Capítulo 2 - Soluções de equações em uma variável.ipynb

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    "# Capítulo 2 - Soluções de equações em uma variável #\n",
    "\n",
    "Os métodos numéricos discutidos neste capítulo visam aproximar soluções para equações de uma variável quando as soluções exatas destas equações não podem ser obtidas através de métodos algébricos (solução explícita), como é o caso da equação $90000 e^{\\lambda} + \\frac{10500}{\\lambda} (e^{\\lambda} -1) - 125000 = 0$.\n",
    "\n",
    "Estes métodos utilizam teoremas matemáticos para criar algoritmos iterativos que convergem para a solução da equação quando os parâmetros iniciais são adequadamente escolhidos. Cada método apresentado neste capítulo possui suas especificidades e critérios de convergência.\n",
    "\n",
    "Os métodos serão implementados para resolver uma das questões mais básicias da aproximação numérica: a determinação de raízes (ou zeros) de uma função do tipo $f(x) = 0$."
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    "## Método da Bissecção #\n",
    "\n",
    "Em computação, os processos que usam da técnica de divisão contínua de conjuntos ao meio para obter um resultado são chamados de procedimentos de busca binária. O Método da Bissecção é um desses procedimentos, e baseia-se no **Teorema do Valor Intermediário** para buscar a raiz de uma função em um determinado intervalo $[a,b]$ através de sucessivas subdivisões deste. \n",
    "\n",
    "---\n",
    "<p style=\"text-align: center;\">**Teorema do Valor Intermediário**</p>\n",
    "\n",
    "Supondo que $f$ é uma função contínua definida no intervalo $[a, b]$, e que $f(a)$ e $f(b)$ possuem sinais opostos, ou seja, $f(a) . f(b) < 0$, então existirá pelo menos um número $p$ em $(a, b)$ com $f(p) = 0$. \n",
    "\n",
    "---\n",
    "\n",
    "O Método da Bissecção inicia com a definição de um intervalo $[a, b]$ para o qual uma determinada função $f$ atenda aos critérios do teorema do valor intermediário, garantindo assim a existência de uma raiz da equação $f(x)=0$.\n",
    "\n",
    "Definido tal intervalo calcula-se o valor da função $f(x)$ em seu ponto médio $(p)$, sendo:\n",
    "\n",
    "$${p}={a}_{n}+\\frac{{b}_{n}-{a}_{n}}{2}=\\frac{{a}_{n}+{b}_{n}}{2}$$.\n",
    "\n",
    "a cada subdivisão, o ponto médio do intervalo, $p$, é testado, podendo uma das 3 seguintes situações ocorrer:\n",
    "\n",
    "<p style=\"text-align: center;\"> 1. $f(p)=0$; </p>\n",
    "<p style=\"text-align: center;\"> 2. $f(p)>0$; </p>\n",
    "<p style=\"text-align: center;\"> 3. $f(p)<0$. </p>\n",
    "\n",
