{"id":5885,"date":"2016-06-18T11:25:55","date_gmt":"2016-06-18T14:25:55","guid":{"rendered":"https:\/\/www.antonioguilherme.web.br.com\/blog\/?p=5885"},"modified":"2022-09-08T08:38:43","modified_gmt":"2022-09-08T11:38:43","slug":"metodo-de-newton-raphson","status":"publish","type":"post","link":"https:\/\/www.antonioguilherme.web.br.com\/blog\/matematica\/metodo-de-newton-raphson\/","title":{"rendered":"M\u00e9todo de Newton-Raphson"},"content":{"rendered":"<hr \/>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Resolver equa\u00e7\u00f5es n\u00e3o lineares faz parte de importantes problemas pr\u00e1ticos da engenharia. Existem diversas metodologias para resolver este problema, mas a comunidade cient\u00edfica consagrou o <a href=\"https:\/\/en.wikipedia.org\/wiki\/Newton%27s_method\" target=\"_blank\" rel=\"noopener\">m\u00e9todo de Newton-Raphson<\/a>.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00f5es n\u00e3o-lineares n\u00e3o possuem solu\u00e7\u00f5es anal\u00edticas, e os m\u00e9todo num\u00e9ricos continuam a \u00fanica forma de resolv\u00ea-las.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">De acordo com o m\u00e9todo de Newton-Raphson, dada uma fun\u00e7\u00e3o qualquer <em>f(i)<\/em>, que possua derivada <em>f'(i)<\/em>, o valor de i que resolve a equa\u00e7\u00e3o <em>f(i)<\/em>=0 ser\u00e1 dado por:<\/span><\/p>\n<figure id=\"attachment_5870\" aria-describedby=\"caption-attachment-5870\" style=\"width: 307px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/newton_rapson_1.png\"><img decoding=\"async\" class=\"wp-image-5870\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/newton_rapson_1.png\" alt=\"equa\u00e7\u00e3o\" width=\"307\" height=\"100\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/newton_rapson_1.png 423w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/newton_rapson_1-300x98.png 300w\" sizes=\"(max-width: 307px) 100vw, 307px\" \/><\/a><figcaption id=\"caption-attachment-5870\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">equa\u00e7\u00e3o 1<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">k \u00e9 o n\u00famero da itera\u00e7\u00e3o;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><em>f<\/em> \u00e9 a fun\u00e7\u00e3o;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><em>f&#8217;<\/em> \u00e9 a derivada da fun\u00e7\u00e3o;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a vari\u00e1vel que desejamos encontrar;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">k \u00e9 o n\u00famero da itera\u00e7\u00e3o.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Este m\u00e9todo necessita de um valor inicial para iniciar o processo de busca da solu\u00e7\u00e3o. Dependendo da fun\u00e7\u00e3o e desse valor inicial, o processo pode divergir e n\u00e3o encontrar a solu\u00e7\u00e3o correta.<\/span><\/p>\n<hr \/>\n<h5 id='exemplo-1'  id=\"boomdevs_1\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplo #1<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Determine a taxa de desconto (i) de uma s\u00e9rie de n pagamentos (P<sub>mt<\/sub>) iguais e valor presente V<sub>p<\/sub>.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">As fun\u00e7\u00f5es <a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/\">f(i)<\/a> e f'(i) abaixo resolvem este problema.<\/span><\/p>\n<figure id=\"attachment_5880\" aria-describedby=\"caption-attachment-5880\" style=\"width: 573px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7.png\"><img decoding=\"async\" class=\"wp-image-5880\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7.png\" alt=\"Equa\u00e7\u00e3o\" width=\"573\" height=\"100\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7.png 773w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7-300x52.