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Les commandes principales consistent à positionner la tortue, lever ou baisser le crayon (pour écrire ou non lorsque la tortue se déplace) et à commander les mouvements de la tortue (avancer/reculer, tourner à gauche/droite d'un certain angle).\n\nPour mieux comprendre et découvrir les commandes accessibles, étudier l'exemple de démonstration ci-dessous :"},{"metadata":{"trusted":true},"cell_type":"code","source":"import turtle as tt # import du module turtle dans Basthon\n\ntt.speed(3)         # vitesse moyenne (maxi = 10)\n\ntt.penup()          # lève le crayon (pour ne pas écrire pendant le déplacement)\ntt.setposition(-100, 100) \n# origine (0, 0) au centre du cadre de dessin (dimensions 600x600)\ntt.pendown()        # abaisse le crayon (pour voir la trace de la tortue)\n\ntt.forward(20)      # avance de 20\ntt.left(60)         # virage de 60° vers la gauche\ntt.forward(100)    \ntt.right(120)       # virage de 120° vers la droite\ntt.pencolor(\"red\")  # couleurs autorisées  \"blue\", \"yellow\", \"brown\", \"black\", \"purple\", \"green\"\ntt.forward(100)  \ntt.left(60)       \ntt.forward(100)     # recule de 100\ntt.backward(70)  \ntt.right(90)   \ntt.pencolor(\"green\")  \ntt.forward(300)  \n\ntt.penup()\ntt.home()           # retour à l'origine\n\ntt.done()           # indispensable pour activer la tortue dans Basthon","execution_count":3,"outputs":[{"output_type":"display_data","data":{"image/svg+xml":"<svg xmlns=\"http://www.w3.org/2000/svg\" width=\"640\" height=\"480\" preserveaspectratio=\"xMidYMid meet\" viewbox=\"0 0 640 480\"><animate id=\"af_834c721b65114680add0597f36cbe268_0\" attributename=\"opacity\" attributetype=\"CSS\" from=\"1\" to=\"1\" begin=\"0s\" dur=\"1ms\" fill=\"freeze\"></animate><g transform=\"translate(320 240)\"></g><g transform=\"translate(320 240)\"><line x1=\"0\" y1=\"0\" x2=\"0\" y2=\"0\" style=\"stroke: black;stroke-width: 1\"><animate id=\"af_834c721b65114680add0597f36cbe268_2\" attributename=\"x2\" 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Fractales : courbe de Koch</h2>\n\nLa courbe de Koch est une fractale reposant sur la construction récursive suviante :\n1. Étape 1 : Tracer un segment de longueur a. \n\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_0.png)\n\n2. Étape 2 : Diviser le segment en 3 parties de même longueur. Construire un triangle équilatéral ayant pour base le segment médian de la première étape, et en supprimer la base.\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_1.png)\n\n3. Étape 3 : Reprendre l'étape 2 sur chacun des segments créés.\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_2.png)\n\n4. Et ainsi de suite...\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_3.png)\n"},{"metadata":{},"cell_type":"markdown","source":"On peut construire récursivement cette courbe. \n\nLa fonction de tracer prend deux paramètres en entrée :\n* la longeur $a$ du segment.\n* l'étape $n$ de \"profondeur\" de récursivité. \n\nPar exemple, à la profondeur $n=0$, on trace un simple segment : ceci constituera la condition d'arrêt des appels récursifs. À la profondeur $n=1$, le tracé donne la figure de l'étape 2."},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-warning\">ex 1 : Dessiner avec la tortue (pas de récursivité ici)  la figure correspondant à l'étape 2 (décrite ci-avant).</h2>\nÉcrire une fonction etape2(a) pour réaliser cette tache."},{"metadata":{"trusted":false},"cell_type":"code","source":"tt.speed(10)\n\n\ndef etape2(a):\n    # VOTRE CODE CI-DESSOUS\n    #...\n    \n# test:\na = 50\netape2(a)\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-warning\">ex 2 : Courbe de Koch : fonction récursive.</h2>\n\nEn vous inspirant de la logique du code de la fonction précédente (en la \"rendant récursive\"), écrire une fonction koch(a, n) récursive qui :\n\n- prend comme paramètres un nombre entier a représentant la longueur du segment et un entier n égal au nombre de récursions souhaité.\n- construit la courbe de Koch en divisant récursivement chacun des segments\n\n*Rappel* : si n=0, le tracé est un simplement segment de longueur a."},{"metadata":{"trusted":false},"cell_type":"code","source":"tt.speed(10)\ntt.penup()\ntt.setposition(-300, 0) \ntt.pendown()\n\ndef koch(a, n):\n    # VOTRE CODE CI-DESSOUS\n    #...\n\nkoch(360, 3)\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\">Et pour s'amuser encore un peu...</h2>\nExécuter le code ci-dessous :"},{"metadata":{"trusted":false},"cell_type":"code","source":"a = 180\n\ntt.speed(10)\ntt.penup()\ntt.setposition(-a/2, a/3) \ntt.pendown()\n\ndef koch(a, n):\n    if n == 0:\n        tt.forward(a)\n    else:\n        koch(a/3, n-1)\n        tt.left(60)\n        koch(a/3, n-1)\n        tt.right(120)\n        koch(a/3, n-1)\n        tt.left(60)\n        koch(a/3, n-1)\n\ndef flocon(a, n):\n    for i in range(3):\n        koch(a, n)\n        tt.right(120)\n        \nflocon(a, 3)\n\ntt.penup()\ntt.home()\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"","execution_count":null,"outputs":[]}],"metadata":{"kernelspec":{"display_name":"Python 3.8.1 64-bit ('python38': conda)","language":"python","name":"python38164bitpython38conda56991d5ad1414e06a4dcd344400cf456"}},"nbformat":4,"nbformat_minor":2}