    "A ocorrência da situação 1 (ou se atende aos critérios de tolerância estabelecidos) significa que $p$ é a raiz da equação $f(x)=0$ e o método é encerrado. Já as situações 2 e 3 indicam a necessidade de uma nova iteração que avalie os pontos $a$, $b$, e o ponto médio entre eles, $p$: se $f(a) . f(p) < 0$ significa que a raiz da função está no intervalo $[a_n,b_n]=[a, p]$. Neste caso, $a_n$ mantém o valor de $a$ e $b_n$ assume o valor de $p$; caso o contrário, se $f(p) . f(b) < 0$, $a_n$ assume o valor de $p$ e $b_n$ mantém o valor de $b$.\n",
    "\n",
    "Como o intervalo $[{a}_{n}, {b}_{n}]$ atende aos critérios do Teorema do Valor Intermediário, a cada iteração $n$ o intervalo estudado vai fechando em direção à raiz, até que o ponto $p$ resulte em $f(p) = 0$ ou que os critérios de tolerância sejam atendidos para uma aproximação considerada suficientemente boa ($f(p) \\approx 0$ ou outro critério de parada do método). \n",
    "\n",
    "O método da bisseccção pode adotar diversos critérios de parada. Alguns deles são:\n",
    "\n",
    "* $\\frac{{b}_{n}-{a}_{n}}{2}<\\varepsilon$ (tamanho limite da bissecção do intervalo $[{a}_{n}, {b}_{n}]$ após $n$ iterações)\n",
    "\n",
    "* $\\left|{p}_{n}-{p}_{n-1}\\right|<\\varepsilon$\n",
    "\n",
    "* $\\frac{\\left|{p}_{n}-{p}_{n-1}\\right|}{{p}_{n}}<\\varepsilon$, com ${p}_{n}\\neq0$ (seu uso é o mais indicado, por se aproximar de um erro relativo)\n",
    "\n",
    "* $\\left|f({p}_{n})\\right|<\\varepsilon$\n",
    "\n",
    "* $n>N_0$ ($n$ atingiu o número máximo de iterações $N_0$. Comumente usado para evitar loops infinitos em métodos iterativos que não convergem)\n",
    "\n",
    "---\n",
    "\n",
    "<p style=\"text-align: center;\"> **Teorema - limitante para o erro de aproximação no método da Bissecção** </p>\n",
    "\n",
    "Sejam $f \\in C[a, b]$ e $f(a).f(b)<0$, o método da Bissecção gera uma sequência $\\left \\{ {p}_{n} \\right \\}_{n=1}^{\\infty}$ que se aproxima de um zero $p$ de $f$ com $$\\left|{p}_{n}-{p}\\right| \\leqslant  \\frac{b-a}{2^{n}}\\textrm{, quando n}\\geqslant 1$$\n",
    "**Demonstração:**\n",
    "\n",
    "![Teorema_2_1](https://lh3.googleusercontent.com/_FFpatuJfuNL_cEpb7gg_bwrAq3MJ5km93j_FArpcJ42LhzeFP3m2zmREekOTec-5wu7ACc0JmY03oK9HaqcT9TEVWLn3otczck9dmxTlbuSNi-GecizvgRqbA2DCWoR6lTMONXHE1_x8RzaaazmWewsAGazGK0ns4PXf8PwGIBlt_lG2detMNWfQWHKXHokZrj_mv1ixECIrNw_HEjpNwUk2ijqMGnp1w5B6D8JTW3pdKpmRFGza7JcQVBicPnbyhPSYk0DDyQYXDumo_VhOJE7M4i0HxNkBq1ajTfvzBJBv6e5G4Y77F-4Qi59WanvW_x_fTKF0ZNEvaXE-4AOhfQurer0Sy2X-a4c-fb4uQvkdKLCOeqPT82lyzMljnEfoSlp87bxTyqusW2aC3bCXtWAnP6N5DaRRJ6TiyIdrD79__WRLp9Vsqk7BkTNZPqvWMWZqn5IkVJMM1JbwtrNdbMk8UsxLBdROapWqLqh74ddqqsRGPXol6uogU1FvnWnf-nGzJoSnthcZF6uGi8-Pstb881bBA4Rr9EeoRwHooyO96QpiUpyigjneDEOd2mLrfUFq1NxNhYb9fL1ArCQ7-6C48S8mk__AkQAop9rRpu9g5-jm3f13mZxDTNGXoizIfaQau9A7L2cUUh35fyIf820vcRLVZho=w273-h143-no)\n",
