diff --git a/cours.md b/cours.md
index 78f47b455051b6bd11101407f90f206166c97817..07122eb92189e7aa616432f5d65722ddff6185de 100644
--- a/cours.md
+++ b/cours.md
@@ -1114,22 +1114,348 @@ $f(x)=ax^3+bx^2+cx+d$.
# Optimisation
-## Généralités
+## La régression linéaire
-## Position du problème
+Lors d'une régression linéaire, le but est de trouver la droite, $y(x)=a\cdot x + b$, qui passe au mieux au travers d'un nuage de $N$ points $(x_i, y_i)$,
+$i=1,...,N$ (voir @fig:reg).
-Un problème d'optimisation se représente mathématiquement de la façon suivante
+{#fig:reg width=70%}
+Pour déterminer l'équation de cette droite, nous devons donc trouver les coefficients $a$ et $b$ tels que la droite
+passe au plus proche des points. Nous devons d'abord définir ce que signifie mathématiquement "passe au mieux par au travaers du nuage de points".
+Une façon de mesurer la "qualité" d'une droite est de mesurer la somme des distances au carré entre les points $(x_i,y_i)$ et
+la droite $y(x)=a\cdot x + b$ pour des valeurs de $a$ et $b$ données, soit
+$$
+E(a,b)=\sum_{i=1}^N (y(x_i)-y_i)^2.
+$$
+Nous cherchons par conséquent à minimiser $E(a,b)$ sous la contrainte que $y(x)$ est une droite. Pour simplifier encore plus le problème mathématique,
+nous pouvons rajouter comme contrainte que la droite $y(x)$ passe par le point $(0,0)$, on a donc que $y(x)=a\cdot x$ (l'ordonnée à l'origine est nulle, $b=0$) et que
+$$
+E(a)=\sum_{i=1}^N (y(x_i)-y_i)^2,
+$$
+est indépendant de $b$. En résumé nous cherchons à résoudre le problème mathématique
\begin{align}
-&\mbox{minimisation de }f(\ux),\\
-&\mbox{où }\ux\in\mathcal{X}\subseteq \real^n.
+&\min_{a\in\real} E(a) = \min_{a \in\real} \sum_{i=1}^N (y(x_i)-y_i)^2,\\
+&\mbox{où }y(x)=a\cdot x, \quad \mbox{(contrainte)}.
\end{align}
-Ici $\ux$ est un vecteur à $n$ dimensions,
-\begin{equation}
-\ux=(x_0,...,x_{n-1}),
-\end{equation}
-qui représente les variables de notre problème d'optimisation et qui satisfont les *contraintes*, représentées par $\mathcal{X}$.
-La fonction $f$, appelée fonction objectif, fonction de coût, ... est la fonction que nous cherchons à *minimiser* en ajustant $\ux$.
+On peut réécrire la fonction $E(a)$ comme
+\begin{align}
+E(a)&=\sum_{i=1}^N \left(y^2(x_i)-2\cdot y_i\cdot y(x_i)+y_i^2\right)=\sum_{i=1}^N \left(a^2\cdot x_i^2-2\cdot a\cdot x_i\cdot y_i+y_i^2\right),\nonumber\\
+ &=a^2\sum_{i=1}^Nx_i^2 + 2a\sum_{i=1}^Nx_iy_i+\sum_{i=1}^Ny_i^2.
+\end{align}
+Les $x_i$ et $y_i$ étant connus, nous cherchons $a$, tel que $E(a)$ soit minimal. $E(a)$ est en fait l'équation d'une parabole: elle a la forme
+$$
+E(a)=B\cdot a^2-2C\cdot a + D,
+$$
+avec $B=\sum_{i=1}^Nx_i^2$, $C=\sum_{i=1}^Nx_iy_i$, et $D=\sum_{i=1}^N y_i^2$. $B$ étant forcément positif cette parabole sera **convexe** et donc
+nous sommes assurés qu'il existe un minimum pour $E(a)$. Une façon de déterminer $a$, tel que $E(a)$ est minimal est d'utiliser la dérivée.
+On a l'équation $E'(a)=0$ à résoudre:
+\begin{align}
+E'(a)&=0,\nonumber\\
+2\cdot B\cdot a-2\cdot C&=0,\nonumber\\
+a &= \frac{C}{B}=\frac{\sum_{i=1}^Nx_iy_i}{\sum_{i=1}^Nx_i^2}.
