From 66c3bbfa94d8a41e58adf154be25e6d86fee8e30 Mon Sep 17 00:00:00 2001 From: "martial.simon" Date: Sun, 13 Apr 2025 19:54:19 +0200 Subject: init: initial commit --- .../ma51 x.T A x sur un maillage en Numpy .ipynb | 32043 +++++++++++++++++++ 1 file changed, 32043 insertions(+) create mode 100644 PVCM/cama/fr/ma51 x.T A x sur un maillage en Numpy .ipynb (limited to 'PVCM/cama/fr/ma51 x.T A x sur un maillage en Numpy .ipynb') diff --git a/PVCM/cama/fr/ma51 x.T A x sur un maillage en Numpy .ipynb b/PVCM/cama/fr/ma51 x.T A x sur un maillage en Numpy .ipynb new file mode 100644 index 0000000..9933335 --- /dev/null +++ b/PVCM/cama/fr/ma51 x.T A x sur un maillage en Numpy .ipynb @@ -0,0 +1,32043 @@ +{ + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.10" + } + }, + "nbformat": 4, + "nbformat_minor": 2, + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import numpy.linalg as lin\n", + "import matplotlib.pylab as plt\n", + "import plotly.offline as py\n", + "import plotly.graph_objects as go\n", + "\n", + "%matplotlib inline\n", + "%config InlineBackend.figure_format = 'retina'\n", + "\n", + "np.set_printoptions(precision=3, linewidth=150, suppress=True)\n", + "plt.style.use(['seaborn-whitegrid','data/cours.mplstyle'])" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "# Calculons ${\\bf x}^T \\, A \\, {\\bf x} $ avec Numpy\n", + "\n", + "Le but est de tracer la courbe $J_A({\\bf x})$ avec ${\\bf x} \\in ℝ^2$ pour que les pauvres petits humains que nous\n", + "sommes puissent voir quelque chose.\n", + "\n", + "$$\n", + "J_A({\\bf x}) = {\\bf x}^T \\, A \\, {\\bf x}\n", + "$$\n", + "\n", + "Plus tard on prendra ${\\bf x} \\in ℝ^n$.\n", + "\n", + "Pour une valeur **x** ce calcule est simple : `x.T @ A @ x` mais on veut faire ce calcul pour un ensemble de\n", + "**x**." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "On commence par construire un maillage c.a.d. un ensemble de points ${\\bf x}$ pour lesquels on calculera \n", + "$J_A({\\bf x})$. Pour cela on utilise `np.meshgrid`. Regardons ce que cela donne." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "shape of M: (2, 4, 3)\n", + "[[[-1. 0. 1.]\n", + " [-1. 0. 1.]\n", + " [-1. 0. 1.]\n", + " [-1. 0. 1.]]\n", + "\n", + " [[-1. -1. -1.]\n", + " [ 0. 0. 0.]\n", + " [ 1. 1. 1.]\n", + " [ 2. 2. 2.]]]\n", + "\n", + "shape of M: (4, 3, 2)\n" + ] + }, + { + "data": { + "text/plain": [ + "array([[[-1., -1.],\n", + " [ 0., -1.],\n", + " [ 1., -1.]],\n", + "\n", + " [[-1., 0.],\n", + " [ 0., 0.],\n", + " [ 1., 0.]],\n", + "\n", + " [[-1., 1.],\n", + " [ 0., 1.],\n", + " [ 1., 1.]],\n", + "\n", + " [[-1., 2.],\n", + " [ 0., 2.],\n", + " [ 1., 2.]]])" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x = np.linspace(-1,1,3)\n", + "y = np.linspace(-1,2,4)\n", + "\n", + "mesh = np.meshgrid(x,y) # donne les x puis les y du maillage\n", + "M = np.array(mesh)\n", + "print(\"shape of M:\", M.shape)\n", + "print(M)\n", + "M = M.transpose([1,2,0]) # je préfère avoir un tableau 4x3 de points\n", + "print(\"\\nshape of M:\", M.shape)\n", + "M" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "Je veux appliquer ${\\bf x}^T \\, A \\, {\\bf x}$ à chaque point de mon maillage. Je commence avec la matrice identité\n", + "pour que je puisse facilement vérifier les calculs.\n", + "\n", + "En premier je calcule ${\\bf x}^T \\, A$ pour tous les points du maillage.\n", + "\n", + "### Cas test avec A = 2 Id\n", + "\n", + "Je regarde sur un tout petit maillage comment écrire les calculs. Je prends pour A deux fois\n", + "la matrice identité pour pouvoir vérifier les calculs de tête." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[[-2., -2.],\n", + " [ 0., -2.],\n", + " [ 2., -2.]],\n", + "\n", + " [[-2., 0.],\n", + " [ 0., 0.],\n", + " [ 2., 0.]],\n", + "\n", + " [[-2., 2.],\n", + " [ 0., 2.],\n", + " [ 2., 2.]],\n", + "\n", + " [[-2., 4.],\n", + " [ 0., 4.],\n", + " [ 2., 4.]]])" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A = 2 * np.diag([1, 1])\n", + "MA = np.einsum(\"ijk, ka -> ija\", M, A) # voici un cas où la notation d'Einstein simplifie bien la vie\n", + "MA # k est en double et pas dans le résultat, donc on somme sur lui" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "Je retrouve `2 M` ce qui est cohérent.\n", + "\n", + "Dans un autre cas on pourrait vérifier certains points :" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([ 0., -2.])" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# vérification sur un point particulier\n", + "M[0,1] @ A" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "Notez que j'ai écrit ${\\bf x} \\, A$ et non ${\\bf x}^T \\, A$ mais lorsqu'on a un vecteur Numpy s'adapte et privilégie le produit matrice vecteur qui donne un vecteur. Ainsi `m[0,1] @ A == m[0,1].T @ A` car, comme\n", + "dit la doc, \"Transposing a 1-D array returns an unchanged view of the original array\". \n", + "\n", + "Si on veux avoir une différence entre un vecteur vertical et un vecteur horizontal il faut utiliser\n", + "des `array` 2D de taille 1xn ou nx1." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[ 4., 2., 4.],\n", + " [ 2., 0., 2.],\n", + " [ 4., 2., 4.],\n", + " [10., 8., 10.]])" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np.einsum(\"ijk, ijk -> ij\", MA, M) # comme k n'est pas dans le résultat, c'est sur lui qu'on fait la somme" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "Comme A est 2 fois l'identité on doit trouver pour tout point 2 fois sa norme au carré. C'est bon.\n", + "\n", + "### Un vrai cas\n", + "\n", + "On prend un maillage de 100x100 et une matrice A aléatoire :" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[-0.02 0.05]\n", + " [ 0.03 -0.02]]\n" + ] + } + ], + "source": [ + "np.random.seed(2)\n", + "A = np.random.randint(-10,10, size=(2,2)) / 100\n", + "print(A)\n", + "\n", + "x = np.linspace(-20,20,100)\n", + "y = np.linspace(-20,20,100)\n", + "\n", + "mx, my = np.meshgrid(x,y) \n", + "M = np.array(np.meshgrid(x,y)).transpose([1,2,0])\n", + "\n", + "MA = np.einsum(\"ijk, ka -> ija\", M, A)\n", + "mz = np.einsum(\"ijk, ijk -> ij\", MA, M)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/vnd.plotly.v1+json": { + "config": { + 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "trace = go.Surface(x=mx, y=my, z=mz)\n", + "fig = go.Figure(data=[trace])\n", + "py.iplot(fig, show_link=False)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "On a un superbe point selle.\n", + "\n", + "Bien sûr avec une autre graine aléatoire nous aurions autre chose. Imaginez ce que vous pourriez avoir si toutes\n", + "les valeurs de A sont positives." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "## Optimisons\n", + "\n", + "Si on doit faire ce calcul de nombreuses fois en grande dimension, il serait bien de trouver la solution\n", + "optimale. Si la notation d'Einstein aide bien ici, elle n'est pas toujours la plus rapide. Donc comparons avec\n", + "d'autres méthodes.\n", + "\n", + "### Passer par une fonction J" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.0" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def J(x):\n", + " return x @ A @ x\n", + "\n", + "mz2 = np.array([J(x) for line in M for x in line]).reshape([M.shape[0], M.shape[1]])\n", + "\n", + "np.max(np.square(mz2 - mz)) # pour vérifier qu'on obtient la même chose" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "143 µs ± 1.99 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n" + ] + } + ], + "source": [ + "# comparons les temps calculs :\n", + "\n", + "%timeit np.einsum(\"ijk, ijk -> ij\", np.einsum(\"ijk, ka -> ija\", M, A), M)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "19.7 ms ± 494 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" + ] + } + ], + "source": [ + "%timeit np.array([J(x) for line in M for x in line]).reshape([M.shape[0], M.shape[1]])" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "On est plus de 100 fois plus rapide avec la sommation d'Einstein." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "Notons qu'avec `apply_along_axis` c'est pire :" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "34.6 ms ± 989 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" + ] + } + ], + "source": [ + "%timeit np.array([np.apply_along_axis(J, 1, line) for line in M]).reshape([M.shape[0], M.shape[1]])" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "### Utiliser `np.tensordot`\n", + "\n", + "Un tensor est une matrice en N dimension donc voici une fonction qui devrait pouvoir servir puisque notre ensemble de point est en 3 dimensions." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "100 µs ± 1.15 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n" + ] + } + ], + "source": [ + "%%timeit # pour calculer le temps d'execution de toute la cellule %%\n", + "\n", + "MA = np.tensordot(M, A, axes=(2,1)) # on somme sur l'axe 2 de M (les points) et l'axe 1 de A (les colonnes)\n", + "np.einsum(\"ijk, ijk -> ij\", MA, M) # je n'arrive pas à faire cette opération autrement" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "lang": "fr" + }, + "source": [ + "On a un gain de 30 % par rapport à `einsum`.\n", + "\n", + "Pour avoir de bonne mesures il faudrait utiliser des cas plus gros que 100x100 et lancer les tests\n", + "sur une machine qui ne fait que cela, donc en mode single user sous Linux.\n", + "\n", + "### Conclusion\n", + "\n", + "Grosso modo on a :\n", + "\n", + "| Méthode | Temps |\n", + "| --- | --: |\n", + "|`einsum` | 140 µs|\n", + "|`tensordot` | 100 µs|\n", + "|boucle `for` | 20 000 µs|" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ] +} \ No newline at end of file -- cgit v1.2.3