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 23 "\n local s,an,bn,te
mp,n;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 15 "\n an:=a;bn:=b; " }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 6 "\nn:=0;" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 33 "\n while is(abs(an-bn)>epsilon) do" }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 46 "\n     n:=n+1:temp:=sqrt(an*bn);bn:=(an+b
n)/2; " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 13 "\n  an:=temp; " }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nod:" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 17 "\n['an'=an,'n'=n];" }{MPLTEXT 1 205 0 "" }{MPLTEXT 
1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "
> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
205 25 "limite(0.5,3,1/1000000); " }{TEXT 207 19 "limite approch\351e \+
\340 " }{TEXT 206 7 "10^(-6)" }{TEXT 207 10 " pr\350s avec" }{TEXT 
206 15 " a0=0.5 et b0=3" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#7$/I#anG6\"$\"+rVmv9!\"*/I\"nGF&\"\"%" }{TEXT 208 0 "" 
}}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {TEXT 207 17 "M\351thode r\351cursive" }{MPLTEXT 1 205 0 "" 
}}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 42 "limite:=proc(a,b,epsi
lon) option remember;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 12 "\n loc
al s,u;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 44 "\n if is(abs(a-b)<=e
psilon) then s:=[a,b,0]; " }{TEXT 207 80 "la recursion doit pouvoir s'
arr\352ter pour donner une valeur effective ici [a0,b0]" }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 8 "\n  else " }{MPLTEXT 1 205 0 "" }{MPLTEXT 
1 205 43 "\n    s:=limite(sqrt(a*b),(a+b)/2,epsilon); " }{TEXT 207 22 
" on en d\351duit an et bn" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 27 "
\n    s:=[s[1],s[2],s[3]+1];" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "
\nfi:" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\n s;" }{MPLTEXT 1 205 
0 "" }{MPLTEXT 1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 25 "limite(0.5,3,1/1000000); " }{TEXT 207 19 "limite ap
proch\351e \340 " }{TEXT 206 7 "10^(-6)" }{TEXT 207 10 " pr\350s avec"
 }{TEXT 206 15 " a0=0.5 et b0=3" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 
1 "" {XPPMATH 20 "6#7%$\"+rVmv9!\"*$\"+sVmv9F&\"\"%" }{TEXT 208 0 "" }
}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "
> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {TEXT 204 
10 "Question c" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 203 "" 0 "" 
{TEXT 203 13 "M\351thode na\357ve" }{TEXT 203 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 205 19 "erreur:=proc(a,b,n)" }{MPLTEXT 1 205 
0 "" }{MPLTEXT 1 205 22 "\n local souvenir,i,un;" }{MPLTEXT 1 205 0 ""
 }{MPLTEXT 1 205 16 "\n souvenir:=b-a;" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 21 "\nfor i from 1 to n do" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 54 "\n un:=iteration(a,b,i);souvenir:=souvenir,un[2]-un
[1];" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nod;" }{MPLTEXT 1 205 
0 "" }{MPLTEXT 1 205 7 "\n[sol];" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 5 "\nend:" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 1 "\n" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 205 21 "evalf(erreur(1,3,5));" }
{MPLTEXT 1 205 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#7($\"\"#\"\"!$
\"*#>\\zE!\"*$\"('o:[F)$\"%b:F)$!\"\"F)$F&F&" }{TEXT 208 0 "" }}}
{EXCHG {PARA 202 "" 1 "" {TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {TEXT 207 17 "M\351t
hode it\351rative" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 19 "erreur:=proc(a,b,n)" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 33 "\n  local s,an,bn,temp,k,souvenir;" }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 15 "\n an:=a;bn:=b; " }{TEXT 207 1 " " }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 17 "\nsouvenir:=bn-an;" }{MPLTEXT 
1 205 0 "" }{MPLTEXT 1 205 24 "\n for k from 1 to n do  " }{TEXT 207 
0 "" }{TEXT 207 5 "\n    " }{MPLTEXT 1 205 35 "  temp:=sqrt(an*bn);bn:
=(an+bn)/2; " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 14 "\n  an:=temp;  \+
" }{TEXT 207 1 " " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 28 "\n  souven
ir:=souvenir,bn-an;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nod:" }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 13 "\n [souvenir];" }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 21 "evalf(erreur(1,3,5));" }{MPLTEXT 1 205 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#7($\"\"#\"\"!$\"*#>\\zE!\"*$\"('o:[F)