+\end{align}
+
+---
+
+Exemple +.#
+
+Soient les 4 points $(0, 0.1)$, $(1, 0.3)$, $(2, 0.3)$ et $(3, 0.4)$. La fonction d'erreur $E(a)$ s'écrit
+$$
+E(a)=14\cdot a^2-4.2\cdot a + 0.35.
+$$
+On peut la représenter comme sur la @fig:e_a et on constate qu'elle possède un minimum proche de $a=0$.
+
+![La fonction $E(a)=14a^2-4.2a+0.35$ pour $a\in[-1,1]$. On voit bien qu'elle possède un minimum proche de $a=0$.](figs/e_a.svg){#fig:e_a width=70%}
+
+En résolvant $E'(a)=0$, on obtient $a=4.2/24=0.15$. On a que l'équation de la droite passant par $(0,0)$ et au plus proche de nos 4 points est
+$$
+y(x)=0.15\cdot x.
+$$
+On peut observer le résultat de la régression sur la @fig:regression_ex, où on voit les 4 points (en noir), ainsi que la droite obtenue (en trait bleu).
+
+{#fig:regression_ex width=70%}
+
+---
+
+La régression linéaire est un problème **d'optimisation continu**. On voit que contrairement au problème du voyageur du commerce,
+l'ensemble des solutions est $a\in\real$ et non une suite discrète de villes. Ce genre de problème, bien que possédant un espace de recherche infini,
+est bien souvent plus simple à résoudre, car il possède un cadre théorique mieux défini.
+
+Pour le résoudre, nous avons commencé, comme pour le problème du voyageur du commerce par faire un modèle mathématique.
+Nous avons construit une fonction à minimiser, $E(a)$, et ajouté une contraite, la forme de $y(x)$. Puis, il a suffit de trouver le minimum de $E(a)$
+sous la contrainte et le tour était joué.
+
+## L'optimisation mathématique
+
+Suite à ces deux exemples, nous allons essayer de définir de façon assez théorique comment formuler mathématiquement un problème d'optimisation.
+Il existe deux types disctincts de problèmes d'optimisation:
+
+1. L'optimisation continue.
+2. L'optimisation discrète (souvent appelée optimisation combinatoire).
+
+Dans ce chapitre nous ne parlerons que del'optimisation continue.
+
+### L'optimisation continue
+
+L'optimisation continue ou *programme mathématique continu* est un programme d'optimisation soumis à certaines contraintes.
+On peut l'exprimer de la façon suivante.
+
+Soit $f:\real^n\rightarrow\real$ une fonction objectif (ou fontion de coût), on cherche $\vec x_0\in\real^n$, tel que $f(\vec x_0)\leq f(\vec x)$ pour $\vec x$ certaines conditions: **les contraintes**. Celles-ci sont en général des égalités strictes ou des inégalités qui peuvent s'exprimer de la façon suivante.
+Soient $m$ fonctions $g_i:\real^n\rightarrow\real$
+\begin{align}
+&g_i(\vec x)\leq 0,\quad i=1,...,m.
+\end{align}
+Si $m=0$ on a à faire à un problème d'optimisation sans contraintes. On peut résumer tout cela comme
+\begin{align*}
+&\min_{\vec x\in\real^n}f(\vec x),\\
+&g_i(\vec x)\leq 0,\quad i=1,...,m,\\
+&\mbox{pour }m\geq 0.
+\end{align*}
+Les contraintes limitent l'espace des solutions et forment un sous-ensemble, noté $A$, de $\real^n$ ($A\subseteq\real^n$).
+
+Une des difficultés pour déterminer le minimum d'une fonction coût est l'existence de plusieurs minima locaux.
+Un **minimum local**, $\vec x^\ast\in A$, est tel que pour une région proche de $\vec x^\ast$, on a que $f(\vec x)\geq f(\vec x^\ast)$.
+Un exemple d'une telle fonction, est une fonction de Ackley. En une dimension, elle est de la forme (voir la @fig:ackley)
+$$
+f(x)=-20e^{-0.2*\sqrt{0.5x^2}}-e^{0.5(\cos(2\pi x))}+e+20.
