175 lines
11 KiB
Plaintext
175 lines
11 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "83556834-6ac7-4721-8121-4189de0b13d0",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import math\n",
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"from matplotlib . colors import ListedColormap\n"
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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": "a151b08e-688f-4a07-95c1-c0ad95232e0e",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"def f(x):\n",
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" y = np.array([\n",
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" x[0]**5 - 10*x[0]**3 * x[1]**2 + 5 * x[0] * x[1]**4 - 1,\n",
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" -5 * x[0]**4 * x[1] + 10 * x[0]**2 * x[1]**3 - x[1]**5\n",
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" ])\n",
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" \n",
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" J = np.array([\n",
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" [5*x[0]**4 - 10*3*x[0]**2 * x[1]**2 + 5 * x[1]**4, -10*x[0]**3 * 2 * x[1] + 5 * x[0] * 4 * x[1]**3 ],\n",
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" [-5 * 4 * x[0] ** 3 * x[1] + 10 * x[0] * 2 * x[1]**3, -5 * x[0]**4 + 10 * x[0]**2 * x[1] ** 2 * 3 - 5*x[1]**4]\n",
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" ])\n",
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" \n",
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" return y, J"
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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": "214d1930-4c0e-4aab-ae5e-bdc44dc3cd90",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"def NewtonRn(fct,x0,tol=1e-6):\n",
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" k = 0\n",
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" x = x0\n",
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" f, df =fct(x)\n",
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" # f, df = fct(x)\n",
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" # print(\"f=\",f,\"df=\", df, \"|f|=\",np.linalg.norm(f), tol)\n",
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" while np.linalg.norm(f) > tol:\n",
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" f, df = fct(x)\n",
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" dfI = np.linalg.inv(df)\n",
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" x = x - f@dfI\n",
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" \n",
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" k+=1\n",
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" if k > 10:\n",
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" break\n",
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" \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": 5,
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"id": "59abd31c-26b5-4a71-9668-ecef255d54a4",
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"metadata": {
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"tags": []
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},
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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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"[0.30901699 0.95105652]\n"
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]
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}
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],
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"source": [
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"p = np.array([1, 1])\n",
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"\n",
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"# print(f(p))\n",
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"\n",
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"x0 = np.array([1,1])\n",
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"xroot = NewtonRn(f, x0)\n",
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"print(xroot)"
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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": 7,
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"id": "4370d5c6-b098-484b-a577-9ce86ada4f02",
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"metadata": {
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"tags": []
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},
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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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"Erreur pour x0 = [0. 0.]\n"
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]
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},
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{
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"data": {
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"image/png": 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",
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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"# fonction a utiliser si on souhaite tracer les bassins d’attraction d’une fonction f\n",
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"def plot_newton_fractal (f , roots , h =0.007 , domain =( -2 , 2 , -2 , 2) ) :\n",
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" colors = ['b', 'r', 'g', 'y','m'] # Liste de couleurs\n",
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" xmin , xmax , ymin , ymax = domain\n",
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" nbpixelsY = math . ceil (( ymax - ymin ) / h )\n",
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" nbpixelsX = math . ceil (( xmax - xmin ) / h )\n",
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" image = np . zeros (( nbpixelsY , nbpixelsX ) )\n",
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" for j in range ( nbpixelsX ) :\n",
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" for i in range ( nbpixelsY ) :\n",
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" x0 = np . array ([ xmin + j *h , ymin + i * h ])\n",
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" try:\n",
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" r = NewtonRn (f , x0 ) #On appelle la methode ici !\n",
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" if r is not False : image [i , j ] = np . argmin ( np . linalg . norm ( roots -r , axis =1) )\n",
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" except:\n",
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" print(\"Erreur pour x0 = \", x0)\n",
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" plt . imshow ( image , cmap = ListedColormap ( colors [: len( roots ) ]) , origin ='lower')\n",
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" plt . axis ('off')\n",
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" plt . show ()\n",
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"# Definition des racines et trace des bassins d’attraction\n",
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"roots = np . array([[ -0.809 , -0.587] ,[0.309 , -0.951] ,[1.000 ,0.000] ,[0.309 ,0.951] ,[ -0.809 ,0.587]])\n",
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"x = plot_newton_fractal (f , roots, h=0.05 )\n"
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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": null,
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"id": "c3811318-c08f-4fcc-8e6a-f35d9958548f",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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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": "3.11.5"
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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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}
|