Exercice 3 Question a) restart: perturbee:=proc(u1,n) option remember; M\351thode r\351cursive
local u;
if n=1 then [u1]; else
u:=perturbee(u1,n-1);[op(u),evalf(u[n-1]*(u[n-1]+1/n))];
fi;
end: perturbee(1,5); NiM3JyIiIiQiKysrKys6ISIqJCIrKysrXUZGJyQiKysrK10jKUYnJCIrKytEcnAhIik= perturbee(1,10); NiM3LCIiIiQiKysrKys6ISIqJCIrKysrXUZGJyQiKysrK10jKUYnJCIrKytEcnAhIikkIisxOVhyWyEiJyQiK3NNPHRCISIjJCIrZEImPmomIiImJCIrTigpKT08JCIjPyQiKzp5MzE1IiNd Question b) test:=proc(u1)
local l,u,n,suite,sol,maxi;
maxi:=10: pour \351viter une boucle infinie on se fixe un nombre maximal (maxi ) d'it\351rations
n:=1;l:=1; Par d\351faut nous affectons la valeur l=1
u:=u1;
la limite potentielle l est d\351termin\351e lors d'une des conditions ci-dessous pour u1
if is(u<=1-1/n) then l:=0;
elif is(u>=1) then l:=+infinity; fi;
N'ayons pas de r\351ponse d\351terminn\351e pour u1, nous essayons les valeurs successives (jusqu'\340 l'indice maxi)
while is(l=1 and n<maxi) do
u:=u*(u+1/n);n:=n+1;
if is(u<=1-1/n) then l:=0;
elif is(u>=1) then l:=+infinity; fi;
od;
Apr\350s ces nombreuses tentatives, on peut esp\351rer conclure sans trop se tromper
if l=+infinity then sol:=true else sol:=false; fi;
end:
test(0.3);test(1); NiNJJmZhbHNlR0kqcHJvdGVjdGVkR0Yk NiNJJXRydWVHSSpwcm90ZWN0ZWRHRiQ= Question c) alpha:=proc(epsilon)
local a,b,m,n,t;
n:=0:a:=0;b:=1; intialement (n=0) on travaille sur l'intervalle [a,b]=[0,1]
while is(b-a>epsilon) do
n:=n+1:m:=(a+b)/2;t:=test(m);
On remplace [a,b] par le segmet coup\351 en son milieu m
if t then b:=m c'est le segment [a,m] si pour u1=m la suite u tend vers +infini, ( car alors alpha<m)
else a:=m fi; c'est le segment [m,b] si pour u1=m la suite u ne tend pas vers +infini ( car alors alpha>=m)
od;
m,n;
end: u:=alpha(1/10):'alpha'=evalf(u[1]); 'n'=u[2]; NiMvSSZhbHBoYUc2IiQiKysrK3ZWISM1 NiMvSSJuRzYiIiIl u:=alpha(1/100):'alpha'=evalf(u[1]); 'n'=u[2]; NiMvSSZhbHBoYUc2IiQiKytdN2BXISM1 NiMvSSJuRzYiIiIo u:=alpha(1/1000):'alpha'=evalf(u[1]); 'n'=u[2]; NiMvSSZhbHBoYUc2IiQiK0QxKkdZJSEjNQ== NiMvSSJuRzYiIiM1 u:=alpha(1/10000):'alpha'=evalf(u[1]); 'n'=u[2]; NiMvSSZhbHBoYUc2IiQiKyp5JFFvVyEjNQ== NiMvSSJuRzYiIiM5 Digits:=12: u:=alpha(10^(-10)):'alpha'=evalf(u[1]); 'n'=u[2]; NiMvSSZhbHBoYUc2IiQiLSM+Inlib1chIzc= NiMvSSJuRzYiIiNN with(plots): Warning, the name changecoords has been redefined e:=10^(-4):u:=perturbee(alpha(e)):n:=nops(u):plot([seq([k,u[k]],k=1..n)],0..n,0..e,style=point); 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 e:=10^(-10):u:=perturbee(alpha(e)):n:=nops(u):plot([seq([k,u[k]],k=1..n)],0..n,0..e,style=point); 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 e:=10^(-15):u:=perturbee(alpha(e)):n:=nops(u):plot([seq([k,u[k]],k=1..n)],0..n,0..e,style=point); 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 e:=10^(-20):u:=perturbee(alpha(e)):n:=nops(u):plot([seq([k,u[k]],k=1..n)],0..n,0..e,style=point); 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