    "\n",
    "Ou seja, a raiz $p$ sempre estará no intervalo $[{a}_{n}, {p}_{n}]$ ou $[{p}_{n}, {b}_{n}]$ (ou no máximo em um dos extremos: ${a}_{n}=p$; ou ${b}_{n}=p$; ou $p={p}_{n}$). Pode-se escrever: $$\\left|{p}_{n}-{p}\\right| \\leqslant  \\frac{{b}_{n}-{a}_{n}}{2}\\textrm{, para todo n}\\geqslant 1,$$como: $$\\frac{{b}_{n}-{a}_{n}}{2} = \\frac{b-a}{2^{n}},$$a equação fica: $$\\left|{p}_{n}-{p}\\right| \\leqslant (b-a)\\frac{1}{2^{n}}$$.\n",
    "\n",
    "Esta equação pode ser escrita como uma equação de taxa de convergência $\\left|{\\alpha}_{n}-{\\alpha}\\right| \\leqslant K\\left| {\\beta}_{n}\\right| \\therefore {\\alpha}_{n} = \\alpha + O({\\beta}_{n})$. Assim a sequência $\\left \\{ {p}_{n} \\right \\}_{n=1}^{\\infty}$ converge para a raiz $p$ com uma taxa de convergência de $O(\\frac{1}{2^n})$, ou seja: $${p}_{n} = p + O(\\frac{1}{2^{n}})$$\n",
    "\n",
    "como exemplo: $$\\left|{p}_{9}-{p}\\right| \\leqslant  \\frac{2-1}{2^{9}} \\approx 2 x 10^{-3},$$é o limitante (erro máximo) para o método da bissecção com 9 iterações e $a=1$ e $b=2$.\n",
    "\n",
    "---\n",
    "\n",
    "### Vantagem do Método da Bissecção #\n",
    "\n",
    "A maior vantagem da utilização deste método para buscar a solução de uma equação do tipo $f(x)=0$ é que, atendidas as condições estabelecidas para sua aplicação, a convergência para uma solução é garantida. Por esse motivo importante, este método é bastante utilizado como um primeiro passo na busca por uma solução.\n",
    "\n",
    "### Desvantagens do Método da Bisseção #\n",
    "\n",
    "O Método da Bissecção possui algumas desvantagens notáveis quando comparado com outros métodos, sendo as mais importantes:\n",
    "\n",
    "* Convergência lenta\n",
    "\n",
    "Quando comparado a outros métodos, este método costuma precisar de uma quantidade maior de iterações para alcançar um resultado satisfatório e com uma boa precisão. \n",
    "\n",
    "Observe que, como destacado anteriormente, a sequência {$p_n$}$_{n=1}^\\infty$ converge para o valor da raiz $p$, com uma taxa de convergência de O$(\\frac{1}{2^n})$. Quando comparamos com a sequência ${(\\beta_n)}_{n=1}^{\\infty}$ definida na seção \"Convergência\" do capítulo anterior, percebemos que o valor aqui assumido pela variável n é baixo (2), o que indica uma baixa taxa de convergência para o resultado.\n",
    "\n",
    "$$Colocar \\, um \\,  exemplo \\, de \\, convergência \\, lenta$$\n",
    "\n",
    "* Resultados interessantes para o problema podem ser descartados\n",
    "\n",
    "Devido à maneira como o método funciona, e às suas condições iniciais e critérios de parada, é possível que soluções com erros menores sejam descartadas. Uma outra possibilidade é quando há mais de uma solução no intervalo, sendo uma delas ignorada durante as iterações.\n",