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+{"cells":[{"metadata":{},"cell_type":"markdown","source":"<h1 class=\"alert alert-success\">Récursivité </h2>"},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\"> 1. Introduction </h2>\n\nLa **récursivité** est un concept qui est très proche de la notion mathématiques de la récurrence. On dit qu’une fonction est récursive si elle s’appelle elle-même.\nPour résoudre un problème ou effectuer un calcul, on se ramène à la résolution d’un problème similaire mais de complexité moindre. On recommence ainsi jusqu’à obtenir un problème élémentaire que l’on sait résoudre. \n\n!!! note Définition\nOn qualifie de **récursive** toute fonction qui s'appelle elle-même.\n!!!\n\nLa plus-part des langages de programmation actuels permettent de mettre en œuvre directement cette réduction du problème, en autorisant une fonction à s’appeler elle-même : on parle alors de fonction récursive. On trouvera souvent deux versions possibles d'un même programme : itérative et récursive.\n\nCependant deux éléments sont indispensables à la terminaison d’une fonction récursive :\n\n- il est nécessaire qu’il y ait une condition d’arrêt ;\n- il ne doit pas y avoir de suite infinie d’appels récursifs.\n\n### Exemple\nOn souhaite calculer la somme $S=\\sum_{i=1}^n i =1 +2+3+\\cdots+(n-1) + n$. Ci-dessous une fonction recursive qui calcule la somme des $n$ premiers termes dont on peut visualiser les appels"},{"metadata":{"trusted":true},"cell_type":"code","source":"from rcviz import viz # pour visualiser les appels \n@viz\ndef somme(n):\n    if n < 1:\n        return 0\n    else:\n        return n + somme(n - 1)\n\nsomme(4) # calcule 1 + 2 + 3 + 4","execution_count":1,"outputs":[{"output_type":"stream","text":"Traceback (most recent call last):\n  File \"<input>\", line 1, in <module>\nModuleNotFoundError: The module 'rcviz' is included in the Pyodide distribution, but it is not installed.\nYou can install it by calling:\n  await micropip.install(\"rcviz\") in Python, or\n  await pyodide.loadPackage(\"rcviz\") in JavaScript\nSee https://pyodide.org/en/stable/usage/loading-packages.html for more details.\n","name":"stderr"},{"output_type":"error"}]},{"metadata":{},"cell_type":"markdown","source":"Un sous-programme est dit récursif s’il fait appel à lui-même, comme ici.\nDétaillons la procédure compte :\n\n- Pour calculer la somme des `n` termes, on dépcompose le problème comme calculer la somme des `n-1` termes plus `n`;\n- le test `n < 0` est une condition d’arrêt, obligatoire sinon le sous-programme peut boucler indéfiniment;\n- les appels récursifs continuent jusqu’à ce que le paramètre passé à la procédure prenne la valeur 0;\n- la dernière valeur retournée est donc 0.\n\nOn peut visualiser les appels avec :"},{"metadata":{"trusted":true,"scrolled":true},"cell_type":"code","source":"somme.callgraph() # pour visualiser les appels récusifs","execution_count":2,"outputs":[{"output_type":"stream","text":"Traceback (most recent call last):\n  File \"<input>\", line 1, in <module>\nNameError: name 'somme' is not defined\n","name":"stderr"},{"output_type":"error"}]},{"metadata":{},"cell_type":"markdown","source":"On remarque qu'il est possible d'exécuter cette fonction avec un paramètre qui n'est pas un entier positif. Pour interdire ce cas de figure on peut utiliser l'instruction `assert` :"},{"metadata":{"trusted":true},"cell_type":"code","source":"def somme(n):\n    assert n >= 0 and isinstance(n, int), \"n doit être un entier naturel\"\n    if n < 1:\n        return 0\n    else:\n        return n + somme(n - 1)","execution_count":20,"outputs":[]},{"metadata":{"trusted":true},"cell_type":"code","source":"somme(-2)","execution_count":21,"outputs":[{"output_type":"stream","text":"Traceback (most recent call last):\n  File \"<input>\", line 1, in <module>\n  File \"<input>\", line 2, in somme\nAssertionError: n doit être un entier naturel\n","name":"stderr"},{"output_type":"error"}]},{"metadata":{"trusted":true,"scrolled":true},"cell_type":"code","source":"somme(2.8)","execution_count":22,"outputs":[{"output_type":"stream","text":"Traceback (most recent call last):\n  File \"<input>\", line 1, in <module>\n  File \"<input>\", line 2, in somme\nAssertionError: n doit être un entier naturel\n","name":"stderr"},{"output_type":"error"}]},{"metadata":{"trusted":true},"cell_type":"code","source":"somme(8)","execution_count":23,"outputs":[{"output_type":"execute_result","execution_count":23,"data":{"text/plain":"36"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"<h3 style=\"color:teal;background-color:azure;\" > <i class=\"fa fa-pencil\" aria-hidden=\"true\"> </i> &nbsp; Exercice 1 : Somme des éléments d'une liste </h3>\n\nEcrire une fonction récursive ```somme_liste``` qui :\n\n- prend en paramètre une liste d'entiers `L`\n- retourne la somme des éléments de la liste `L`\n\n\n**Coup de pouce :** \n\n*Condition d'arrêt* : si la liste contient un seul élément, la somme de ses élément vaut la valeur de cet élément : donc la fonction renvoie ce nombre.\n\n*Récursion* : renvoyer la valeur du 1er élément additionnée de la somme du reste de la liste."},{"metadata":{"trusted":true},"cell_type":"code","source":"# VOTRE CODE CI-DESSOUS\n","execution_count":null,"outputs":[]},{"metadata":{"trusted":true},"cell_type":"code","source":"# test \ntab = [5, 4, 7, 2]\nsomme(tab)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3 style=\"color:teal;background-color:azure;\" > <i class=\"fa fa-pencil\" aria-hidden=\"true\"> </i> &nbsp; Exercice 2 : Miroir d’une chaîne de caractère </h3>\n\nÉcrire une fonction ```miroir``` récursive qui :\n\n- prend comme paramètre une chaîne de caractères `s`\n- renvoie le « miroir » de la chaîne; par exemple, le miroir de \"abcde\" est \"edcba\".\n\n\n**Coup de pouce :** \n\n*Condition d'arrêt* : si la chaine contient un seul caractère ou aucun, son miroir est elle-même.\n\n*Récursion* : Le principe consiste à permutter le 1er et le dernier carcatère, et à recopier le miroir  de la sous-chaine interne (entre le 2ème caractère et l'avant-dernier) au milieu de ces 2 caractères. Il faut donc renvoyer une chaine qui est la concaténation du dernier caratère, du miroir de la sous-chaine interne, et du 1er caractère.\n\n**Aide** : on pourra utiliser la syntaxe ```s[1:-1]``` qui permet d'extraire une sous-chaine de caractères entre le second élément de l'avant dernier."