+$$
+
+{#fig:ackley width=70%}
+
+On constate la présence d'un grand nombre de minima locaux qui rendent la recherche du minimum global (se trouvant en $x=0$) particulièrement compliqué à déterminer.
+
+L'optimisation continue est très communément utilisée en apprentissage automatique (machine learning), en particulier pour
+optimiser les poids des réseaux de neurones.
+
+## Optimisation continue
+
+Dans cette section, nous allons considérer des problèmes purement continus.
+Nous allons dans un premier temps considérer une fonction opbjectif, $f$,
+$$
+f:D\rightarrow\real,\quad D\subseteq \real,
+$$
+dont nous allons chercher le minimum (pour autant qu'il existe). Nous allons supposer que
+$f$ est une fonction continue et dérivable.
+
+### Minimum local/global
+
+Comme vous le savez, le minimum (ou le maximum) d'une fonction, se situe à un endroit où sa dérivée est nulle.
+On recherche donc, $x$, tel que
+$$
+f'(x)=0.
+$$
+Mais cette contrainte sur $f'(x)$ n'est pas suffisante pour garantir de trouver un minimum.
+En effet, si $f'(x)=0$, peut également vouloir dire qu'on se trouve sur un point d'inflexion
+ou sur un maximum.
+On peut assez facilement, discriminer ces deux cas, en considérant la deuxième dérivée de $f$.
+En effet, nous avons à faire à un minimum seulement si
+$$
+f''(x)>0.
+$$
+Les cas où $f''(x)=0$ est un point d'inflexion et $f''(x)<0$ est un maximum.
+
+Un autre problème beaucoup plus compliqué à résoudre est de déterminer un minimum **global**.
+En effet, comme pour la fonction de Ackley (voir la @fig:ackley), une fonction peut posséder un grand nombre de minimam **locaux** (où
+$f'(x)=0$ et $f''(x)>0$) mais qui n'est pas un mimumum global.
+
+Mathématiquement un *minimum local* se définit comme $x^\ast$ tel qu'il existe $\delta>0$ et que $f(x^\ast)\leq f(x)$, pour
+$x\in[x^\ast-\delta,x^\ast+delta]$. Un *minimum global* est un $x^\ast$ tel que $\forall x\in D$, $f(x^\ast)\leq f(x)$.
+
+En fait, il n'existe pas de méthode pour déterminer un minimum global, pour n'importe quelle fonction.
+Nous somme assurés de le trouver, uniquement si $f$ est une fonction convexe partout ($f''(x)>0 \ \forall x$).
+
+## Algorithme de recherche d'un zéro d'une fonction
+
+Comme nous venons de le voir, lors de la recherche d'un minimum, il est nécessaire de trouver le point $x^\ast$
+où $f'(x^\ast)=0$. Le problème est donc de déterminer les zéros de la fonction $f'(x)$. Pour avoir un maximum de généralité,
+nous allons considérer une fonction $g(x)$ et chercher ses zéros, soit
+$$
+\{x\in\real|g(x)=0\}.
+$$
+Dans des cas simples (des fonctions polynomiales de degré 2 ou 3, ou des fonctions inversibles) on peut trouver
+analytiquement les zéros. En revanche, pour des fonctions plus complexes, ou "implicites" (on ne peut pas mettre
+l'équation $g(x)=0$ sous la forme $x=...$) la détermination des zéros est beaucoup plus difficile et nécessite l'utilisation
+de **méthodes itératives**. Nous allons en voir quelques unes.
+
+### Méthode de la bissection
+
+{#fig:bissection_method width=50%}
+
+Afin de déterminer le zéro d'une fonction, une des méthodes les plus simple est la méthode de la bissection.
+Il s'agit de choisir deux points, $a_1$ et $b_1$, $b_1>a_1$, tels que le signe de $g(a_1)$ et $g(b_1)$ est différent.
+Si cela est le cas, nous aommes assurés de l'existence d'au moins un zéro si la fonction $g(x)$ est continue
+(en vertu du théorème de la valeur intermédiaire). Ensuite, nous allons calculer la valeur se situant "au milieu"
+entre $a_1$ et $b_1$
+$$
+c_1=\frac{b_1+a_1}{2}.