    "\n",
    "$$Colocar \\, um \\,  exemplo \\, de \\, função \\, trigonométrica$$\n",
    "\n",
    "* Limitação da aplicação do método \n",
    "\n",
    "O método da Bissecção falha para uma equação que não troca de sinal em sua imagem, ou seja, que o seu gráfico não cruza o eixo x, mas somente o tangencia nos pontos da suas raízes. Exemplos destas equações são $f(x)=3x^{4}$ e $g(x)=-2x^{2}$.\n",
    "\n",
    "$$Colocar \\, um \\, exemplo$$\n",
    "\n",
    "* Sensibilidade quanto ao intervalo escolhido\n",
    "\n",
    "Como o método trabalha com sucessivas subdivisões de um intervalo, a escolha do intervalo inicial possui um impacto considerável no número de iterações que serão necessárias para se chegar a um resultado preciso. Além disso, escolhendo-se um intervalo que contenha mais de uma raiz, o método ignorará pelo menos uma delas.\n",
    "\n",
    "$$Colocar \\, um \\,  exemplo$$\n",
    "\n",
    "### Observações quanto à aplicação do Método da Bissecção #\n",
    "\n",
    "Critérios de parada e tamanho do intervalo escolhido\n",
    "\n",
    "### Implementação do método da Bissecção em linguagem Python #\n",
    "\n",
    "O programa abaixo implementa o método da Bissecção para a obtenção de uma raiz de uma função, gerando uma tabela com os parâmetros do método nas sucessivas subdivisões do intervalo $[a, b]$ que contém a raiz, e a ilustração gráfica do método, representando a função (linha verde) e as subdivisões do intervalo $[a, b]$ em linhas azuis e vermelhas."
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    "Adicionar vantagens e desvantagens do método da bissecção. Um dos problemas do método da bissecção é que ele pode ignorar uma raiz, por exemplo. Conselhos para otimização."
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      "Digite a função f(x): 4*cos(x) - 2\n",
      "Tolerância para o tamanho do intervalo (b-a)/2 (digite 0 para desativar): 0.0001\n",
      "Tolerância para o erro relativo de p (ponto médio do intervalo) (digite 0 para desativar): 0.0001\n",
      "Tolerância para o valor de f(p) ~= 0 (digite 0 para desativar): 0\n",
      "Número máximo de iterações: 20\n",
      "Defina a: 0\n",
      "Defina b: 1.5\n",
      "\n",
      "n\t a\t         b\t         p\t         f(p)\t         (b-a)/2\t e. relat. p\tMundança\n",
      "1 \t 0.0000000000 \t 1.5000000000 \t 0.7500000000 \t 0.9267554755 \t 0.7500000000 \t 1.0000000000   f(a)*f(p)>0.0, a=p\n",
      "2 \t 0.7500000000 \t 1.5000000000 \t 1.1250000000 \t -0.2752939328 \t 0.3750000000 \t 0.3333333333   f(a)*f(p)<0.0, b=p\n",
