{"id":544,"date":"2018-09-01T15:11:11","date_gmt":"2018-09-01T13:11:11","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=544"},"modified":"2023-12-30T17:10:17","modified_gmt":"2023-12-30T16:10:17","slug":"la-methode-de-horner","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/","title":{"rendered":"La m\u00e9thode de H\u00f6rner"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">La m\u00e9thode de H\u00f6rner va nous permettre de trouver les coefficients du polyn\u00f4me&nbsp;<em>Q<\/em> tel que :&nbsp;\\[P(x)=(x-a)Q(x)\\] o\u00f9 <em>P<\/em> est un polyn\u00f4me dont une racine est \u00e9gale \u00e0 <em>a<\/em>.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Bien entendu, il existe d&rsquo;autres m\u00e9thodes, comme la <em>division euclidienne de polyn\u00f4mes&nbsp;<\/em>ou encore la <em>m\u00e9thode des coefficients ind\u00e9termin\u00e9s<\/em>, mais nous allons voir que la m\u00e9thode de H\u00f6rner a deux avantages sur les autres : sa rapidit\u00e9 et le fait que l&rsquo;on puisse la programmer ais\u00e9ment.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"658\" height=\"167\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2.png\" alt=\"m\u00e9thode de H\u00f6rner\" class=\"wp-image-547\" style=\"object-fit:cover\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2.png 658w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2-300x76.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2-600x152.png 600w\" sizes=\"auto, (max-width: 658px) 100vw, 658px\" \/><\/a><\/figure>\n<\/div>\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Methode_de_Horner_un_exemple_simple\" >M\u00e9thode de H\u00f6rner: un exemple simple<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Generalisation\" >G\u00e9n\u00e9ralisation<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Schematisation\" >Sch\u00e9matisation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Application\" >Application<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Methode_de_Horner_programme_Python\" >M\u00e9thode de H\u00f6rner: programme Python<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Une_implementation_elementaire\" >Une impl\u00e9mentation \u00e9l\u00e9mentaire<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2018\/09\/01\/la-methode-de-horner\/#Methode_de_Horner_en_POO\" >M\u00e9thode de H\u00f6rner en POO<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Methode_de_Horner_un_exemple_simple\"><\/span>M\u00e9thode de H\u00f6rner: un exemple simple<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Consid\u00e9rons le polyn\u00f4me :&nbsp;\\[ P(x)=x^4-3x^3+7x^2-4x-12,\\]&nbsp;dont une racine \u00e9vidente est <em>a&nbsp;<\/em>= 2.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Nous allons r\u00e9fl\u00e9chir \u00e0 une m\u00e9thode qui nous permet de trouver les coefficients de <em>Q<\/em>(<em>x<\/em>), tel que <em>P<\/em>(<em>x<\/em>) = (<em>x&nbsp;<\/em>&#8211; 2)<em>Q<\/em>(<em>x<\/em>), \u00e0 l&rsquo;aide d&rsquo;une division euclidienne.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/division-euclidienne.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"162\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/division-euclidienne-300x162.png\" alt=\"division euclidienne de polyn\u00f4mes\" class=\"wp-image-545\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/division-euclidienne-300x162.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/division-euclidienne-600x324.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/division-euclidienne.png 604w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On peut alors remarquer que :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(-1 = -3 + 2\\times2 \\)<\/li>\n\n\n\n<li>\\( 5=7+2\\times(-1)\\)<\/li>\n\n\n\n<li>\\(6=-4+2\\times 5\\)<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Generalisation\"><\/span>G\u00e9n\u00e9ralisation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Posons \\(P(x)=\\displaystyle\\sum_{k\\leq n} p_kx^k\\), \\(Q(x)=\\displaystyle\\sum_{k&lt;n} q_kx^k\\) et&nbsp;<em>a<\/em> une racine de <em>P<\/em>.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">D&rsquo;apr\u00e8s le raisonnement pr\u00e9c\u00e9dent, on peut \u00e9crire :&nbsp;\\[ \\begin{cases} q_{n-1}=p_n\\\\ q_k=p_{k+1}+aq_{k+1}\\quad\\forall\\ 0\\leq k &lt; n\\end{cases} \\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Schematisation\"><\/span>Sch\u00e9matisation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"729\" height=\"175\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner.png\" alt=\"m\u00e9thode de H\u00f6rner\" class=\"wp-image-546\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner.png 729w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-300x72.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-600x144.png 600w\" sizes=\"auto, (max-width: 729px) 100vw, 729px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Application\"><\/span>Application<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"658\" height=\"167\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2.png\" alt=\"m\u00e9thode de H\u00f6rner\" class=\"wp-image-547\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2.png 658w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2-300x76.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2018\/09\/tableau-horner-2-600x152.png 600w\" sizes=\"auto, (max-width: 658px) 100vw, 658px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Soit :&nbsp;\\[ P(x)=3x^5-4x^4+8x^3-3x^2-2x-2.\\]&nbsp;Une racine de&nbsp;<em>P<\/em> est <em>a&nbsp;<\/em>= 1, d&rsquo;o\u00f9 :<br>D&rsquo;o\u00f9 :&nbsp;\\[&nbsp;P(x)=(x-1)(3x^4-x^3+7x^2+4x+2).&nbsp;\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Methode_de_Horner_programme_Python\"><\/span>M\u00e9thode de H\u00f6rner: programme Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_implementation_elementaire\"><\/span>Une impl\u00e9mentation \u00e9l\u00e9mentaire<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Voici un programme Python qui n\u00e9cessite tr\u00e8s peu de m\u00e9moire puisque sa complexit\u00e9 est lin\u00e9aire.