},{"metadata":{"trusted":true},"cell_type":"code","source":"# VOTRE CODE CI-DESSOUS","execution_count":null,"outputs":[]},{"metadata":{"trusted":true},"cell_type":"code","source":"# test\nchaine = 'quelle belle chaine'\nmiroir(chaine)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\">2. Rappels sur Turtle</h2>\n\nTurtle est un module Python permettant de dessiner sur un canevas. Le crayon est dirigé par une tortue !\n\nJupyter propose une implémentation de ce module (très légèrement modifié). Les commandes principales consistent à positionner la tortue, lever ou baisser le crayon (pour écrire ou non lorsque la tortue se déplace) et à commander les mouvements de la tortue (avancer/reculer, tourner à gauche/droite d'un certain angle).\n\nPour mieux comprendre et découvrir les commandes accessibles, étudier l'exemple de démonstration ci-dessous :"},{"metadata":{"trusted":false},"cell_type":"code","source":"import turtle as tt # import du module turtle\n\ntt.speed(3)         # vitesse moyenne (maxi = 10)\n\ntt.penup()          # lève le crayon (pour ne pas écrire pendant le déplacement)\ntt.setposition(-100, 100) \n# origine (0, 0) au centre du cadre de dessin (dimensions 600x600)\ntt.pendown()        # abaisse le crayon (pour voir la trace de la tortue)\n\ntt.forward(20)      # avance de 20\ntt.left(60)         # virage de 60° vers la gauche\ntt.forward(100)    \ntt.right(120)       # virage de 120° vers la droite\ntt.pencolor(\"red\")  # couleurs autorisées  \"blue\", \"yellow\", \"brown\", \"black\", \"purple\", \"green\"\ntt.forward(100)  \ntt.left(60)       \ntt.forward(100)     # recule de 100\ntt.backward(70)  \ntt.right(90)   \ntt.pencolor(\"green\")  \ntt.forward(300)  \n\ntt.penup()\ntt.home()           # retour à l'origine\n\ntt.done()           # indispensable pour activer la tortue dans Basthon","execution_count":3,"outputs":[{"output_type":"display_data","data":{"image/svg+xml":"<svg xmlns=\"http://www.w3.org/2000/svg\" width=\"640\" height=\"480\" preserveaspectratio=\"xMidYMid meet\" viewbox=\"0 0 640 480\"><animate id=\"af_834c721b65114680add0597f36cbe268_0\" attributename=\"opacity\" attributetype=\"CSS\" from=\"1\" to=\"1\" begin=\"0s\" dur=\"1ms\" fill=\"freeze\"></animate><g transform=\"translate(320 240)\"></g><g transform=\"translate(320 240)\"><line x1=\"0\" y1=\"0\" x2=\"0\" y2=\"0\" style=\"stroke: black;stroke-width: 1\"><animate id=\"af_834c721b65114680add0597f36cbe268_2\" attributename=\"x2\" attributetype=\"XML\" 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fill=\"freeze\"></animateTransform></polygon></g></svg>"},"metadata":{}}]},{"metadata":{},"cell_type":"markdown","source":"**Pour résumer:**\n\n| Fonction |Exemple|Commentaire|\n|:-------- |:------|:----------|\n|forward(x)|forward(150)|Trace un trait de 150 points|\n|backward(x)|backward(150)|Trace un trait “à reculons” de 150 points|\n|left(n)|left(60)|Tourne sur place la tortue de 60° à gauche|\n|right(n)|right(60)|Tourne sur place la tortue de 60° à droite|\n|setposition(x, y) |setposition(-100, 100)|va à la position (-100,100) sachant que l'origine (0;0) est au centre du cadre de dessin (dimensions 600x600)|\n|width(x)|width(5)|Change l’épaisseur à 5 points|\n|color(\"c\")|color(\"yellow\")|Change la couleur du trait (mais aucun trait n’est tracé à ce moment). Notez les guillemets !|\n|penup()|penup()|Lève la tortue (permet de se déplacer sans dessiner)|\n|pendown()|pendown()|Baisse le stylo|\n|home()|home()|retour à l'origine|"},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\">3. Fractales : courbe de Koch</h2>\n\nLa courbe de Koch est une fractale reposant sur la construction récursive suviante :\n1. Étape 1 : Tracer un segment de longueur a. \n\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_0.png)\n\n2. Étape 2 : Diviser le segment en 3 parties de même longueur. Construire un triangle équilatéral ayant pour base le segment médian de la première étape, et en supprimer la base.\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_1.png)\n\n3. Étape 3 : Reprendre l'étape 2 sur chacun des segments créés.\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_2.png)\n\n4. Et ainsi de suite...\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_3.png)\n"},{"metadata":{},"cell_type":"markdown","source":"On peut construire récursivement cette courbe. \n\nLa fonction de tracer prend deux paramètres en entrée :\n* la longeur $a$ du segment.\n* l'étape $n$ de \"profondeur\" de récursivité. \n\nPar exemple, à la profondeur $n=0$, on trace un simple segment : ceci constituera la condition d'arrêt des appels récursifs. À la profondeur $n=1$, le tracé donne la figure de l'étape 2."},{"metadata":{},"cell_type":"markdown","source":"<h3 style=\"color:teal;background-color:azure;\" > <i class=\"fa fa-pencil\" aria-hidden=\"true\"> </i> &nbsp; Exercice 3 : Faire l'étape 2 </h3>\n\n\n1. Écrire une fonction `etape2(a)` qui permet de dessiner avec la tortue (pas de récursivité ici) la figure correspondant à l'étape 2 (décrite ci-avant).\n"},{"metadata":{"trusted":false},"cell_type":"code","source":"import turtle as tt # import du module turtle\ntt.speed(10) # pour dessiner plus rapidement\n\ndef etape2(a):\n    # VOTRE CODE CI-DESSOUS\n    #...\n    \n# test:\na = 50\netape2(a)\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2. En vous inspirant de la logique du code de la fonction précédente (en la \"rendant récursive\"), écrire une fonction koch(a, n) récursive qui :\n\n    - prend comme paramètres un nombre entier a représentant la longueur du segment et un entier n égal au nombre de récursions souhaité.\n    - construit la courbe de Koch en divisant récursivement chacun des segments\n\n    *Rappel* : si n=0, le tracé est un simplement segment de longueur a."