176 lines
5.0 KiB
Plaintext
176 lines
5.0 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "075806aa",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "c9235b01",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"A l'itété 0, on trouve x=[-0.2 -0.42]\n",
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"A l'itété 1, on trouve x=[ 0.148 -0.2112]\n",
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"A l'itété 2, on trouve x=[ 0.27328 -0.136032]\n",
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"A l'itété 3, on trouve x=[ 0.3183808 -0.10897152]\n",
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"A l'itété 4, on trouve x=[ 0.33461709 -0.09922975]\n",
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"A l'itété 5, on trouve x=[ 0.34046215 -0.09572271]\n",
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"A l'itété 6, on trouve x=[ 0.34256637 -0.09446018]\n",
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"A l'itété 7, on trouve x=[ 0.34332389 -0.09400566]\n",
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"A l'itété 8, on trouve x=[ 0.3435966 -0.09384204]\n",
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"A l'itété 9, on trouve x=[ 0.34369478 -0.09378313]\n"
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]
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}
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],
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"source": [
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"# Un premier exemple de descente par coordonnées\n",
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"\n",
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"# Itéré intial\n",
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"x = np.array([-1/2, -1])\n",
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"\n",
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"maxiter = 10\n",
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"for k in range(maxiter):\n",
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" #On mets (de façon optimale) x1\n",
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" x[0] = (6*x[1]+4)/10\n",
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" #On mets (de façon optimale) x2\n",
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" x[1] = (6*x[0]-3)/10\n",
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" print(f\"A l'itété {k}, on trouve x={x}\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "999a5a80",
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"metadata": {},
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"outputs": [],
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"source": [
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"def CDquadratic(Q,c,p,x,maxiter=10**2):\n",
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" # Ce code applique la méthode de descente par coordonnées\n",
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" # au problème min_x 0.5 x'*Q*x-c'*x+p\n",
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" # Le x fournit en paramètre est l'itéré initial\n",
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" # Q nous est donnée sous forme symétrique\n",
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" n = len(x)\n",
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" # Une itération coûte O(n^2)\n",
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" for k in range(maxiter):\n",
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" # Dans cette boucle, on mets les variables à jour\n",
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" for i in range(n):\n",
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" somme = 0\n",
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" for j in range(n):\n",
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" if not j==i:\n",
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" somme += Q[i,j]*x[j]\n",
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" x[i] = (c[i]-somme)/Q[i,i]\n",
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" print(f\"A l'itété {k}, on trouve x={x}\")\n",
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" return x"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "d7a850ff",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"A l'itété 0, on trouve x=[-0.2 -0.42]\n",
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"A l'itété 1, on trouve x=[ 0.148 -0.2112]\n",
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"A l'itété 2, on trouve x=[ 0.27328 -0.136032]\n",
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"A l'itété 3, on trouve x=[ 0.3183808 -0.10897152]\n",
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"A l'itété 4, on trouve x=[ 0.33461709 -0.09922975]\n",
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"A l'itété 5, on trouve x=[ 0.34046215 -0.09572271]\n",
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"A l'itété 6, on trouve x=[ 0.34256637 -0.09446018]\n",
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"A l'itété 7, on trouve x=[ 0.34332389 -0.09400566]\n",
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"A l'itété 8, on trouve x=[ 0.3435966 -0.09384204]\n",
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"A l'itété 9, on trouve x=[ 0.34369478 -0.09378313]\n"
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]
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}
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],
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"source": [
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"Q = np.array([[10, -6],[-6, 10]])\n",
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"c = np.array([4, -3])\n",
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"p = 0\n",
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"x = np.array([-1/2, -1])\n",
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"xopt = CDquadratic(Q,c,p,x,10)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "960500d8",
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"metadata": {},
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"outputs": [],
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"source": [
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"def leastsquares(A,b,x,maxiter=10**2):\n",
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" Q = 2*A.T@A\n",
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" c = 2*A.T@b\n",
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" p = b.T@b\n",
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" x = CDquadratic(Q,c,p,x,maxiter)\n",
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" return x"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "560043ef",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"A l'itété 0, on trouve x=[ 0. -0.5]\n",
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"A l'itété 1, on trouve x=[ 0.25 -0.625]\n",
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"A l'itété 2, on trouve x=[ 0.3125 -0.65625]\n",
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"A l'itété 3, on trouve x=[ 0.328125 -0.6640625]\n",
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"A l'itété 4, on trouve x=[ 0.33203125 -0.66601562]\n",
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"A l'itété 5, on trouve x=[ 0.33300781 -0.66650391]\n",
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"A l'itété 6, on trouve x=[ 0.33325195 -0.66662598]\n",
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"A l'itété 7, on trouve x=[ 0.33331299 -0.66665649]\n",
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"A l'itété 8, on trouve x=[ 0.33332825 -0.66666412]\n",
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"A l'itété 9, on trouve x=[ 0.33333206 -0.66666603]\n"
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]
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}
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],
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"source": [
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"A = np.array([[1,0], [0,1], [1,1]])\n",
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"b = np.array([3,2,-3])\n",
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"x = np.array([1.0, 0.0]) # Attention au entiers\n",
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"xopt = leastsquares(A,b,x,10)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "optinonlin",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.13.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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