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-rw-r--r--semestre 4/algo/0- Rentrée.pdf3772
-rw-r--r--semestre 4/algo/0- Rentrée.typ106
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diff --git a/semestre 4/algo/0- Rentrée.pdf b/semestre 4/algo/0- Rentrée.pdf
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diff --git a/semestre 4/algo/0- Rentrée.typ b/semestre 4/algo/0- Rentrée.typ
new file mode 100644
index 0000000..84c5d9d
--- /dev/null
+++ b/semestre 4/algo/0- Rentrée.typ
@@ -0,0 +1,106 @@
+#import "@local/template:1.0.0": *
+#import "@local/callout:1.0.0": *
+
+#show: doc.with(
+ authors: (
+ (name: "William Hergès", affiliation: "Sorbonne Université", email: "william@herges.fr"),
+ ),
+ page_title: "Introduction",
+)
+
+= Introduction
+
+Un algorithme est un moyen de décomposer un problème en plein de petites tâches.
+Une recette de cuisine est un algorithme.
+
+On cherche à trouver un moyen pour analyser les algorithmes et pour choisir le meilleur.
+
+Plan du semestre :
+1. Théorie sur les algorithmes ;
+2. Structure des données.
+
+Contrôle continu en TD, partiel et examen.
+Utilise la règle du max.
+
+Les QCM sur Moodle ne sont pas notés.
+Il y a souvent des QCM aux exams.
+
+= Rappels mathématiques
+
+Raisonnements valides :
+1. calculatoire
+2. raisonnements logiques
+ - directe
+ - absurde
+ - contraposé
+3. par récurrence
+
+C'est du blabla pour nous rappeler comment on fait des maths.
+
+Ne pas oublier que~:
+$ sum^n_(i=1) i = (n(n-1))/2 $
+et que~:
+$ forall x in RR, quad sum^n_(i=0) x^i = (x^(n+1)-1)/(x-1) $
+
+= Solution générale des suites
+
+== Suites récurrente linéaire homogène d'ordre 2
+
+#defn[
+ Une _suite récurrente linéaire homogène d'ordre 2_ est une suite définie par une relation de récurrence
+ $ u_n = a u_(n-1) + b u_(n-2) $
+ avec des conditions initiales
+ $ u_0 = a_0 quad u_1 = a_1 $
+ où $a$, $b$, $a_0$ et $a_1$ sont des nombres réels.
+]
+
+#defn[
+ On définit le _polynôme caractéristique_ associé à la suite récurrente précédente le polynôme~:
+ $ r^2 + a r - b $
+]
+
+Par exemple, le polynôme associé à la suite de Fibonacci est $ r^2-r-1 $
+
+=== Méthode pour déterminer la solution
+
+On calcule les racines du _polynôme caractéristique_.
+
+*Si le polynôme possède deux racines* $r_1$, et $r_2$, alors la solution générale de la récurrence est de la forme~:
+$ u_n = alpha r_1^n + beta r_2^n $
+On détermine ensuite $alpha$ et $beta$ à l'aide des conditions initiales~:
+$ alpha + beta = a_0 \
+alpha r_1 + beta r_2 = a_1 $
+
+*Si le polynôme possède une racine* $r$, alors la solution générale est de la forme~:
+$ u_n = alpha r^n + beta n r^n $
+$alpha$ et $beta$ sont déterminés comme dans le cas précédent.
+
+== Autres cas
+
+Voir le diapo, car le prof n'a pas détaillé et est allé très vite.
+
+= Notations de Landau
+
+#defn[
+ Soient $f$ et $g$ deux fonctions de $NN$ dans $RR_+$.
+
+ On écrit $f in O(g)$ si :
+ $ exists D > 0, exists n_0 >= 0, quad forall n > n_0, quad f(n) <= D g(n) $
+
+ On écrit $f in Omega (g)$ si :
+ $ exists C > 0, exists n_0 >= 0, quad forall n > n_0, quad C g(n) <= f(n) $
+
+ On écrit $f in Theta (g)$ si :
+ $ exists D >, exists C > 0, exists n_0 >= 0, quad forall n > n_0, quad C g(n) <= f(n) <= D g(n) $
+]
+Autrement dit
+- si $f in O(g)$, alors $f$ croit plus lentement que $g$~;
+- si $f in Omega (g)$, alors $f$ croit plus vite que $g$, i.e. $g in O(f)$~;
+- si $f in Theta (g)$, alors $f$ croit aussi vite que $g$, i.e. $f in O(g)$ et $f in Omega (g)$.
+
+#props[
+ Dans le cas où $(f(n))/(g(n))$ adment une limite dans $n$ tend vers $+inf$, si~:
+ - $lim (f(n))/(g(n)) = 0$, alors $f in O(g)$~;
+ - $lim (f(n))/(g(n)) = +inf$, alors $f in Omega (g)$~;
+ - $lim (f(n))/(g(n)) = l$ (où $l in RR backslash {0}$), alors $f in Theta (g)$.
+]
generated by cgit v1.2.3 (git 2.25.1) at 2026-05-03 11:46:36 +0000