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7-768x134.png 768w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/a><figcaption id=\"caption-attachment-5880\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 2<\/span><\/figcaption><\/figure>\n<figure id=\"attachment_5889\" aria-describedby=\"caption-attachment-5889\" style=\"width: 505px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_rapson_2.png\"><img decoding=\"async\" class=\"wp-image-5889\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_rapson_2.png\" alt=\"Equa\u00e7\u00e3o\" width=\"505\" height=\"100\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_rapson_2.png 682w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_rapson_2-300x59.png 300w\" sizes=\"(max-width: 505px) 100vw, 505px\" \/><\/a><figcaption id=\"caption-attachment-5889\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 3<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">V<sub>p<\/sub> \u00e9 o valor presente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">P<sub>mt<\/sub> \u00e9 o valor do pagamento peri\u00f3dico;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 o n\u00famero de pagamentos peri\u00f3dicos.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Considerando o chute inicial igual a 5%, a tabela e a figura abaixas mostram que o m\u00e9todo convergiu em 3 itera\u00e7\u00f5es.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><span id=\"tablepress-44-description\" class=\"tablepress-table-description tablepress-table-description-id-44\">n=20, Vp\/Pmt=10, i(0)=5%<\/span>\n\n<table id=\"tablepress-44\" class=\"tablepress tablepress-id-44\" aria-describedby=\"tablepress-44-description\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">k<\/th><th class=\"column-2\">i(k) [%]<\/th><th class=\"column-3\">f(ik)<\/th><th class=\"column-4\">f'(ik)<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">0<\/td><td class=\"column-2\">5,0<\/td><td class=\"column-3\">-0,12<\/td><td class=\"column-4\">2,82<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">1<\/td><td class=\"column-2\">9,4<\/td><td class=\"column-3\">0,10<\/td><td class=\"column-4\">6,95<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">2<\/td><td class=\"column-2\">7,9<\/td><td class=\"column-3\">0,01<\/td><td class=\"column-4\">5,93<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">3<\/td><td class=\"column-2\">7,8<\/td><td class=\"column-3\">0,00<\/td><td class=\"column-4\">5,83<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-44 from cache --><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/Newton_rap_2.png\"><img decoding=\"async\" class=\"size-full wp-image-5921 aligncenter\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/Newton_rap_2.png\" alt=\"Newton_rap_2\" width=\"600\" height=\"400\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/Newton_rap_2.png 600w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/Newton_rap_2-300x200.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/a>O que aconteceria se o chute inicial tivesse sido menor do que 3?<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O que aconteceria se o chute inicial fosse zero?<\/span><\/p>\n<h5 id='exemplo-2'  id=\"boomdevs_2\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplo #2<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Determine a taxa interna de retorno de um fluxo de caixa gen\u00e9rico.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A determina\u00e7\u00e3o da taxa interna de retorno (TIR) requer a solu\u00e7\u00e3o da equa\u00e7\u00e3o abaixo.<\/span><\/p>\n<figure id=\"attachment_3148\" aria-describedby=\"caption-attachment-3148\" style=\"width: 287px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir.png\"><img decoding=\"async\" class=\"wp-image-3148\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir.png\" alt=\"\" width=\"287\" height=\"100\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir.png 482w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir-300x105.png 300w\" sizes=\"(max-width: 287px) 100vw, 287px\" \/><\/a><figcaption id=\"caption-attachment-3148\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 4<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A substitui\u00e7\u00e3o de vari\u00e1vel abaixo facilita a solu\u00e7\u00e3o do problema.<\/span><\/p>\n<figure id=\"attachment_6073\" aria-describedby=\"caption-attachment-6073\" style=\"width: 565px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_1.png\"><img decoding=\"async\" class=\"wp-image-6073\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_1.png\" alt=\"tir_1\" width=\"565\" height=\"200\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_1.png 582w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_1-300x106.png 300w\" sizes=\"(max-width: 565px) 100vw, 565px\" \/><\/a><figcaption id=\"caption-attachment-6073\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 5<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Desta forma, o problema de encontrar a TIR transformou-se na determina\u00e7\u00e3o das ra\u00edzes (os valores de x) de um polin\u00f4mio de grau <em>N<\/em> com coeficientes reais.