$\"%b:F)$!\"\"F)$F&F&" }{TEXT 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {TEXT 207 17 "M\351t
hode r\351cursive" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 22 "val_suite:=proc(a,b,n)" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 19 "\n  option remember;" }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 12 "\n local s,u;" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 26 "\n if n=0 then s:= [[a,b]] " }{TEXT 207 80 "la recursion doit \+
pouvoir s'arr\352ter pour donner une valeur effective ici [a0,b0]" }
{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 8 "\n  else " }{MPLTEXT 1 205 0 ""
 }{MPLTEXT 1 205 29 "\n     u:=val_suite(a,b,n-1); " }{TEXT 207 82 "on
 fait appel \340 notre propre proc\351dure pour connaitre les valeurs \+
a(n-1) et b(n-1)" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 62 "\n     s:=[
op(u),[sqrt(u[n][1]*u[n][2]),(u[n][1]+u[n][2])/2]]; " }{TEXT 207 22 " \+
on en d\351duit an et bn" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\nfi
:" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 4 "\n s;" }{MPLTEXT 1 205 0 ""
 }{MPLTEXT 1 205 5 "\nend:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "
> " 0 "" {MPLTEXT 1 205 17 "val_suite(a,b,2);" }{MPLTEXT 1 205 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#7%7$I\"aG6\"I\"bGF&7$*$*&F%\"\"\"F'F+
#F+\"\"#,&F%F,F'F,7$*$*&F*F,F.F+F,,(F)F,F%#F+\"\"%F'F3" }{TEXT 208 0 "
" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 205 19 "erreur:=proc(a,b,n)" }{MPLTEXT 1 205 
0 "" }{MPLTEXT 1 205 28 "\n  local suite,k,sol,suiten;" }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 26 "\n suite:=val_suite(a,b,n);" }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 14 "\n sol:=NULL;  " }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 25 "\n for k from 1 to n+1 do " }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 22 "\n   suiten:=suite[k]; " }{MPLTEXT 1 205 0 "" }
{MPLTEXT 1 205 36 "\n   sol:=sol,suiten[2]-suiten[1];   " }{MPLTEXT 1 
205 0 "" }{MPLTEXT 1 205 4 "\nod:" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 
205 8 "\n [sol];" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 5 "\nend:" }
{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" 
}}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 21 "evalf(erreur(1,3,5));
" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#7($\"\"#\"\"
!$\"*#>\\zE!\"*$\"('o:[F)$\"%b:F)$!\"\"F)$F&F&" }{TEXT 208 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 203 "" 
0 "" {TEXT 203 153 "Commentons ces valeurs, il apparait des choses \35
1tranges comme la valeur n\351agtive pour b4-a4 alors que celle-ci dev
rait \352tre positive (suites adjacentes) " }{TEXT 203 0 "" }}{PARA 
203 "" 0 "" {TEXT 203 147 "Par ailleurs la diff\351rence b5-a5 est sus
pecte car on imagine mal la suite devenir stationnaire comme nous le m
ontre le calcul des valeurs suivantes" }{TEXT 203 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 205 22 "evalf(erreur(1,3,15));" }{MPLTEXT 1 205 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#72$\"\"#\"\"!$\"*#>\\zE!\"*$\"('o:[F)
$\"%b:F)$!\"\"F)$F&F&F0F0F0F0F0F0F0$\"\"\"F)F1F." }{TEXT 208 0 "" }}}
{EXCHG {PARA 203 "" 0 "" {TEXT 203 87 "En fait les choses deviennent p
lus claires lorsque l'on change la pr\351cision des calculs" }{TEXT 
203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 205 11 "Digits:=50;" }{MPLTEXT 1 205 0 
"" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I'DigitsG6\"\"#]" }{TEXT 208 0 
"" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 21 "evalf(erreur(1,3,5
));" }{MPLTEXT 1 205 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#7($\"\"#
\"\"!$\"R'*=YZ>dIjFT\\eObskqA6V#>\\zE!#\\$\"P+bvT**oz+okywcG\"))[%R-e&
o:[F)$\"M_\"fD8pE1<!QP__Q&epCo)\\b:F)$\"FAi;YuLi;f/`[tR\\.Hi\"F)$\"8%p
?\\(3ehH-mw\"F)" }{TEXT 208 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 
203 140 "La convergence \351tant si rapide et les expressions si compl
exes que les r\351sultats approch\351s sont tr\350s sensibles \340 la \+
pr\351cision qu'on se fixe." }{TEXT 203 0 "" }}}{PARA 205 "" 0 "" 
{TEXT 210 0 "" }}{PARA 205 "" 0 "" {TEXT 210 0 "" }}{PARA 205 "" 0 "" 
{TEXT 210 0 "" }}{PARA 205 "" 0 "" {TEXT 210 0 "" }}{PARA 205 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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