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def horner(r,P):\n    Q = [ P[0] ]\n    for k in range( 1 , len(P)-1 ):\n        Q.append( P[k] + r*Q[k-1] )\n        \n    return Q\n<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cela n\u00e9cessite toutefois quelques explications pour comprendre.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cette fonction prend pour arguments la racine du polyn\u00f4me, et la liste des coefficients du polyn\u00f4me, suivant les puissances d\u00e9croissantes. Ainsi, pour le polyn\u00f4me $$P(x)=30x^4-11x^3-219x^2-61x+21$$on pourra \u00e9crire:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; horner(3,&#91;30,-11,-219,-61,21])\n&#91;30, 79, 18, -7]<\/code><\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cela signifie donc que:$$P(x)=(x-3)(30x^3+79x^2+18x-7).$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Methode_de_Horner_en_POO\"><\/span>M\u00e9thode de H\u00f6rner en POO<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Je vous propose ici de cr\u00e9er une classe <em>Polynome<\/em> afin d&rsquo;y ins\u00e9rer une m\u00e9thode <em>horner(racine)<\/em>:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">class Polynome:\n    def __init__(self,*coefs):\n        if type(coefs[0]) == tuple:\n            self.coefs = [ i for i in coefs[0] ]\n        else:   \n            self.coefs = coefs        \n        self.deg = len( self.coefs ) - 1\n        \n    def __str__(self):\n        D = { self.deg-i:self.coefs[i] for i in range( len( self.coefs ) ) }\n        return str(D)\n        \n    def horner(self,r):\n        Q = [ self.coefs[0] ]\n        for k in range( 1 , len(self.coefs)-1 ):\n            Q.append( self.coefs[k] + r*Q[k-1] )\n        \n        return Polynome( tuple(Q) )<\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Bien s\u00fbr, la classe <em>Polynome<\/em> n&rsquo;est pas compl\u00e8te, et se concentre uniquement sur ce qui nous int\u00e9resse ici. Avec l&rsquo;exemple du polyn\u00f4me pr\u00e9c\u00e9dent, cela donne:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>>>> P = Polynome(30,-11,-219,-61,21)\n>>> print( P.horner(3) )\n{3: 30, 2: 79, 1: 18, 0: -7}<\/code><\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cela dit, l&rsquo;interface graphique est importante&#8230; et je ne peux pas laisser cet affichage ainsi! Rectifions donc l&rsquo;affichage du r\u00e9sultat:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">class Polynome:\n    def __init__(self,*coefs):\n        if type(coefs[0]) == tuple:\n            self.coefs = [ i for i in coefs[0] ]\n        else:   \n            self.coefs = coefs        \n        self.deg = len( self.coefs ) - 1\n        \n    def __str__(self):\n        E = { u[0]:\"x\" + chr(u[1]) for u in ((2,0x00B2), (3,0x00B3), (4,0x2074), (5,0x2075), (6,0x2076), (7,0x2077), (8,0x2078), (9,0x2079)) }\n        E[0] = '' \n        E[1] = 'x'\n        if self.deg > 9:\n            for d in range( self.deg - 9 ):\n                E[ 10+d ] = 'x^{'+str(10+d)+'}'\n        start = True\n        D = { self.deg-i:self.coefs[i] for i in range( len( self.coefs ) ) }\n        R = ''\n        for k,v in D.items():\n            if v > 0:\n                if not start:\n                    R += '+'\n                else:\n                    start = False\n                if v == 1:\n                    if k != 0:\n                        R += E[k]\n                    else:\n                        R += '1'\n                else:\n                    R += str(v) + E[k]\n            elif v &lt; 0:\n                if v == -1:\n                    if k != 0:\n                        R += '-' + E[k]\n                    else:\n                        R += '-1'\n                else:   \n                    R += str(v) + E[k]\n        return str(R)\n        \n    def horner(self,r):\n        Q = [ self.coefs[0] ]\n        for k in range( 1 , len(self.coefs)-1 ):\n            Q.append( self.coefs[k] + r*Q[k-1] )\n        \n        return Polynome( tuple(Q) )<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>>>> P = Polynome(1,0,0,0,0,0,0,0,0,1,-1,-1)\n>>> print( P.horner(1) )\nx^{10}+x\u2079+x\u2078+x\u2077+x\u2076+x\u2075+x\u2074+x\u00b3+x\u00b2+2x+1<\/code><\/pre>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">L&rsquo;affichage Unicode des exposants ne va que jusqu&rsquo;\u00e0 9&#8230; Nous sommes donc oblig\u00e9s d&rsquo;afficher les exposants sup\u00e9rieurs \u00e0 9 de la mani\u00e8re classique : \u00ab\u00a0x^{&#8230;}\u00a0\u00bb (les accolades sont l\u00e0 pour la lisibilit\u00e9&#8230;)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La m\u00e9thode de H\u00f6rner va nous permettre de trouver les coefficients du polyn\u00f4me&nbsp;Q tel que :&nbsp;\\[P(x)=(x-a)Q(x)\\] o\u00f9 P est un polyn\u00f4me dont une racine est \u00e9gale \u00e0 a. Bien entendu, il existe d&rsquo;autres m\u00e9thodes, comme la division euclidienne de polyn\u00f4mes&nbsp;ou encore la m\u00e9thode des coefficients ind\u00e9termin\u00e9s, mais nous allons voir [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":9453,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,6,5],"tags":[51,50],"class_list":["post-544","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-informatique","category-mathematiques","category-python","tag-algorithme","tag-factorisation"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>La m\u00e9thode de H\u00f6rner - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"La m\u00e9thode de H\u00f6rner est une m\u00e9thode de factorisation de polyn\u00f4mes \u00e0 partir d&#039;une racine \u00e9vidente. 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