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Isto significa que existem <em>N<\/em> poss\u00edveis solu\u00e7\u00f5es; reais ou complexas. Solu\u00e7\u00f5es complexas n\u00e3o possuem significado financeiro e, consequentemente, s\u00e3o desconsideradas. <span id='easy-footnote-1-5885' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/matematica\/metodo-de-newton-raphson\/#easy-footnote-bottom-1-5885\" title=\" Como os fluxos de caixa l\u00edquidos (&lt;em&gt;F&lt;sub&gt;n&lt;\/sub&gt;&lt;\/em&gt;) tamb\u00e9m s\u00e3o n\u00fameros reais, as poss\u00edveis ra\u00edzes complexas aparecer\u00e3o sempre como pares de n\u00fameros complexos conjugados. \"><sup>1<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Matematicamente, a taxa interna de retorno pode assumir valores positivos ou negativos. Apesar de poss\u00edveis, taxas negativas s\u00e3o indesej\u00e1veis porque representam preju\u00edzo.<span id='easy-footnote-2-5885' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/matematica\/metodo-de-newton-raphson\/#easy-footnote-bottom-2-5885\" title=\" Guardar o dinheiro no colch\u00e3o ou na conta corrente, sem qualquer remunera\u00e7\u00e3o, seriam op\u00e7\u00f5es mais lucrativas. \"><sup>2<\/sup><\/a><\/span>Portanto, consideramos apenas taxas de retorno positivas porque simplifica a solu\u00e7\u00e3o do problema, em decorr\u00eancia da vari\u00e1vel auxiliar <em>x<\/em> ser sempre menor do que um neste caso. Consequentemente, os termos da s\u00e9rie da equa\u00e7\u00e3o 5 convergem rapidamente para zero, e podemos aplicar teoremas matem\u00e1ticos importantes.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Segundo Descartes, o n\u00famero de ra\u00edzes reais e positivas de qualquer polin\u00f4mio depende do n\u00famero de varia\u00e7\u00f5es de sinal dos seus coeficientes. <span id='easy-footnote-3-5885' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/matematica\/metodo-de-newton-raphson\/#easy-footnote-bottom-3-5885\" title=\" desconsidera-se os coeficientes nulos\"><sup>3<\/sup><\/a><\/span> Neste caso, o n\u00famero de taxas de desconto positivas poss\u00edveis depende do n\u00famero de varia\u00e7\u00f5es de sinal das receitas l\u00edquidas. De acordo com esta regra, o n\u00famero de solu\u00e7\u00f5es poss\u00edveis maiores que zero ser\u00e1 menor ou igual que o n\u00famero de varia\u00e7\u00f5es de sinal.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Al\u00e9m disso, a diferen\u00e7a entre o n\u00famero de varia\u00e7\u00f5es de sinal e o n\u00famero de ra\u00edzes reais positivas \u00e9 sempre um n\u00famero par e inteiro. Por exemplo, fluxos de caixa com valor inicial negativo (investimento) seguidos de receitas l\u00edquidas positivas apresentam apenas uma ou nenhuma TIR positiva. Contudo, como a diferen\u00e7a entre o n\u00famero de varia\u00e7\u00f5es e o n\u00famero de solu\u00e7\u00f5es positivas deve ser par, <strong>fluxos de caixa com apenas uma varia\u00e7\u00e3o de sinal possuem necessariamente uma TIR positiva<\/strong>. Esta conclus\u00e3o independe da ordem do polin\u00f4mio, neste caso, do n\u00famero de per\u00edodos do fluxo de caixa.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Taxas de desconto negativas s\u00e3o poss\u00edveis, apesar de indesej\u00e1veis. O n\u00famero de ra\u00edzes reais negativas de determinado polin\u00f4mio depende do n\u00famero de varia\u00e7\u00f5es de sinal dos coeficientes de p(-<em>x<\/em>). A altera\u00e7\u00e3o do sinal de <em>x <\/em>acarreta mudan\u00e7a de sinal apenas dos coeficientes de exponentes \u00edmpares, e obt\u00e9m-se o n\u00famero de ra\u00edzes negativas aplicando as regras anteriores.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">As equa\u00e7\u00f5es abaixo fornecem as fun\u00e7\u00f5es para o m\u00e9todo de Newton-Raphson referentes \u00e0 equa\u00e7\u00e3o 5, e s\u00e3o polin\u00f4mios da vari\u00e1vel <em>x.