From d66d9f9a3b360abd594a43cbc5f39efcdfb1c7a4 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?J=C3=A9r=C3=B4me=20de=20Favereau?=
<jerome.defavereau@uclouvain.be>
Date: Tue, 23 Nov 2021 13:32:44 +0000
Subject: [PATCH] Upload New File
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NM_3_exercice.ipynb | 339 ++++++++++++++++++++++++++++++++++++++++++++
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "narrow-newspaper",
+ "metadata": {},
+ "source": [
+ "## Exercice : méthode de Romberg\n",
+ "\n",
+ "Estimation de \n",
+ "\n",
+ "$$\n",
+ "\\int_0^{\\frac{3\\pi}{4}} \\sin x\\ dx\n",
+ "$$\n",
+ "\n",
+ "Par la méthode de Romberg. Cette intégrale vaut $1 + 1/\\sqrt{2} \\approx 1.7071067811865475$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 38,
+ "id": "accomplished-coordinator",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R11=0.8330405509046936\n"
+ ]
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import math\n",
+ "\n",
+ "def f(x):\n",
+ " return np.sin(x)\n",
+ "\n",
+ "# méthode des trapèzes avec 2 points:\n",
+ "x_0 = 0\n",
+ "x_n = 3 * math.pi / 4\n",
+ "R11 = (f(x_0) + f(x_n))/2 * (x_n - x_0)\n",
+ "print(f\"{R11=}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "annoying-access",
+ "metadata": {},
+ "source": [
+ "Calcul de R21 par la méthode itérative des trapèzes:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 46,
+ "id": "fewer-partition",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R21=1.5049402075046334\n"
+ ]
+ }
+ ],
+ "source": [
+ "R21 = R11/2 + f(x_n/2) * (x_n - x_0)/2\n",
+ "print(f\"{R21=}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "returning-mission",
+ "metadata": {},
+ "source": [
+ "calcul de R22 par Richardson avec $p=2$ puisque l'erreur est dominée par $h^2$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "id": "liable-forestry",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R22=1.7289067597046133\n"
+ ]
+ }
+ ],
+ "source": [
+ "R22 = (4*R21 - R11)/3\n",
+ "print(f\"{R22=}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "streaming-tucson",
+ "metadata": {},
+ "source": [
+ "Calcul de R31 par la méthode itérative des trapèzes:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 42,
+ "id": "cathedral-investigation",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R31=1.6574582026781939\n"
+ ]
+ }
+ ],
+ "source": [
+ "h = (x_n - x_0)/2 # espace entre les nouveaux points\n",
+ "x = x_0 + h/2 # position x du premier point\n",
+ "R31 = R21/2 + (f(x) + f(x+h)) * h/2\n",
+ "print(f\"{R31=}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "sacred-pension",
+ "metadata": {},
+ "source": [
+ "calcul de R32 par Richardson avec $p=2$ puisque l'erreur sur R21 et R31 est dominée par $h^2$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 43,
+ "id": "compliant-zimbabwe",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R32=1.708297534402714\n"
+ ]
+ }
+ ],
+ "source": [
+ "R32 = (4*R31 - R21)/3\n",
+ "print(f\"{R32=}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "elect-desktop",
+ "metadata": {},
+ "source": [
+ "calcul de R33 par Richardson avec $p=4$ puisque l'erreur sur R22 et R32 est dominée par $h^4$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 44,
+ "id": "floppy-alignment",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R33=1.706923586049254\n"
+ ]
+ }
+ ],
+ "source": [
+ "R33 = (16*R32 - R22)/15\n",
+ "print(f\"{R33=}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "surgical-syndrome",
+ "metadata": {},
+ "source": [
+ "Même chose en utilisant les fonctions définies au cours:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 45,
+ "id": "green-league",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R11=0.8330405509046936\n",
+ "R21=1.5049402075046334\n",
+ "R22=1.7289067597046133\n",
+ "R31=1.6574582026781939\n",
+ "R32=1.708297534402714\n",
+ "R33=1.706923586049254\n",
+ "R41=1.6947487165588795\n",
+ "R42=1.7071788878524412\n",
+ "R43=1.7071043114157562\n",
+ "R44=1.7071071800723674\n",
+ "\n",
+ "En utilisant la formule analytique:\n",
+ "INT=1.7071067811865475\n"
+ ]
+ }
+ ],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "from math import sin, pi, sqrt\n",
+ "def trapezes_rec(f, x_0, x_n, i_old, k):\n",
+ " if k == 1: \n",
+ " return (f(x_0) + f(x_n))*(x_n - x_0)/2\n",
+ " else:\n",
+ " n = 2**(k-2)\n",
+ " h = (x_n - x_0)/n\n",
+ " x = x_0 + h/2\n",
+ " sum = 0\n",
+ " for i in range(n):\n",
+ " sum += f(x)\n",
+ " x += h\n",
+ " return i_old/2 + h*sum/2\n",
+ "\n",
+ " \n",
+ "def richardson(i_1, i_2, k):\n",
+ " fact = 2**k\n",
+ " return (fact*i_2 - i_1)/(fact - 1)\n",
+ "\n",
+ "# ligne 1\n",
+ "R11 = trapezes_rec(np.sin, 0, 3*pi/4, 0, 1)\n",
+ "print(f\"{R11=}\")\n",
+ "\n",
+ "# ligne 2\n",
+ "R21 = trapezes_rec(np.sin, 0, 3*pi/4, R11, 2)\n",
+ "print(f\"{R21=}\")\n",
+ "R22 = richardson(R11, R21, 2)\n",
+ "print(f\"{R22=}\")\n",
+ "\n",
+ "# ligne 3\n",
+ "R31 = trapezes_rec(np.sin, 0, 3*pi/4, R21, 3)\n",
+ "print(f\"{R31=}\")\n",
+ "R32 = richardson(R21, R31, 2)\n",
+ "print(f\"{R32=}\")\n",
+ "R33 = richardson(R22, R32, 4)\n",
+ "print(f\"{R33=}\")\n",
+ "\n",
+ "# ligne 4\n",
+ "R41 = trapezes_rec(np.sin, 0, 3*pi/4, R31, 4)\n",
+ "print(f\"{R41=}\")\n",
+ "R42 = richardson(R31, R41, 2)\n",
+ "print(f\"{R42=}\")\n",
+ "R43 = richardson(R32, R42, 4)\n",
+ "print(f\"{R43=}\")\n",
+ "R44 = richardson(R33, R43, 6)\n",
+ "print(f\"{R44=}\")\n",
+ "\n",
+ "print(\"\\nEn utilisant la formule analytique:\")\n",
+ "print('INT={}'.format(1+1/math.sqrt(2)))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "leading-montana",
+ "metadata": {},
+ "outputs": [],
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+ },
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--
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