+$$
+Puis, nous évaluons $g(c_1)$ et si ce n'est pas un zéro, étudions son signe. Si le signe $g(c_1)$ est différent de celui de $g(a_1)$, nous remplaçons
+$b_1$ par $c_1$ et recommençons. Si le signe de $g(c_1)$ est différent de celui de $g(b_1)$, nous remplaçons $a_1$ par $c_1$.
+Nous itérons cette méthode jusqu'à ce que nous ayons atteint une valeur "siffisamment proche" (nous vons une précision acceptable pour nous)
+de zéro. Une façon d'exprimer "proche" est de considérer la taille de l'intervalle $b_1-a_1$ et de le comparer avec une précision $\varepsilon>0$ que nous
+aurons choisie
+$$
+b_1-a_1<\varepsilon.
+$$
+
+Au pire des cas, cette méthode nous rapproche de $(b_1+a_1)/2$ du zéro à chaque itération. Après $n$ itération, nous somme donc à une
+distance maximale du zéro de $(b_1+a_1)/2^n$. On dit que cette méthode est d'ordre $1$ (on divise l'intervalle de recherche par 2 et la précision par 2
+à chaque itération).
+
+---
+
+Exercice (Racice de polynôme) +.#
+
+Déterminer la racine du polynôme $x^4+x^3+x^2-1$ avec $a_1=0.5$ et $b_1=1$ (faire au maximum 6 itérations).
+
+---
+
+### Méthode de la fausse position (*regula falsi*)
+
+{#fig:false_position_method width=50%}
+
+Une méthode un peu plus avancée est la méthode de la fausse position (voir la @fig:false_position_method). Dans cette méthode qui est relativement similaire à celle de la bissection,
+mais au lieu de diviser l'intervalle en deux parts égales à chaque itération on va choisir les point $c$, comme étant le point
+où la droite reliant $g(a_1)$ et $g(b_1)$ coupe l'axe horizontal (le zéro de la droite entre $g(a_1)$ et $g(b_1)$). Le reste de l'algorithme reste exactement le même.
+On choisit deux points, $a_1$ et $b_1$, où le signe de $f$ est différent, puis ont construit la droite passant par $g(a_1)$ et $g(b_1)$
+$$
+y=\frac{g(b_1)-g(a_1)}{b_1-a_1}(x-a_1) + g(a_1).
+$$
+On cherche le point, $c$, où $y(c)=0$
+$$
+\frac{g(b_1)-g(a_1)}{b_1-a_1}(c-a_1) + g(a_1)=0.
+$$
+Cette équation s'inverse aisément et on obtient
+$$
+c_1=a_1-\frac{b_1-a_1}{g(b_1)-g(a_1)}g(a_1).
+$$
+Puis, comme pour la méthode de la bissection, on compare les signes de $g(c_1)$ avec $g(a_1)$ et $g(b_1)$ et on remplace $a_1$ ou $b_1$ par $c_1$
+si $g(c_1)$ a un signe différent de $g(b_1)$ ou $g(a_1)$ respectivement.
+
+Il est important de noter que si la fonction est continue, et que $a_1$ et $b_1$ sont choisis tels que $g(a_1)$ et $g(b_1)$ sont de signes opposés,
+alors cette méthode convergera **toujours**.
+
+La méthode de la fausse position est plus efficace que la méthode de la bissection, elle est superlinéaire (d'ordre plus grand que un).
+
+---
+
+Exercice +.#
+
+Déterminer le zéro positif de la fonction
+$$
+x^2-25=0,
+$$
+à l'aide de la méthode de la fausse position.
+
+---
+
+### Méthode de la sécante
+
+{#fig:secant_method width=50%}
+
+La méthode de la sécante (voir la @fig:secant_method) est très similaire à la méthode de la fausse position. La seule différence se situe dans la dernière étape de l'algorithme.
+Plutôt que choisir de remplacer $a_1$ ou $b_1$ par $c_1$, on remplace toujours la dernière valeur calculée.
+Ainsi après avoir choisi $a < b$, avec $g(a)$ et $g(b)$ avec des signes différents, on calcule
+une suite de $x_i$, avec $x_0=a$, $x_1=b$, tels que
+$$
+x_{i+1}=x_{i-1}-\frac{x_i-x_{i-1}}{g(x_i)-g(x_{i-1})}g(x_{i-1}), \quad i\geq 2.