\u00a0<\/em><\/span><\/p>\n<figure id=\"attachment_6096\" aria-describedby=\"caption-attachment-6096\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_4.png\"><img decoding=\"async\" class=\"wp-image-6075\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_3.png\" alt=\"tir_3\" width=\"300\" height=\"200\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_3.png 352w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/tir_3-300x200.png 300w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-6096\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\"><img decoding=\"async\" class=\"wp-image-6096\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_4.png\" alt=\"TIR_4\" width=\"330\" height=\"100\" \/> Equa\u00e7\u00e3o 6<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Determina-se o valor da TIR a partir da equa\u00e7\u00e3o 5 ap\u00f3s se encontrar o valor de <em>x<\/em>.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O objetivo da equa\u00e7\u00e3o consiste em determinar precisamente o valor de <em>x<\/em> para f(<em>x<\/em>)=0, e a derivada da fun\u00e7\u00e3o existe apenas para acelerar o processo. Por isso, pode-se aproximar o valor da derivada pelos seus fatores iniciais.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A Tabela abaixo apresenta quatro fluxos de caixa com quantidades distintas de trocas de sinal. Observa-se que, dependendo do n\u00famero de invers\u00f5es de sinal no fluxo de caixa, o n\u00famero de solu\u00e7\u00f5es reais positivas varia.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><span id=\"tablepress-46-description\" class=\"tablepress-table-description tablepress-table-description-id-46\">Fluxos de Caixa<\/span>\n\n<table id=\"tablepress-46\" class=\"tablepress tablepress-id-46\" aria-describedby=\"tablepress-46-description\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">n<\/th><th class=\"column-2\">Fluxo #A<\/th><th class=\"column-3\">Fluxo #B<\/th><th class=\"column-4\">Fluxo #C<\/th><th class=\"column-5\">Fluxo #D<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">0<\/td><td class=\"column-2\">-0,1<\/td><td class=\"column-3\">-0,1<\/td><td class=\"column-4\">-0,1<\/td><td class=\"column-5\">-0,1<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">1<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">-1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">1<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">2<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">-1<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">3<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">-1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">1<\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\">4<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">-1<br \/>\n<\/td>\n<\/tr>\n<tr class=\"row-7\">\n\t<td class=\"column-1\">5<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">-1<\/td><td class=\"column-4\">-0,1<\/td><td class=\"column-5\">1<br \/>\n<\/td>\n<\/tr>\n<tr class=\"row-8\">\n\t<td class=\"column-1\">6<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">-1<\/td>\n<\/tr>\n<tr class=\"row-9\">\n\t<td class=\"column-1\">7<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">-1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">1<\/td>\n<\/tr>\n<tr class=\"row-10\">\n\t<td class=\"column-1\">8<\/td><td class=\"column-2\">1<\/td><td class=\"column-3\">1<\/td><td class=\"column-4\">1<\/td><td class=\"column-5\">-1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-46 from cache --><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph.png\"><img decoding=\"async\" class=\"size-large wp-image-6106 aligncenter\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph-1024x412.png\" alt=\"TIR_Graph\" width=\"840\" height=\"338\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph-1024x412.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph-300x121.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph-768x309.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph-1200x483.png 1200w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/TIR_Graph.png 1920w\" sizes=\"(max-width: 840px) 100vw, 840px\" \/><\/a><\/span><\/p>\n<hr \/>\n<h2 id='refer\u00eancias'  id=\"boomdevs_3\"><span style=\"font-family: verdana, geneva, sans-serif;\">Refer\u00eancias<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">HAUSER, J.R., <strong>Numerical Methods for Nonlinear Engineering Models<\/strong>, Springer, 2009.