},{"metadata":{"trusted":false},"cell_type":"code","source":"import turtle as tt # import du module turtle\n\ntt.speed(10)\ntt.penup()\ntt.setposition(-300, 0) \ntt.pendown()\n\ndef koch(a, n):\n    # VOTRE CODE CI-DESSOUS\n    #...\n\nkoch(360, 3)\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\">Et pour s'amuser encore un peu...</h2>\nExécuter le code ci-dessous :"},{"metadata":{"trusted":false},"cell_type":"code","source":"a = 180\n\ntt.speed(10)\ntt.penup()\ntt.setposition(-a/2, a/3) \ntt.pendown()\n\ndef koch(a, n):\n    if n == 0:\n        tt.forward(a)\n    else:\n        koch(a/3, n-1)\n        tt.left(60)\n        koch(a/3, n-1)\n        tt.right(120)\n        koch(a/3, n-1)\n        tt.left(60)\n        koch(a/3, n-1)\n\ndef flocon(a, n):\n    for i in range(3):\n        koch(a, n)\n        tt.right(120)\n        \nflocon(a, 3)\n\ntt.penup()\ntt.home()\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"markdown","source":"---\n\n#### Remarque générale\n\nUne partie de ce notebook reprend le cours sur la récursivité du lycée Faustin Flerettiré http://nsinfo.yo.fr/nsi_term_programmation_recursivite.html. Le notebook est sous license Creative Commons [BY-NC-SA](https://creativecommons.org/licenses/?lang=fr)\n![Licence Creative Commons](https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png)"}],"metadata":{"kernelspec":{"display_name":"Python 3.8.1 64-bit ('python38': conda)","language":"python","name":"python38164bitpython38conda56991d5ad1414e06a4dcd344400cf456"}},"nbformat":4,"nbformat_minor":2}
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-{"cells":[{"metadata":{},"cell_type":"markdown","source":"<h1 class=\"alert alert-success\">Récursivité - partie 2</h2>"},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\">1. Rappels sur Turtle</h2>\n\nTurtle est un module Python permettant de dessiner sur un canevas. Le crayon est dirigé par une tortue !\n\nJupyter propose une implémentation de ce module (très légèrement modifié). Les commandes principales consistent à positionner la tortue, lever ou baisser le crayon (pour écrire ou non lorsque la tortue se déplace) et à commander les mouvements de la tortue (avancer/reculer, tourner à gauche/droite d'un certain angle).\n\nPour mieux comprendre et découvrir les commandes accessibles, étudier l'exemple de démonstration ci-dessous :"},{"metadata":{"trusted":true},"cell_type":"code","source":"import turtle as tt # import du module turtle dans Basthon\n\ntt.speed(3)         # vitesse moyenne (maxi = 10)\n\ntt.penup()          # lève le crayon (pour ne pas écrire pendant le déplacement)\ntt.setposition(-100, 100) \n# origine (0, 0) au centre du cadre de dessin (dimensions 600x600)\ntt.pendown()        # abaisse le crayon (pour voir la trace de la tortue)\n\ntt.forward(20)      # avance de 20\ntt.left(60)         # virage de 60° vers la gauche\ntt.forward(100)    \ntt.right(120)       # virage de 120° vers la droite\ntt.pencolor(\"red\")  # couleurs autorisées  \"blue\", \"yellow\", \"brown\", \"black\", \"purple\", \"green\"\ntt.forward(100)  \ntt.left(60)       \ntt.forward(100)     # recule de 100\ntt.backward(70)  \ntt.right(90)   \ntt.pencolor(\"green\")  \ntt.forward(300)  \n\ntt.penup()\ntt.home()           # retour à l'origine\n\ntt.done()           # indispensable pour activer la tortue dans Basthon","execution_count":3,"outputs":[{"output_type":"display_data","data":{"image/svg+xml":"<svg xmlns=\"http://www.w3.org/2000/svg\" width=\"640\" height=\"480\" preserveaspectratio=\"xMidYMid meet\" viewbox=\"0 0 640 480\"><animate id=\"af_834c721b65114680add0597f36cbe268_0\" attributename=\"opacity\" attributetype=\"CSS\" from=\"1\" to=\"1\" begin=\"0s\" dur=\"1ms\" fill=\"freeze\"></animate><g transform=\"translate(320 240)\"></g><g transform=\"translate(320 240)\"><line x1=\"0\" y1=\"0\" x2=\"0\" y2=\"0\" style=\"stroke: black;stroke-width: 1\"><animate id=\"af_834c721b65114680add0597f36cbe268_2\" attributename=\"x2\" 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Fractales : courbe de Koch</h2>\n\nLa courbe de Koch est une fractale reposant sur la construction récursive suviante :\n1. Étape 1 : Tracer un segment de longueur a. \n\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_0.png)\n\n2. Étape 2 : Diviser le segment en 3 parties de même longueur. Construire un triangle équilatéral ayant pour base le segment médian de la première étape, et en supprimer la base.\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_1.png)\n\n3. Étape 3 : Reprendre l'étape 2 sur chacun des segments créés.\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_2.png)\n\n4. Et ainsi de suite...\n![Image de losange](https://githepia.hesge.ch/info_sismondi/3AM.OS/-/raw/main/recursivite/koch_3.png)\n"},{"metadata":{},"cell_type":"markdown","source":"On peut construire récursivement cette courbe. \n\nLa fonction de tracer prend deux paramètres en entrée :\n* la longeur $a$ du segment.\n* l'étape $n$ de \"profondeur\" de récursivité. \n\nPar exemple, à la profondeur $n=0$, on trace un simple segment : ceci constituera la condition d'arrêt des appels récursifs. À la profondeur $n=1$, le tracé donne la figure de l'étape 2."},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-warning\">ex 1 : Dessiner avec la tortue (pas de récursivité ici)  la figure correspondant à l'étape 2 (décrite ci-avant).</h2>\nÉcrire une fonction etape2(a) pour réaliser cette tache."},{"metadata":{"trusted":false},"cell_type":"code","source":"tt.speed(10)\n\n\ndef etape2(a):\n    # VOTRE CODE CI-DESSOUS\n    #...\n    \n# test:\na = 50\netape2(a)\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-warning\">ex 2 : Courbe de Koch : fonction récursive.