+$$
+
+La méthode de la sécante ne garantit pas la convergence, contrairement à la méthode de la bissection et de la fausse position.
+En revanche elle est plus efficace, lorsque qu'elle converge, que ces deux méthodes.
+
+---
+
+Exercice +.#
+
+Déterminer le zéro positif de la fonction
+$$
+x^2-25=0,
+$$
+à l'aide de la méthode de la sécante.
+
+---
+
+### Méthode de Newton (ou *Newton-Raphson")
+
+La méthode de Newton est également une méthode itérative, qui nécessite que la fonction $g(x)$ soit non seulement continue mais également dérivable.
+Revenons sur la méthode de la sécante. Il s'agissait de choisir deux points, $a < b$, et de déterminer la droite, $y(x)$, passant par $g(a)$ et $g(b)$,
+\begin{equation*}
+y=\frac{g(b)-g(a)}{b-a}(x-a) + g(a).
+\end{equation*}
+Il se trouve que $g(b)-g(a)/(b-a)$ n'est autre qu'une approximation avec une formule de différences finies
+de la dérivée de $g$ et $a$, $g'(a)$. Si la fonction $f$ est dérivable, on peut simplement remplacer ce terme par $g'(a)$
+et on obtient
+$$
+y=g'(a)(x-a) + g(a).
+$$
+Puis on détermine $c$, tel que $y(c)=0$
+$$
+0=g'(a)(c-a) + g(a),
+$$
+et on obtient
+$$
+c=a - \frac{g(a)}{g'(a)}.
+$$
+
+On peut donc généraliser l'algorithme. En partant d'un point $x_0=a$, on construit la suite
+$$
+x_{i+1}=x_n-\frac{g(x_i)}{g'(x_i)}, \ i\geq 0.
+$$
+On s'arrête lorsque le zéro est déterminé avec une précision suffisante, ou que la variation entre deux itérations successives est assez petite. Ce qui revient à choisir un $\varepsilon>0$, tel que
+$$
+|g(x_n)| < \varepsilon,\quad |x_n-x_{n-1}| < \varepsilon.
+$$
+
+Lorsque qu'elle converge la mtéhode de Newton est la plus efficace de toutes celles que nous avons vues. On dit qu'elle est d'ordre $2$.
+En revanche les contraintes pour sa convergence sont plus strictes que pour les méthodes vues précédemment.
+
+---
+
+Remarques (non-convergence ou convergence lente) +.#
+
+Il y a un certain nombre de cas où la méthode de Newton ne converge pas.
+
+1. S'il existe un $n$ tel que $g'(x_n)=0$ alors la suite diverge.
+2. La suite peut entrer dans un cycle.
+3. La dérivée est mal définie proche du zéro (ou sur le zéro).
+4. Elle peut converger très lentement si la dérivée de la fonction est nulle sur le zéro.
+5. A chaque point de départ ne correspond qu'un zéro. Si la fonction possède plusieurs zéros, il n'y a pas moyen de le savoir avec un seul point de départ. Il faut alors en essayer plusieurs.
+
+---
+
+---
+
+Exercice +.#
+
+Déterminer le zéro de la fonction
+$$
+x^2-25=0,
+$$
+à l'aide de la méthode de Newton.
+
+---
+
+### Résumé
+
+A l'aide des méthodes vues ci-dessus, on peut déterminer un zéro d'une fonction (s'il existe).
+Ces méthodes sont également utilisables pour calculer le minimum d'une fonction comme nous l'avons discuté plus haut.
+Il suffit de remplacer $g(x)$ par $f'(x)$ et le tour est joué.
+
+---
+
+Exercice +.#
+
+Écrire l'algorithme de Newton pour le cas de la minimisation d'une fonction $f(x)$ quelconque, mais continuement dérivable 2 fois.
+
+---
Équations différentielles ordinaires
diff --git a/figs/Bisection_method.svg b/figs/Bisection_method.svg
new file mode 100644
index 0000000000000000000000000000000000000000..eb3f3458f62e5147d4dbb9111a8aaa03ee524d9d
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