From d30d0f49864c03709b4f401bb2a51a0cb49629c2 Mon Sep 17 00:00:00 2001
From: Denis Monnerat <monnerat@u-pec.fr>
Date: Tue, 6 May 2025 08:14:15 +0200
Subject: [PATCH] tp Lagrange

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 num/tp3/README.md         |  87 ++++++++++++++++++++++++++++++++++++++
 num/tp3/img/lagrange1.png | Bin 0 -> 28214 bytes
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 create mode 100644 num/tp3/README.md
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diff --git a/num/tp3/README.md b/num/tp3/README.md
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+# TP3 Interpolation polynomiale
+
+## Lagrange
+On a vu dans le module d'algèbre linéaire que les polynômes de Lagrange associés à des points d'interpolations $(x_1,x_2,\ldots,x_n)$ s'écrivent
+\[
+	L_i(x)=\frac{\prod_{j\not=i}(x-x_j)}{\prod_{j\not=i}(x_i-x_j)}
+\]
+
+Le polynôme d'interpolation s'écrit 
+\[
+	P_n(x)=\sum_{i=1}^{i=n} f_i.L_i(x)
+\]
+
+Ce polynôme  prend la valeur $f_i$ au point $x_i$, pour $i = 1,\ldots,n$.
+
+Scilab permet de construire un polynôme avec la fonction `poly`, soit à partir de ses racines :
+```
+--> poly([1,2,3],"x","roots")
+ans  =
+
+  -6 +11x -6x² +x³
+```
+soit à partir de ses coefficients :
+```
+--> poly([1,2,3],"x","coeff")
+ ans  =
+
+  1 +2x +3x²
+```
+
+Pour évaluer un polynôme en un point avec scilab, on utilise la fonction `horner`
+```
+--> p=poly([1,2,3],"x","coeff")
+ p  =
+
+  1 +2x +3x²
+
+--> horner(p,2)
+ ans  =
+
+   17.
+```
+
+1. Voici un code qui calcule le polynome d'interpolation pour 5 points (lesquels ?)
+```
+n=5
+x=linspace(-5,5,n)
+F=[1,3,5,6,7]
+for i=1:n
+y=x; y(i)=[]
+P(i)=poly(y,"x")
+P(i)=P(i)/horner(P(i),x(i))
+end
+Pn=F*P
+plot2d([-5:0.1:5],horner(Pn,[-5:0.1:5]))
+plot2d(x,F,-2)
+```
+
+<div align="center">
+<img src="./img/lagrange1.png">
+</div>
+
+1. Modifier le code pour interpoler la fonction $f(x)=\frac{1}{1+x^2}$. Faites afficher le polynôme interpolateur et la fonction $f$ pour $n=5,10,15,20$. Que constate-t-on ?
+2. Faites varier le nombre de points d'interpolation et estimer l'erreur commise 
+\[ 
+	\lVert P_n - f\rVert_{\infty} = \sup_{x\in [-5,5]} |P_n(x)-f(x)|
+\]
+et tracer la en fonction de $n$
+
+## Approximation avec les polynômes de Bernstein.
+On cherche à approcher une fonction sur $[0,1]$ par les polynômes de Bernstein :
+\[
+	B_n(f)(x) = \sum_{k=0}^{k=n} \binom{n}{k}f(\frac{k}{n}) x^k (1-x)^{n-k}
+\]
+
+On va utiliser le fait que pour $x$ fixé dans $[0,1]$ les coefficients $\binom{n}{k}x^k(1-x)^{n-k}$ sont les probabilités d'une loi 
+binomiale. utilisera la fonction  scilab `binomial`.
+
+
+On prendra `nr=200` points pour le tracer des courbes. On cherche à approcher la fonction 
+\[
+	g(x)=\frac{\sin 5\pi x}{1+10x^2}
+\]
+
+
+1. Tracer sur un même graphique la fonction $g$ et $B_3(f)$
+2. Tracer sur un même graphique $g$ et $B_n(f)$ pour $n$ `5:20:200`. Que constate-t-on ?
diff --git a/num/tp3/img/lagrange1.png b/num/tp3/img/lagrange1.png
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