{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"font-size:8pt\">*ENSAM-Bordeaux, Mathématiques et informatique. Date : le 12/10/19. Auteur : Éric Ducasse. Version : 1.0*</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\"> <span style=\"color:#800000\"> *On pourra faire exécuter ce notebook cellule par cellule $\\left(\\mathtt{Maj+Entrée}\\right)$ ou intégralement par $\\mathtt{\\;Kernel\\rightarrow Restart\\;\\&\\;Run\\;All}$.* </span> </div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sb\n",
    "sb.init_printing() # Pour avoir de jolies formules mathématiques à l'écran"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(f\"Version de sympy : {sb.__version__}\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# **Réécriture d'expressions mathématiques**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Avec* sympy, *on utilise un* **Dictionnaire** (*classe* dict) *de* Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Cela consiste à pouvoir remplacer une ou plusieurs sous-expressions par d'autres.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voir aussi le tutoriel **sympy**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://docs.sympy.org/latest/tutorial/index.html#tutorial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "et en particulier :"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://docs.sympy.org/latest/modules/core.html?highlight=replace#sympy.core.basic.Basic.replace <br /> et <br />\n",
    "https://docs.sympy.org/latest/modules/core.html?highlight=replace#sympy.core.basic.Basic.xreplace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">1. *Manipulation de dictionnaires*</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\">1.1 **Création et ajout d'items**</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ensemble non ordonné d'associations **Clef/Objet**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dico = {\"c\":\"truc\",\"b\":sb.pi,\"a\":[-2.34,5.67]} ; dico"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dict( ((\"c\",\"truc\"),(\"b\",sb.pi),(\"a\",-2.34)) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dico = dict()\n",
    "for key,val in ((\"c\",\"truc\"),(\"b\",sb.pi),(\"a\",-2.34)) :\n",
    "    dico[key] = val\n",
    "dico"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dict.fromkeys([1,2,3],99)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\">1.2 **Méthodes disponibles**</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "[m for m in dir(dico) if not m.startswith(\"__\")]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\"> 1.3 ** Accès à la</span>  <span style=\"color:#008800\"> *valeur* </span> <span style=\"color:purple\"> par la </span> <span style=\"color:#008800\">*clef*</span> **"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dico[\"b\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "try :\n",
    "    dico[\"d\"]\n",
    "except :\n",
    "    print(\"Erreur d'exécution !\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "[ dico.get(k) for k in [\"a\",\"b\",\"c\",\"d\"] ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\"> 1.4 ** Itérables **"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### a/ *Liste des clefs*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Attention** : un dictionnaire n'est pas ordonné !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dico.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sorted(dico.keys())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "list(dico)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b/ *Liste des valeurs*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dico.values()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "list(dico.values())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### c/ *Itérations sur un dictionnaire*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dico.items()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for k,v in dico.items() :\n",
    "    print(f\"'{k}' -> {v}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:blue\">2. *Substitution dans une expression symbolique*</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\">2.1 **Programmation standard, avec affectation**</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y = sb.symbols(\"x,y\")\n",
    "a = x*(y-1)\n",
    "b = (a+x).expand()\n",
    "[a,b]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\">2.2 **Réécriture en utilisant une règle, avec la méthode xreplace**</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a,x,y = sb.symbols(\"a,x,y\")\n",
    "regle = {a:x*(y-1)}\n",
    "b = a+x\n",
    "print(\"L'application de la règle donne :\",b.xreplace(regle).expand())\n",
    "[a,b]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Variante avec la méthode $\\mathtt{replace}$, qui n'utilise pas de dictionnaire mais ne peut faire qu'un seul remplacement à la fois :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "b.replace(a,x*(y-1)).expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\">2.3 **Exemple de séparation expression littérale/application numérique**</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Volume et surface d'un cylindre de rayon $r$ et de hauteur $h$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r,h = sb.symbols(\"r,h\", real=True, positive=True)\n",
    "V = sb.pi*r**2*h\n",
    "S = ( 2 * sb.pi*r**2 + 2*sb.pi*r * h ).factor()\n",
    "(V,S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Deux applications numériques (sans affectation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "AN1,AN2 = {r:2e-3,h:3e-2},{r:2.5e-3,h:2e-2}\n",
    "for regle in (AN1,AN2) : print([float(e.xreplace(regle)) for e in (r,h,V,S)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "AN1,AN2 = {r:2e-3,h:3e-2},{r:2.5e-3,h:2e-2}\n",
    "for regle in (AN1,AN2) :\n",
    "    print(f\"r = {1e3*regle[r]:.1f} mm ; h = {1e3*regle[h]:.1f} mm -> \" + \\\n",
    "          f\"V ~ {1e9*float(V.xreplace(regle)):.2f} mm^3 et \" + \\\n",
    "          f\"S ~ {1e6*float(S.xreplace(regle)):.2f} mm^2\")\n",
    "(V,S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Minimimisation de la surface $S$, pour un volume $V$ donné"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "vol = sb.symbols(\"V\", real=True, positive=True)\n",
    "regle_h = sb.solve(vol - V, h, dict=True)[0] ; regle_h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S_de_V = S.xreplace(regle_h).expand() ; S_de_V "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dSdr = S_de_V.diff(r).factor() ; dSdr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "regle_r_opt = sb.solve(dSdr,r,dict=True)[0] ; regle_r_opt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(h/r).xreplace(regle_h).xreplace(regle_r_opt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Surface minimale pour $2\\,r=h$ (diamètre et hauteur identiques)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  <span style=\"color:purple\">2.4 Outil de réécriture plus puissant : voir le notebook</span> <span style=\"color:blue;font-family:Courier New;font-weight:bold\">UEF-MINI_CM_substitutions_patterns.ipynb</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\"> <span style=\"color:#800000\"> *Ce dernier point ne sera pas exigible* </span> </div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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