<\/span><\/p>\n<hr \/>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">[wpdevart_youtube width=&#8221;640&#8243; height=&#8221;385&#8243; autoplay=&#8221;0&#8243; theme=&#8221;dark&#8221; loop_video=&#8221;0&#8243; enable_fullscreen=&#8221;1&#8243; show_related=&#8221;1&#8243; show_popup=&#8221;0&#8243; thumb_popup_width=&#8221;213&#8243; thumb_popup_height=&#8221;128&#8243; show_title=&#8221;1&#8243; show_youtube_icon=&#8221;1&#8243; show_annotations=&#8221;1&#8243; show_progress_bar_color=&#8221;white&#8221; autohide_parameters=&#8221;1&#8243; set_initial_volume=&#8221;true&#8221; initial_volume=&#8221;72&#8243; disable_keyboard=&#8221;1&#8243;]04wFBrlYh_w[\/wpdevart_youtube]<\/span><\/p>\n<div id=\"gtx-trans\" style=\"position: absolute; left: 543px; top: 7936.64px;\">\n<div class=\"gtx-trans-icon\"><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Resolver equa\u00e7\u00f5es n\u00e3o lineares faz parte de importantes problemas pr\u00e1ticos da engenharia. Existem diversas metodologias para resolver este problema, mas a comunidade cient\u00edfica consagrou o m\u00e9todo de Newton-Raphson. Equa\u00e7\u00f5es n\u00e3o-lineares n\u00e3o possuem solu\u00e7\u00f5es anal\u00edticas, e os m\u00e9todo num\u00e9ricos continuam a \u00fanica forma de resolv\u00ea-las. De acordo com o m\u00e9todo de Newton-Raphson, dada uma fun\u00e7\u00e3o qualquer [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":5920,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"templates\/template-full-width.php","format":"aside","meta":{"_uag_custom_page_level_css":"","footnotes":""},"categories":[93],"tags":[199],"class_list":["post-5885","post","type-post","status-publish","format-aside","has-post-thumbnail","hentry","category-matematica","tag-newton-raphson","post_format-post-format-aside"],"uagb_featured_image_src":{"full":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1.png",800,200,false],"thumbnail":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1-150x150.png",150,150,true],"medium":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1-300x75.png",300,75,true],"medium_large":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1-768x192.png",580,145,true],"large":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1.png",580,145,false],"1536x1536":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1.png",800,200,false],"2048x2048":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1.png",800,200,false],"post-thumbnail":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1.png",800,200,false],"twentytwenty-fullscreen":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2016\/06\/newton_1.png",800,200,false]},"uagb_author_info":{"display_name":"admin","author_link":"https:\/\/www.antonioguilherme.web.br.com\/blog\/author\/admin\/"},"uagb_comment_info":0,"uagb_excerpt":"Resolver equa\u00e7\u00f5es n\u00e3o lineares faz parte de importantes problemas pr\u00e1ticos da engenharia. Existem diversas metodologias para resolver este problema, mas a comunidade cient\u00edfica consagrou o m\u00e9todo de Newton-Raphson. Equa\u00e7\u00f5es n\u00e3o-lineares n\u00e3o possuem solu\u00e7\u00f5es anal\u00edticas, e os m\u00e9todo num\u00e9ricos continuam a \u00fanica forma de resolv\u00ea-las. De acordo com o m\u00e9todo de Newton-Raphson, dada uma fun\u00e7\u00e3o qualquer&hellip;","_links":{"self":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/posts\/5885","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/comments?post=5885"}],"version-history":[{"count":45,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/posts\/5885\/revisions"}],"predecessor-version":[{"id":43122,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/posts\/5885\/revisions\/43122"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/media\/5920"}],"wp:attachment":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/media?parent=5885"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/categories?post=5885"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/tags?post=5885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}