{"id":7433,"date":"2021-07-06T20:00:10","date_gmt":"2021-07-07T01:00:10","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/analisisnumerico\/?p=7433"},"modified":"2026-04-05T20:05:21","modified_gmt":"2026-04-06T01:05:21","slug":"s1eva2021paoi_t1-funcion-recursiva-y-raices-de-ecuaciones","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s1eva30\/s1eva2021paoi_t1-funcion-recursiva-y-raices-de-ecuaciones\/","title":{"rendered":"s1Eva2021PAOI_T1 Funci\u00f3n recursiva y ra\u00edces de ecuaciones"},"content":{"rendered":"\n<p><strong>Ejercicio<\/strong>: <a href=\"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-1eva30\/1eva2021paoi_t1-funcion-recursiva-raices-ecuaciones\/\" data-type=\"post\" data-id=\"7422\">1Eva2021PAOI_T1 Funci\u00f3n recursiva y ra\u00edces de ecuaciones<\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Literal a<\/h2>\n\n\n\n<p>Evaluando las sucesi\u00f3n de la forma recursiva:<\/p>\n\n\n\n<pre class=\"wp-block-code alignwide\"><code>xi\n&#091;-0.45       -0.4383     -0.4458     -0.441      -0.4441 \n -0.4421     -0.4434     -0.4425     -0.4431     -0.4427\n -0.4429     -0.4428     -0.4429     -0.4428     -0.4429]\nerrores\n&#091; 1.1745e-02 -7.5489e-03  4.8454e-03 -3.1127e-03  1.9986e-03\n -1.2837e-03  8.2430e-04 -5.2939e-04  3.3996e-04 -2.1833e-04\n  1.4021e-04 -9.0044e-05  5.7826e-05 -3.7136e-05  0.0000e+00]<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2021\/07\/1Eva2021PAOIT1Sucesion.png\" alt=\"1Eva2021PAOIT1Sucesion\" class=\"wp-image-18805\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">literal b<\/h2>\n\n\n\n<p>Se puede afirmar que converge, observe la diferencia entre cada dos valores consecutivos de la sucesi\u00f3n ... (continuar de ser necesario)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">literal c<\/h2>\n\n\n\n<p>Para el algoritmo se requiere la funci\u00f3n f(x) y su derivada f'(x)<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> f(x) = x +ln(x+2) <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> f'(x) = 1 + \\frac{1}{x+2} <\/span>\n\n\n\n<p>x<sub>0<\/sub> = -0.45<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">itera = 1<\/h3>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> x_{i+1} = x_i -\\frac{f(x_i)}{f'(x_i)} <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> x_{1} = x_0 -\\frac{f(x_0)}{f'(x_0)} = -0.45 -\\frac{-0.45+ln(-0.45+2)}{1 + \\frac{1}{-0.45+2}} <\/span>\n\n\n\n<p>x<sub>1<\/sub> = -0.44286<\/p>\n\n\n\n<p>error = |x<sub>1<\/sub>-x<sub>0<\/sub>| = |-0.44286 -(-0.45)| = 0.007139<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">itera = 2<\/h3>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> x_{2} = -0.4428 -\\frac{-0.4428 + ln(-0.45+2)}{1 + \\frac{1}{-0.4428+2}} <\/span>\n\n\n\n<p>x<sub>1<\/sub> = -0.44286<\/p>\n\n\n\n<p>error = |x<sub>1<\/sub>-x<sub>0<\/sub>| = |-0.44285 -(-4.4286)| = 6.4394e-06<\/p>\n\n\n\n<p>con lo que se cumple el valor de tolerancia y no se requiere otra iteraci\u00f3n<\/p>\n\n\n\n<p>la ra\u00edz se encuentra en x = -0.44286<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Soluci\u00f3n con algoritmo<\/h3>\n\n\n\n<pre class=\"wp-block-code alignwide\"><code>xi\n&#091;-0.45       -0.4383     -0.4458     -0.441      -0.4441 \n -0.4421     -0.4434     -0.4425     -0.4431     -0.4427\n -0.4429     -0.4428     -0.4429     -0.4428     -0.4429]\nerrores\n&#091; 1.1745e-02 -7.5489e-03  4.8454e-03 -3.1127e-03  1.9986e-03\n -1.2837e-03  8.2430e-04 -5.2939e-04  3.3996e-04 -2.1833e-04\n  1.4021e-04 -9.0044e-05  5.7826e-05 -3.7136e-05  0.0000e+00]\n&#091;'xi', 'xnuevo', 'tramo']\n&#091;&#091;-4.5000e-01 -4.4286e-01  7.1392e-03]\n &#091;-4.4286e-01 -4.4285e-01  6.4394e-06]]\nra\u00edz en:  -0.44285440100759543\ncon error de:  6.439362322474551e-06\n&gt;&gt;&gt; <\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2021\/07\/1Eva2021PAOIRaizNewtonRaphson.png\" alt=\"1Eva2021PAOIRaizNewtonRaphson\" class=\"wp-image-18806\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Algoritmo en Python<\/h2>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# literal a sucesi\u00f3n en forma recursiva\ndef secuenciaL(n,x0):\n    if n == 0:\n        xn = x0\n    if n&gt;0:\n        xn = np.log(1\/(2+secuenciaL(n-1,x0)))\n    return(xn)\nx0 = -0.45\nn = 15\nxi = np.zeros(n,dtype=float)\nxi&#x5B;0] = x0\nerrado = np.zeros(n,dtype=float)\nfor i in range(1,n,1):\n    xi&#x5B;i] = secuenciaL(i,x0)\n    errado&#x5B;i-1] = xi&#x5B;i] - xi&#x5B;i-1]\n   \nnp.set_printoptions(precision=4)\nprint('xi: ')\nprint(xi)\nprint('errado: ')\nprint(errado)\n\n#Grafica literal a y b\nplt.plot(xi)\nplt.plot(xi,'o')\nplt.xlabel('n')\nplt.ylabel('xn')\nplt.show()\n\n# M\u00e9todo de Newton-Raphson\n# Ejemplo 1 (Burden ejemplo 1 p.51\/pdf.61)\n\nimport numpy as np\n\n# INGRESO\nfx  = lambda x: x+np.log(x+2)\ndfx = lambda x: 1+ 1\/(x+2)\n\nx0 = -0.45\ntolera = 0.0001\n\na = -0.5\nb = -0.2\nmuestras = 21\n# PROCEDIMIENTO\ntabla = &#x5B;]\ntramo = abs(2*tolera)\nxi = x0\nwhile (tramo&gt;=tolera):\n    xnuevo = xi - fx(xi)\/dfx(xi)\n    tramo  = abs(xnuevo-xi)\n    tabla.append(&#x5B;xi,xnuevo,tramo])\n    xi = xnuevo\n\n# convierte la lista a un arreglo.\ntabla = np.array(tabla)\nn = len(tabla)\n\n# para la gr\u00e1fica\nxj = np.linspace(a,b,muestras)\nfj = fx(xj)\n\n# SALIDA\nprint(&#x5B;'xi', 'xnuevo', 'tramo'])\nnp.set_printoptions(precision = 4)\nprint(tabla)\nprint('raiz en: ', xi)\nprint('con error de: ',tramo)\n\nplt.plot(xj,fj)\nplt.axhline(0, color='grey')\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.show()\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">litera d: tarea<\/h2>\n","protected":false},"excerpt":{"rendered":"<p>Ejercicio: 1Eva2021PAOI_T1 Funci\u00f3n recursiva y ra\u00edces de ecuaciones Literal a Evaluando las sucesi\u00f3n de la forma recursiva: literal b Se puede afirmar que converge, observe la diferencia entre cada dos valores consecutivos de la sucesi\u00f3n ... (continuar de ser necesario) literal c Para el algoritmo se requiere la funci\u00f3n f(x) y su derivada f'(x) x0 [&hellip;]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-mn-ejemplo","format":"standard","meta":{"footnotes":""},"categories":[46],"tags":[58,54],"class_list":["post-7433","post","type-post","status-publish","format-standard","hentry","category-mn-s1eva30","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/7433","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=7433"}],"version-history":[{"count":4,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/7433\/revisions"}],"predecessor-version":[{"id":23843,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/7433\/revisions\/23843"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=7433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=7433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=7433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}