</h2>\n\nEn vous inspirant de la logique du code de la fonction précédente (en la \"rendant récursive\"), écrire une fonction koch(a, n) récursive qui :\n\n- prend comme paramètres un nombre entier a représentant la longueur du segment et un entier n égal au nombre de récursions souhaité.\n- construit la courbe de Koch en divisant récursivement chacun des segments\n\n*Rappel* : si n=0, le tracé est un simplement segment de longueur a."},{"metadata":{"trusted":false},"cell_type":"code","source":"tt.speed(10)\ntt.penup()\ntt.setposition(-300, 0) \ntt.pendown()\n\ndef koch(a, n):\n    # VOTRE CODE CI-DESSOUS\n    #...\n\nkoch(360, 3)\n\ntt.done()","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h2 class=\"alert alert-info\">Et pour s'amuser encore un peu...</h2>\nExécuter le code ci-dessous :"},{"metadata":{"trusted":true},"cell_type":"code","source":"a = 180\n\ntt.speed(10)\ntt.penup()\ntt.setposition(-a/2, a/3) \ntt.pendown()\n\ndef koch(a, n):\n    if n == 0:\n        tt.forward(a)\n    else:\n        koch(a/3, n-1)\n        tt.left(60)\n        koch(a/3, n-1)\n        tt.right(120)\n        koch(a/3, n-1)\n        tt.left(60)\n        koch(a/3, n-1)\n\ndef flocon(a, n):\n    for i in range(3):\n        koch(a, n)\n        tt.right(120)\n        \nflocon(a, 3)\n\ntt.penup()\ntt.home()\n\ntt.done()","execution_count":5,"outputs":[{"output_type":"display_data","data":{"image/svg+xml":"<svg xmlns=\"http://www.w3.org/2000/svg\" width=\"640\" height=\"480\" preserveaspectratio=\"xMidYMid meet\" viewbox=\"0 0 640 480\"><animate id=\"af_b853632d840248ebba685813a45dd4bf_0\" attributename=\"opacity\" attributetype=\"CSS\" from=\"1\" to=\"1\" begin=\"0s\" dur=\"1ms\" fill=\"freeze\"></animate><g transform=\"translate(320 240)\"></g><g transform=\"translate(320 240)\"><line x1=\"0\" y1=\"0\" x2=\"0\" y2=\"0\" style=\"stroke: black;stroke-width: 1\"><animate id=\"af_b853632d840248ebba685813a45dd4bf_2\" attributename=\"x2\" attributetype=\"XML\" from=\"0\" to=\"0\" dur=\"1ms\" fill=\"freeze\" begin=\"af_b853632d840248ebba685813a45dd4bf_1.end\"></animate><animate attributename=\"y2\" attributetype=\"XML\" begin=\"af_b853632d840248ebba685813a45dd4bf_1.end\" from=\"0\" to=\"0\" dur=\"1ms\" fill=\"freeze\"></animate></line><line x1=\"0\" y1=\"0\" x2=\"0\" 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-{"cells":[{"metadata":{},"cell_type":"markdown","source":"<h1>Récursivité</h1>\n<ul>\n<h2> Compléments</h2>               \n\n<h3>La fonction <em>somme(n)</em> du cours</h3>\n</ul>"},{"metadata":{},"cell_type":"markdown","source":"Une fois que l’on dispose d’une définition récursive pour une fonction, il est en général assez facile de la programmer en Python.<br/>     \nIl faut néanmoins faire attention à **deux points importants** :     \n\n* Le domaine mathématique d’une fonction, c’est-à dire les valeurs sur lesquelles elle est définie, n’**est pas** toujours le même que l’**ensemble des valeurs du type Python** avec lesquelles elle sera appelée.\n* Le choix d’une **définition récursive plutôt qu’une autre peut dépendre du modèle d’exécution des fonctions récursives**, en particulier quand il s’agit de prendre en compte des contraintes d’efficacité.\n\n<h4>Exemple</h4>\n\nLe code de la fonction <code> somme(n)</code> rappelé ci-dessous,ne se **comporte pas** exactement comme la fonction mathématique.\n\n<ul>"},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme(n):\n        if n == 0:\n            return 0\n        else:\n            return n + somme(n - 1)","execution_count":1,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme(5)","execution_count":2,"outputs":[{"data":{},"execution_count":2,"metadata":{},"output_type":"execute_result"}]},{"metadata":{},"cell_type":"markdown","source":"La principale différence est que la **fonction mathématique est uniquement définie pour des entiers naturels**,     \nalors que la fonction <code>somme(n)</code> peut être appelée avec un **entier Python arbitraire**, qui peut être une **valeur négative**.\n\n\nBien que la fonction mathématique ne soit pas définie pour\n**n = −1**, l’appel <code>somme(-1)</code> ne provoque **aucune erreur immédiate**,     \nmais il implique un appel à <code>somme(-2)</code>, qui déclenche un appel à <code>somme(-3)</code>, etc.     \nCe processus **infini** va finir par provoquer une **erreur à l’exécution**.\n</ul>"},{"metadata":{"trusted":false},"cell_type":"code","source":"somme(-1)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"Pour **éviter ce comportement**, c'est de restreindre les appels à la fonction <code>somme(n)</code> aux **entiers positifs ou nuls**.        \nPour cela, on peut utiliser une instruction **assert** comme ci-dessous :"},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme(n):\n    assert n >= 0, \"entrer un entier naturel\"\n    if n == 0:\n        return 0\n    else:\n        return n + somme(n - 1)","execution_count":4,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme(5)","execution_count":5,"outputs":[{"data":{},"execution_count":5,"metadata":{},"output_type":"execute_result"}]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme(-1)","execution_count":6,"outputs":[{"ename":"AssertionError","evalue":"entrer un entier naturel","output_type":"error","traceback":["\u001b[1;31m---------------------------------------------------------------------------\u001b[0m","\u001b[1;31mAssertionError\u001b[0m                            Traceback (most recent call last)","\u001b[1;32m<ipython-input-6-2263d90c8da2>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0msomme\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m","\u001b[1;32m<ipython-input-4-b514195a3042>\u001b[0m in \u001b[0;36msomme\u001b[1;34m(n)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0msomme\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m     \u001b[1;32massert\u001b[0m \u001b[0mn\u001b[0m \u001b[1;33m>=\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m\"entrer un entier naturel\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mn\u001b[0m \u001b[1;33m==\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m     \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n","\u001b[1;31mAssertionError\u001b[0m: entrer un entier naturel"]}]},{"metadata":{},"cell_type":"markdown","source":"Bien que cette solution soit correcte, elle n’est pas encore **complètement satisfaisante**.     \n\nEn effet, pour tout appel <code>somme(n)</code> avec **n >= 0**, chaque appel récursif **commencera par faire le test associé à l’instruction assert**,          \nalors que **chaque valeur** de n, par la suite, sera **nécessairement positive**.\n\nUne solution pour éviter ces tests inutiles est de définir **deux fonctions** :\n<ul>\n  \nLa première, <code>somme_bis(n)</code>, implémente la définition récursive de la fonction\nmathématique <code>somme(n)</code> sans **vérifier son argument**.\n\n</ul>    "},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme_bis(n):\n    if n == 0:\n        return 0\n    else:\n        return n + somme_bis(n - 1)","execution_count":7,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"La seconde, <code>somme(n)</code>, est la fonction « *principale* » qui sera appelée par l’utilisateur.    \nCette fonction ne fait que vérifier (**une et une seule fois**) que\nson argument **n est positif** puis, si c’est le cas, elle appelle la fonction\n<code>somme_bis(n)</code>."},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme(n):\n    assert n >= 0, \"entrer un entier naturel\"\n    return somme_bis(n)","execution_count":8,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme(5)","execution_count":9,"outputs":[{"data":{},"execution_count":9,"metadata":{},"output_type":"execute_result"}]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme(-1)","execution_count":10,"outputs":[{"ename":"AssertionError","evalue":"entrer un entier naturel","output_type":"error","traceback":["\u001b[1;31m---------------------------------------------------------------------------\u001b[0m","\u001b[1;31mAssertionError\u001b[0m                            Traceback (most recent call last)","\u001b[1;32m<ipython-input-10-2263d90c8da2>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0msomme\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m","\u001b[1;32m<ipython-input-8-4a3635feae60>\u001b[0m in \u001b[0;36msomme\u001b[1;34m(n)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0msomme\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m     \u001b[1;32massert\u001b[0m \u001b[0mn\u001b[0m \u001b[1;33m>=\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m\"entrer un entier naturel\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0msomme_bis\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n","\u001b[1;31mAssertionError\u001b[0m: entrer un entier naturel"]}]},{"metadata":{},"cell_type":"markdown","source":"<h4>Conclusion</h4>     \n<br/>\n<div style=\"background-color:#DDD5ED;\">\n    \nUn calcul peut être décrit à l’aide d’une <font color='red'> <strong>définition récursive</strong></font>.    \nL’avantage de cette technique est que l’implémentation est souvent plus proche de la définition.    \n    L’écriture d’une <font color='red'>  **fonction récursive** </font>nécessite de distinguer les <font color='red'> **cas de base**</font>, pour lesquels on peut donner un résultat facilement, et les<font color='red'> **cas récursifs**</font>, qui font appel à la définition en cours.\n\nIl faut faire attention à ce que la fonction en Python ne s’applique que sur le<font color='red'> **domaine**</font> de la fonction mathématique, par exemple en utilisant l’instruction<font color='red'> **assert**</font>.    \n\nEnfin, il faut comprendre le modèle d’exécution des fonctions récursives pour choisir la définition qui<font color='red'> **limite le nombre d’appels récursifs**</font>.    \n(Voir les ***exercices 8 et 9*** sur les puissances)\n\n</div>"},{"metadata":{},"cell_type":"markdown","source":"<ul><h2> Fiche 2 </h2>               \n<h3> Exercice 1</h3>\n\nFonction  **Factorielle**\n\nL'un des exemples les plus classiques est une fonction mathématique, la fonction factorielle.          \nOn dit « factorielle n » et on note avec un point d'exclamation n!.         \nCette fonction est définie « par récurrence » sur les entiers naturels : 0! = 1 et pour tout entier naturel n non nul, \nn! = n × (n – 1)!.        \nAutrement dit, pour tout n non nul, n! = n × (n — 1) × (n — 2) × … × 2 × 1.         \n\nEn utilisant le langage  **Python** \n\n1) Ecrire une fonction itérative calculant factorielle n, ayant pour paramètre un entier positif, et retourne le calcul.      \na) En utilisant le boucle « for »"},{"metadata":{"trusted":false},"cell_type":"code","source":"def fact_for(n):\n    \"\"\"\n    Calcule la factorielle de n\n    paramètre:\n        n : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n   ...","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"fact_for(5)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"b) En utilisant le boucle « while »"},{"metadata":{"trusted":false},"cell_type":"code","source":"def fact_while(n):\n    \"\"\"\n    Calcule la factorielle de n\n    paramètre:\n        n : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n   ...","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"fact_while(5)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2) Fonction récursive"},{"metadata":{"trusted":false},"cell_type":"code","source":"def fact_rec(n):\n    \"\"\"\n    Calcule la factorielle de n\n    paramètre:\n        n : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n    if n == 0:\n        return 1\n    else:\n        return n * fact_rec(n - 1)","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"fact_rec(5)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 2</h3>\n\n**Calcul de pgcd de deux nombres entiers**\n\nÉcrire en Python une fonction récursive pgcd (a , b) renvoyant le plus grand diviseur commun de deux \nnombres a et b.        \nPour cela, on utilisera le résultat mathématique suivant : « pgcd(a , b) = pgcd(b , r), où a =  bq + r » \n"},{"metadata":{"trusted":false},"cell_type":"code","source":"def pgcd(a, b):\n    \"\"\"\n    Calcule la pgcd de a et b\n    paramètre:\n        a : (int) entier >= 0\n        b : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n ...","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"pgcd(220, 12)","execution_count":null,"outputs":[]},{"metadata":{"collapsed":true},"cell_type":"markdown","source":"<h3> Exercice 3</h3>\n\n**Nombres d'adhérents** \n\nUne association a remarqué que d'une année à l'autre, \n<ul> <li>elle perd 5 % de ses adhérents ; </li>\n    <li>elle gagne 200 adhérents.</li>\n</ul>\n\n1)\tEn admettant que le nombre d'adhérents de cette association était égal à 2000 au Ier janvier 2019, écrire en Python une fonction récursive nommée nombre(n) affichant le nombre théorique d'adhérents après n années, n ≥ 1.     \n2)\tDans ce même programme, afficher le nombre théorique d'adhérents au cours des 20 prochaines années. \n"},{"metadata":{"trusted":false},"cell_type":"code","source":"def nombre(n):\n    \"\"\"\n    Calcule le nombre théorique d'adhérents\n    paramètre:\n        n : (int) entier >= 1\n    retour:\n        un flotant positif\n    \"\"\"\n   ","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"nombre(10)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2) Nombres d'adhérents au cours des 20 prochaines années"},{"metadata":{"trusted":false},"cell_type":"code","source":"for ...:\n    print...","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 4</h3>\n\n**Suite de Fibonacci** \n\nLa suite de Fibonacci est la suite numérique définie par :             \nF0 = F1 = 1         ,            pour tout n ≥ 0, Fn+2 = Fn+l + Fn            \n\nÉcrire en Python une fonction récursive fibo(n) permettant d'afficher le terme Fn , où n ≥ 0, puis afficher les termes de F0 à F20.\n"},{"metadata":{"trusted":false},"cell_type":"code","source":"def fibo(n):\n    \"\"\"\n    Calcule la valeur de la suite de Fibonacci pour n\n    paramètre:\n        n : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n...","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 5</h3>"},{"metadata":{"trusted":false},"cell_type":"code","source":"# Version récursive normale\ndef somme_rec(a, b):\n    \"\"\"\n    Calcule la somme de a et b\n    paramètre:\n        a : (int) entier >= 0\n        b : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n....\n\n\n# Version récursive terminale\ndef somme_rec_term(a, b):\n    \"\"\"\n    Calcule la somme de a et b\n    paramètre:\n        a : (int) entier >= 0\n        b : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n....","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme_rec(3, 5)","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme_rec_term(3, 5)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 6</h3>\n1)  Version itérative"},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme(a, b):\n    \"\"\"\n    Calcule la somme de a et b\n    paramètre:\n        a : (int) entier >= 0\n        b : (int) entier >= 0\n    retour:\n        un entier positif\n    \"\"\"\n\n.....\n\nsomme(-3, 5)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2)  Version récursive normale"},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme_rec(a, b):\n    \"\"\"\n    Calcule la somme de a et b\n    paramètre:\n        a : (int) entier relatif\n        b : (int) entier relatif\n    retour:\n        un entier relatif\n    \"\"\"\n....\n    \nsomme_rec(-3, 5) ","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"3)  Version récursive terminale"},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme_rec_term(a, b):\n    \"\"\"\n    Calcule la somme de a et b\n    paramètre:\n        a : (int) entier relatif\n        b : (int) entier relatif\n    retour:\n        un entier relatif\n    \"\"\"\n....\n\nsomme_rec_term(-3, 5)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 7</h3>\n\n**Fonction paire**   "},{"metadata":{"trusted":false},"cell_type":"code","source":"def pair(n):\n    \"\"\"\n    Détermine si un nombre entier positif est pair\n    paramètre:\n        n : (int) entier >= 0\n    retour:\n        un booléen\n    \"\"\"\n....\n    \npair(8)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 8</h3>              \n\n**Fonction puissance**       \n1)  Version récursive normale"},{"metadata":{"trusted":false},"cell_type":"code","source":"def puissance(x, n):\n    \"\"\"\n    Calcule la puissance de x à l'exposant n\n    paramètre:\n        x : (flottant) \n        n : (int) entier relatif\n    retour:\n        un flottant\n    \"\"\"\n....\n\npuissance(7, 28)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2)  Version récursive terminale - On utilise un accumulateur"},{"metadata":{"trusted":false},"cell_type":"code","source":"def puissance(x, n, acc):\n    \"\"\"\n    Calcule la puissance de x à l'exposant n\n    paramètre:\n        x : (float) \n        n : (int) entier relatif\n        acc: (int) accumulateur\n    retour:\n        un flottant\n    \"\"\"\n    if n == 0:\n        return acc\n    else:\n        return puissance(x, n-1, x*acc)\n\npuissance(7, 28, 1)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 9</h3>              \n\n**Fonction puissance deuxième version**\n\nOn peut cependant utiliser une autre définition de la fonction mathématique\n<code>puissance(x, n)</code> qui **réduit** drastiquement le nombre d’appels récursifs emboîtés.     \n\nD'après l'énoncé, on peut décomposer la fonction **puissance** en utilisant la fonction **carre**.\nOn peut le faire avec **deux cas récursifs** qui distinguent la parité de ***n*** et deux cas de base (***n = 0*** et ***n = 1***), comme ci-dessous :\n\n*puissance(x, n) =*      \n\n   1 &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp; &#160; &#160;&ensp;si n = 0,             \n   x &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp; &#160; &#160;&ensp;si n = 1,               \n   carre(puissance(x, n//2)) &emsp;&emsp;&emsp;&ensp;&ensp;  si n > 1 et n est pair,              \n   x * carre(puissance(x, (n - 1)//2)) &ensp;  si n > 1 et n est impair.\n\n<br/>\n\nPour le codage en Python\n<ul>\n    \nL’appel à la fonction <code>carre(x)</code> est simplement remplacé par la multiplication <code>r * r</code>.    \nLe test de parité est réalisé par un **test à zéro du reste de la division entière par 2** (soit <code>r % 2 == 0</code>).\n\n</ul>    "},{"metadata":{"trusted":false},"cell_type":"code","source":"def puissance(x,n):\n    \"\"\"\n    Calcule la puissance de x à l'exposant n\n    paramètre:\n        x : (flottant) \n        n : (int) entier relatif\n    retour:\n        un flottant\n    \"\"\"\n....","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"puissance(7, 28)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 10</h3>              \n\n**Fonction somme sur les élément d'une tableau ou liste en python**\n\n1) On donne ici trois solutions possibles"},{"metadata":{"trusted":false},"cell_type":"code","source":"def somme(liste):\n    \"\"\"\n   Calcule la somme des éléments d'un tableau\n    paramètre:\n        liste : (list) une tableau de nombres\n    retour:\n        un flottant\n    \"\"\"\n....","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"ex_liste = [4, 7, 2]","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme1(ex_liste)","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme2(ex_liste)","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"somme3(ex_liste)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2) Explication du déroulement     \n\nOn utilise la fonction <code>somme(liste)</code> et la liste <code>ex_liste = [4, 7, 2]</code> en paramètre.\n..."},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 11</h3>              \n\n**Fonction mystere**\n\n1)...\n"},{"metadata":{"trusted":false},"cell_type":"code","source":"mystere(18)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"2) On va utiliser une fonction à deux paramètres.\nD'après le cours de première, il suffit d'ajouter **2^n si r est négatif**.      \nEn utilisant la fonction <code>mystere(n)</code> de l'énoncé, on obtient :"},{"metadata":{"trusted":false},"cell_type":"code","source":"def binaire(r, n):  # on suppose -2**(n-1) <= n < 2**(n-1)\n    \"\"\"\n    Détermine l'écriture binaire d'un nombre entier relatif\n    paramètre:\n        r : (int) entier relatif\n        n : (int) un entier naturel stritement positif\n    retour:\n        chaîne de caractères\n    \"\"\"\n ...","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"binaire(18, 8)","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"binaire(-18, 8)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"3) Résultat codé avec n caractères. On peut avoir eventuellement des 0 au début."},{"metadata":{"trusted":false},"cell_type":"code","source":"def binaire(r, n, cpt = 0):  # on suppose -2**(n-1) <= n < 2**(n-1)\n    \"\"\"\n    Détermine l'écriture binaire d'un nombre entier relatif\n    paramètre:\n        r : (int) entier relatif\n        n : (int) un entier naturel stritement positif\n        cpt: (int) compteur\n    retour:\n        chaîne de caractères\n    \"\"\"\n","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"binaire(-18, 16)","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"binaire(7, 8)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 12</h3>              \n\n**Fonction inverse**.    \nCette fonction inverse une chaîne de caractères."},{"metadata":{"trusted":false},"cell_type":"code","source":"def inverse(ch):\n    \"\"\"\n    Inverse l'ordre d'une chaîne de caractères\n    paramètre:\n        ch : (str) chaîne de caractères\n    retour:\n        chaîne de caractères\n    \"\"\"\n....","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"inverse(\"azerty\") ","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"<h3> Exercice 13</h3>              \n\n**Fonction nettoie**.    \nCette fonction elimine les doublons dans une liste.\n\nVersion itérative de l'énoncé  :"},{"metadata":{"trusted":false},"cell_type":"code","source":"def nettoie(liste):\n    \"\"\"\n    Elimine les doublons d'une liste\n    paramètre:\n        liste : (list) une liste de nombres\n    retour:\n        une liste\n    \"\"\"\n    n = len(liste)\n    k = 0\n    while k < n - 1:\n        if liste[k] != liste[k+1]:\n            k = k + 1\n        else:\n            del liste[k]\n            n = len(liste)","execution_count":null,"outputs":[]},{"metadata":{},"cell_type":"markdown","source":"Version récursive :"},{"metadata":{"trusted":false},"cell_type":"code","source":"def nettoie_rec(liste, k = 0):\n    \"\"\"\n    Elimine les doublons d'une liste\n    paramètre:\n        liste : (list) une liste de nombres\n    retour:\n        une liste\n    \"\"\"\n....","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"ex_liste = [1, 1, 2, 6, 6, 6, 8, 8, 9, 10]\nnettoie(ex_liste)\nex_liste","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"ex_liste = [1, 1, 2, 6, 6, 6, 8, 8, 9, 10]\nnettoie_rec(ex_liste)\nex_liste","execution_count":null,"outputs":[]},{"metadata":{"trusted":false},"cell_type":"code","source":"","execution_count":null,"outputs":[]}],"metadata":{"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"}},"nbformat":4,"nbformat_minor":2}
\ No newline at end of file