      "3 \t 0.7500000000 \t 1.1250000000 \t 0.9375000000 \t 0.3672203004 \t 0.1875000000 \t 0.2000000000   f(a)*f(p)>0.0, a=p\n",
      "4 \t 0.9375000000 \t 1.1250000000 \t 1.0312500000 \t 0.0549872772 \t 0.0937500000 \t 0.0909090909   f(a)*f(p)>0.0, a=p\n",
      "5 \t 1.0312500000 \t 1.1250000000 \t 1.0781250000 \t -0.1080751777 \t 0.0468750000 \t 0.0434782609   f(a)*f(p)<0.0, b=p\n",
      "6 \t 1.0312500000 \t 1.0781250000 \t 1.0546875000 \t -0.0260018002 \t 0.0234375000 \t 0.0222222222   f(a)*f(p)<0.0, b=p\n",
      "7 \t 1.0312500000 \t 1.0546875000 \t 1.0429687500 \t 0.0146310707 \t 0.0117187500 \t 0.0112359551   f(a)*f(p)>0.0, a=p\n",
      "8 \t 1.0429687500 \t 1.0546875000 \t 1.0488281250 \t -0.0056511296 \t 0.0058593750 \t 0.0055865922   f(a)*f(p)<0.0, b=p\n",
      "9 \t 1.0429687500 \t 1.0488281250 \t 1.0458984375 \t 0.0044985729 \t 0.0029296875 \t 0.0028011204   f(a)*f(p)>0.0, a=p\n",
      "10 \t 1.0458984375 \t 1.0488281250 \t 1.0473632812 \t -0.0005741332 \t 0.0014648438 \t 0.0013986014   f(a)*f(p)<0.0, b=p\n",
      "11 \t 1.0458984375 \t 1.0473632812 \t 1.0466308594 \t 0.0019627568 \t 0.0007324219 \t 0.0006997901   f(a)*f(p)>0.0, a=p\n",
      "12 \t 1.0466308594 \t 1.0473632812 \t 1.0469970703 \t 0.0006944460 \t 0.0003662109 \t 0.0003497726   f(a)*f(p)>0.0, a=p\n",
      "13 \t 1.0469970703 \t 1.0473632812 \t 1.0471801758 \t 0.0000601899 \t 0.0001831055 \t 0.0001748557   f(a)*f(p)>0.0, a=p\n",
      "14 \t 1.0471801758 \t 1.0473632812 \t 1.0472717285 \t -0.0002569633 \t 0.0000915527 \t 0.0000874202\n",
      "\n",
      "A raiz da função, dentro dos parâmetros de tolerância, é 1.047271728515625\n",
      "\n",
      "O método da Bissecção foi encerrado após 14 iterações.\n"
     ]
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XMjoj4FKtorW0YeAGPVL5Eb307UtqObmlos9Gu44FAHDA9UrbF5Ja3mDMSmtt9ZTbaxmQCfAouYJyaUr7KRr70Fit3rda1T6ppiV7lriOBQDIYE5Lm7V2haQTLjMA3sAYo341+2l9//UqkK2AWoS30EvLXlJicqLraACADOJ6pS017jHGbDbGLDDGVHYdBnCpSqEq+qH/D+pTo4/eXPmmGk9orP2n97uOBQDIAJ5e2jZIKmmtrSZppKQ51xpojBlgjIkyxkQdPXo0wwICGS1bYDaNaztOk9tP1qboTao+prrm7ZjnOhYAIJ15dGmz1sZYa8+mfB4pKdAYU+AaYz+11oZaa0MLFiyYoTkBF7pW7aoNAzaoZO6SajutrZ5Z+Izik+JdxwIApBOPLm3GmCLGGJPyeR1dzMuVRoEU5fOX15q+a/RUnaf0/rr31eCzBvr1xK+uYwEA0oHrS35MlbRG0l3GmAPGmL7GmMeNMY+nDOkoaYsxZrOkDyV1tuzrA/xJUECQPmz1oWY9Oku7T+xWjTE1NH3rdNexAABpzPhiBwoNDbVRUWy4jczn91O/q/PMzlp7YK0G1hqo9x54T1kDs7qOBQC4PpOaQR798iiAm1MyT0mteGyFXmjwgsb8OEZ1x9XV9qPbXccCAKQBShvgYwL9AzW82XAt6LZA0WejFTo2VBM2TXAdCwBwmyhtgI9qWa6lNj2+SXWK1tHYDWP1zMJn2LsUALwYpQ3wYUVzFtU3Pb5RaNFQvb/ufXWY3kHn4s+5jgUAuAWUNsDHBfoH6v2W72tkq5Gat3OeGn3RSIfPHHYdCwBwkyhtQCYxuM5gRXSO0C/HflG98fW05cgW15EAADeB0gZkIm0qtNHK3iuVmJyoBp810De/fuM6EgAglShtQCZT444aWtdvnUrlKaXWk1tr7I9jXUcCAKQCpQ3IhIrnKq5VvVepRdkWGvD1AP1jyT+UbJNdxwIAXAelDcikcgbl1Nwuc/VE6BMasXqEOs3opNiEWNexAADXEOA6AAB3AvwCNKr1KJXLV07PffOcDsQcUETnCBXKXsh1NADAFVhpAzI5Y4yevedZzXx0pjZHb1a9cfXY+goAPBClDYAkqV3Fdlr+2HKdSzin+p/V17e/fes6EgDgMpQ2AJfUKVZH6/qtU9GcRfXApAfYsxQAPAilDcCflMpTSqv7rFajUo30WMRjeuXbV2StdR0LADI9ShuAv8gTnEeRXSPVp3ofvb7idXWf3V1xiXGuYwFApsbZowCuKtA/UOPajlO5fOX0z2X/1P7T+zW702zlz5bfdTQAyJRYaQNwTcYYDWs4TNM6TNP6g+t1z/h7tPvEbtexACBTorQBuKFOVTppWa9lOnnhpOqNq6dV+1a5jgQAmQ6lDUCq1C9RX2v7rlX+bPnVdGJTTf15qutIAJCpUNoApFrZfGW1pu8a1SteT11nddWbK97kzFIAyCCUNgA3JV/WfPqm+zfqHtJdL337kvrM7aP4pHjXsQDA53H2KICbFhQQpIkPT1S5vOX06nevat/pfZr56EzlCc7jOhoA+CxW2gDcEmOM/nX/vzTx4Yla+ftK1R9fX7+d/M11LADwWZQ2ALelR7UeWtxjsaLPRqvuuLpad2Cd60gA4JMobQBuW6NSjbSm7xrlDMqp+yfcr5nbZrqOBAA+h9IGIE3cVeAure27VjWK1NAjXz2id1a/w5mlAJCGKG0A0kzB7AW1rNcyPVL5EQ1dMlRPzH9CicmJrmMBgE/g7FEAaSo4IFhTO0xV2bxl9faqt7X31F5Nf2S6cgXlch0NALwaK20A0pyf8dNbTd/SuIfGaelvS3XvZ/dq/+n9rmMBgFejtAFIN31r9tWCbgv0++nfVXdcXf146EfXkQDAa1HaAKSrZmWa6fs+3yuLfxbd98V9mrdjnutIAOCVKG0A0l3lQpW1tt9aVS5YWWHTwthsHgBuAaUNQIYokqOIlj+2XP1r9VfXWV01bsM415EAwKtw9iiADJMtMJs+bPmh9p/erwHzBig4IFjdQ7q7jgUAXoGVNgAZKiggSDMfnanGpRur15xemrFthutIAOAVKG0AMlzWwKya23mu7il+j7rM7KKvd37tOhIAeDxKGwAnsmfJrshukapRpIY6TO+gxb8udh0JADwapQ2AM7mCcmlh94WqWKCiwqaF6bu937mOBAAei9IGwKl8WfNpcY/FKpWnlNpMbaM1+9e4jgQAHonSBsC5gtkLamnPpSqSo4haTW6lDYc3uI4EAB6H0gbAI9yR8w4t7blUeYLzqEV4C205ssV1JADwKJQ2AB7jztx3almvZQoKCFLTiU2149gO15EAwGNQ2gB4lDJ5y2hpz6WSpKYTm2rPyT2OEwGAZ6C0AfA4dxe4W0t6LFFsYqyaTGii/af3u44EAM5R2gB4pKqFq2pxj8U6deGUmkxsosNnDruOBABOUdoAeKyad9TUwu4LFX02Ws3Cm+nouaOuIwGAM5Q2AB6tXvF6+rrL1/rt5G9qHt5cJ2JPuI4EAE5Q2gB4vEalGmlO5znafmy7Wk5qqZi4GNeRACDDUdoAeIUWZVtoxiMztDF6ox6c8qDOxZ9zHQkAMhSlDYDXeOiuhzS1w1R9v/97tZ3WVrEJsa4jAUCGobQB8CodK3XUhIcn6NvfvlWH6R0UlxjnOhIAZAhKGwCv0z2kuz596FMt2L1AnWd2VkJSgutIAJDuKG0AvFK/mv00stVIzflljnrO6amk5CTXkQAgXQW4DgAAt2pwncGKTYjV0CVDFRwQrPFtx8vP8G9RAL6J0gbAqz3f4HmdTzivV797VVkDsmpU61EyxriOBQBpjtIGwOu90ugVxSbGasTqEQoOCNa7Ld6luAHwOZQ2AF7PGKO3m76t2IRYvbf2PWULzKY3mrzhOhYApClKGwCfYIzR+y3f14XEC3pz5ZvKGpBVL973outYAJBmKG0AfIYxRqPbjFZsYqxe+vYlZQ3MqmfvedZ1LABIE5Q2AD7Fz/jps7DPdCHxgv7+zd8VHBCsJ2s/6ToWANw2ShsAnxPgF6DJ7ScrLilOgyIHKWtAVvWu0dt1LAC4LVzQCIBPCvQP1PSO09WibAv1ndtXU3+e6joSANwWShsAnxUUEKTZnWarUalG6jG7h2Ztn+U6EgDcMqelzRjzmTHmiDFmyzWOG2PMh8aY3caYn4wxNTM6IwDvli0wm+Z1mac6xeqo84zOitwV6ToSANwS1yttX0hqeZ3jrSSVT7kNkDQ6AzIB8DE5suTQgm4LFFI4RO2/bK+le5a6jgQAN81pabPWrpB04jpDwiRNtBetlZTHGHNHxqQD4EtyB+fWou6LVCF/BbWd1lYrf1/pOhIA3BTXK203UkzS/svuH0j5GgDctPzZ8mtxj8W6M/edenDKg1p/cL3rSACQap5e2q62eaC96kBjBhhjoowxUUePHk3nWAC8VeEchbWkxxIVzF5QD0x6QBsPb3QdCQBSxdNL2wFJJS67X1zSoasNtNZ+aq0NtdaGFixYMEPCAfBOxXIV07Key5QrKJdaTGqhrUe2uo4EADfk6aVtrqSeKWeR1pN02lp72HUoAN6vZJ6SWtpzqQL9AtUsvJl2Hd/lOhIAXJfrS35MlbRG0l3GmAPGmL7GmMeNMY+nDImUtEfSbkljJbEXDYA0Uy5fOS3tuVRJyUlqMrGJfjv5m+tIAHBNxtqrvkXMq4WGhtqoqCjXMQB4ic3Rm9V4QmPlCc6jFb1XqHiu4q4jAchcrvYe/r/w9JdHASDdVStSTd/0+EbHY4+r6cSmij4b7ToSAPwFpQ0AJIUWDdWCbgt0MOagmk1spmPnj7mOBAB/QmkDgBT1S9TXvC7z9OvJX9UivIVOxp50HQkALqG0AcBlGpdurNmdZmvr0a1qNbmVzsSdcR0JACRR2gDgL1qWa6npHacr6lCUHpzyoM4nnHcdCQAobQBwNWF3h2ly+8lavX+1wqaF6ULiBdeRAGRylDYAuIZOVTrps7afacmeJeo4vaPik+JdRwKQiVHaAOA6elXvpU8e/ETzd81X15ldlZic6DoSgEyK0gYANzAwdKDee+A9zdw+U73m9FJScpLrSAAyoQDXAQDAGzxd72nFJsTqn8v+qawBWfXpQ5/Kz/DvXgAZh9IGAKk0rOEwxSbG6vUVrys4IFgjW42UManafQYAbhulDQBuwr/v/7diE2L1f2v+T1kDsuo/zf9DcQOQIShtAHATjDH6T/P/KDYxpbgFZtVrjV9zHQtAJkBpA4CbZIzRh60+1IXEC3p9xevKGpBVwxoOcx0LgI+jtAHALfAzfhrTZoxiE1NOTgjMqqfrPe06FgAfRmkDgFvk7+evCQ9PUFxinJ5Z9IyyBmTVwNCBrmMB8FGcrw4AtyHAL0BTOkzRg+Uf1OPzH9eETRNcRwLgoyhtAHCbsvhn0YxHZ6hZmWbqM7ePvtzypetIAHwQpQ0A0kBwQLDmdJqjBiUaqNusbor4JcJ1JAA+htIGAGkke5bsmt91vkKLhurRGY9q4e6FriMB8CGUNgBIQzmDcmph94WqXLCy2n3ZTst+W+Y6EgAfQWkDgDSWJziPvunxjcrmLau2U9tq9b7VriMB8AGUNgBIBwWyFdCSnktULFcxtZ7SWlGHolxHAuDlKG0AkE6K5CiipT2XKn/W/GoR3kKboze7jgTAi1HaACAdFc9VXEt7LlX2LNnVPLy5th3d5joSAC9FaQOAdFY6b2kt7blU/n7+ajaxmXaf2O06EgAvRGkDgAxQIX8FLemxRAnJCWo6sal+P/W760gAvAylDQAySOVClbW4x2LFxMWoycQmOhhz0HUkAF6E0gYAGah6kepa1H2Rjp47qqYTm+qPs3+4jgTAS1DaACCD1SlWR/O7ztf+mP1qHt5cx88fdx0JgBegtAGAAw1LNtTcznO18/hOPTDpAZ26cMp1JAAejtIGAI40LdNUszrN0k9//KTWk1vrTNwZ15EAeDBKGwA41Lp8a03rOE3rD67X3xb+TfFJ8a4jAfBQlDYAcKx9xfaa2G6iJv00SYMjB8ta6zoSAA8U4DoAAEDqWrWrthzZordXva0aRWroidpPuI4EwMOw0gYAHuL1xq/rwfIPasjCIfpu73eu4wDwMJQ2APAQ/n7+mtx+ssrmLauOX3Vk1wQAf0JpAwAPkjs4tyI6RyghKUEPf/mwziecdx0JgIegtAGAh7mrwF2a2mGqNkdvVp+IPpyYAEASpQ0APFKr8q00vNlwfbn1S41YPcJ1HAAegNIGAB7q+frPq0uVLvrn0n9q/s75ruMAcIzSBgAeyhijcW3HqXqR6uo6q6t+OfaL60gAHKK0AYAHyxaYTXM6z1GQf5DCpoWxRymQiVHaAMDD3Zn7Ts18dKb2nNyjrjO7Kik5yXUkAA5Q2gDACzQs2VAftfpIC3Yv0IvLXnQdB4ADbGMFAF5iYOhAbYrepBGrR6ha4WrqUrWL60gAMhArbQDgRT5o9YEa3tlQfeb20Y+HfnQdB0AGorQBgBfJ4p9FMx6doYLZCqrdl+105NwR15EAZBBKGwB4mULZC2lO5zk6dv6YOk7vqPikeNeRAGQAShsAeKGad9TUZ2GfaeW+lfrbgr+5jgMgA3AiAgB4qc5VOmtz9GYNXz1c1YpU0+Ohj7uOBCAdsdIGAF7sjSZvqHX51npqwVNa+ftK13EApCNKGwB4MX8/f01pP0Vl85ZVh+kdtO/0PteRAKQTShsAeLncwbkV0TlCcUlxenjawzqfcN51JADpgNIGAD7grgJ3aUr7KdoUvUl95/aVtdZ1JABpjNIGAD7iwQoP6q2mb2nalmn6z+r/uI4DII1R2gDAh7zQ4AV1qtxJw5YOU+SuSNdxAKQhShsA+BBjjD4L+0zVi1RXl5ldtOPYDteRAKQRShsA+Jhsgdk0p/McBfkHqe20tjp14ZTrSADSAKUNAHzQnbnv1IxHZ2jPyT3qNqubkpKTXEcCcJsobQDgo+4reZ9GthqpyF2Revnbl13HAXCb2MYKAHzY46GPa1P0Jr296m1VK1xNnap0ch0JwC1ipQ0AfNyHrT7UvXfeq94RvbXx8EbXcQDcIkobAPi4LP5ZNOORGSqQrYAe/vJhHTn6/plrAAAgAElEQVR3xHUkALfAaWkzxrQ0xuwwxuw2xvzjKscfM8YcNcZsSrn1c5ETALxd4RyFNbvTbB05d0Qdp3dUfFK860gAbpKz0maM8Zc0SlIrSZUkdTHGVLrK0C+ttdVTbuMyNCQA+JBaRWtpfNvxWrlvpZ5e+LTrOABuksuVtjqSdltr91hr4yVNkxTmMA8A+LyuVbtqaP2hGh01WmOixriOA+AmuCxtxSTtv+z+gZSvXamDMeYnY8wMY0yJjIkGAL7rraZvqWW5lhq8YLBW/r7SdRwAqeSytJmrfM1ecX+epFLW2hBJSyRNuOaDGTPAGBNljIk6evRoGsYEAN/i7+evqR2mqnSe0uowvYP2nd7nOhKAVHBZ2g5IunzlrLikQ5cPsNYet9bGpdwdK6nWtR7MWvuptTbUWhtasGDBNA8LAL4kT3Aeze0yV3FJcWr3ZTudTzjvOhKAG3BZ2n6QVN4YU9oYk0VSZ0lzLx9gjLnjsrttJW3PwHwA4NPuLnC3JrefrI2HN6rf3H6y9soXOwB4EmelzVqbKGmwpEW6WMamW2u3GmNeM8a0TRk2xBiz1RizWdIQSY+5SQsAvqlNhTZ6s8mbmrplqt75/h3XcQBch/HFf1mFhobaqKgo1zEAwCtYa9VlZhdN3zpdkd0i1bJcS9eRgMzmau/z/wt2RACATM4Yo/Ftx6takWrqPKOzdh7f6ToSgKugtAEAlD1Lds3pNEeB/oEKmxam0xdOu44E4AqUNgCAJKlknpKa8cgM7T6xW91mdVNScpLrSAAuQ2kDAFzSqFQjfdDyA83fNV+vfPuK6zgALhPgOgAAwLM8EfqENkVv0lur3lJI4RB1qtLJdSQAYqUNAHAFY4w+av2RGpRooN4RvbUpepPrSABEaQMAXEUW/yya+ehM5c+WX2HTwnT0HNsDAq5R2gAAV1U4R2HN7jRbR84dUcevOiohKcF1JCBTo7QBAK4ptGioxj00Tit+X6GnFz7tOg6QqXEiAgDgurqFdNPmPzbrne/fUbUi1TSg1gDXkYBMiZU2AMANvd30bbUs11KDIwdr9b7VruMAmRKlDQBwQ/5+/prSfopK5SmlDtM7aP/p/a4jAZkOpQ0AkCp5s+ZVROcInU84r3ZftlNsQqzrSECmQmkDAKRaxYIVNbn9ZG04vEH95/WXtdZ1JCDToLQBAG7KQ3c9pDeavKHJP0/Wu2vedR0HyDQobQCAmzbs3mF6pNIjemHJC1q4e6HrOECmQGkDANw0Y4w+D/tcVQpVUecZnbXz+E7XkQCfR2kDANyS7FmyK6JzhAL8AhQ2LUwxcTGuIwE+jdIGALhlpfKU0oxHZ2jX8V3qNqubkm2y60iAz6K0AQBuy/2l7tcHLT/Q1zu/1ivfvuI6DuCz2MYKAHDbnqz9pDZFb9KbK99USOEQPVr5UdeRAJ/DShsA4LYZY/RR649Uv0R99Y7orc3Rm11HAnwOpQ0AkCaCAoI089GZyhucV2HTwnTs/DHXkQCfQmkDAKSZIjmKaE7nOfrj3B965KtHlJCU4DoS4DMobQCANBVaNFRjHxqr5XuX69lFz7qOA/gMTkQAAKS57iHdtTl6s/5vzf+pWpFq6lezn+tIgNdjpQ0AkC6GNxuuFmVb6Mn5T+r7/d+7jgN4PUobACBd+Pv5a1qHaSqZp6Taf9leB2IOuI4EeDVKGwAg3eTNmlcRnSN0LuGc2n3ZTrEJsa4jAV6L0gYASFeVClbS5PaTFXUoSgO+HiBrretIgFeitAEA0l3bu9rq9cava9JPk/TfNf91HQfwSpQ2AECGeLHhi+pYqaOGLhmqRbsXuY4DeB1KGwAgQxhj9HnY56pSqIo6z+ysXcd3uY4EeBVKGwAgw+TIkkNzOs2Rv/FX2LQwxcTFuI4EeA1KGwAgQ5XOW1pfPfKVdh7fqR6zeyjZJruOBHgFShsAIMM1Lt1Y77d8X3N3zNWry191HQfwCmxjBQBwYlDtQdoUvUmvr3hdIYVD1LFSR9eRAI/GShsAwAljjEa1HqVGJRvpk6hPtPvEbteRAI9GaQMAOBMUEKTwduH68fCP6jG7hxKTE11HAjwWpQ0A4FSJ3CX0ceuPtfbAWg1fNdx1HMBjUdoAAM51qdpFXap00b+/+7eiDkW5jgN4JEobAMAjjGo9SkVyFFH3Wd11PuG86ziAx6G0AQA8Qt6sefVF2BfacXyHhi4e6joO4HEobQAAj9G0TFM9XfdpjfphlBbuXug6DuBRKG0AAI/ydrO3ValgJfWJ6KPj54+7jgN4DEobAMCjBAcEa3L7yTp2/pgGfj1Q1lrXkQCPQGkDAHic6kWq6/XGr2vm9pkK/yncdRzAI1DaAAAe6bn6z6nhnQ01OHKw9p7a6zoO4BylDQDgkfz9/DWx3URJUs/ZPZWUnOQ4EeAWpQ0A4LFK5Smlka1GauW+lXp3zbuu4wBOUdoAAB6tZ7Weal+xvV5a9pI2R292HQdwhtIGAPBoxhiNaTNG+bPlV7dZ3XQh8YLrSIATlDYAgMcrkK2APmv7mbYe3aoXl77oOg7gBKUNAOAVWpVvpSdDn9R/1/5Xy35b5joOkOEobQAAr/FOi3dUIX8F9ZrTS6cunHIdB8hQlDYAgNfIFphNk9pNUvTZaA2KHOQ6DpChKG0AAK9Su1htvXLfK5ry8xRN2zLNdRwgw1DaAABeZ1jDYapXvJ6emP+EDsQccB0HyBCUNgCA1wnwC1B4u3DFJ8XrsTmPKdkmu44EpDtKGwDAK5XLV07vPfCelv62VCPXjXQdB0h3lDYAgNfqX7O/2lRooxeWvKCtR7a6jgOkK0obAMBrGWM07qFxyhWUS91nd1d8UrzrSEC6obQBALxa4RyFNfahsdoUvUn/+vZfruMA6YbSBgDwemF3h6lvjb4asXqEVu1b5ToOkC6cljZjTEtjzA5jzG5jzD+ucjzIGPNlyvF1xphSGZ8SAOAN3nvgPZXOW1o9ZvdQTFyM6zhAmnNW2owx/pJGSWolqZKkLsaYSlcM6yvppLW2nKT3JI3I2JQAAG+RMyinwtuFa9/pffrbwr+5jgOkOZcrbXUk7bbW7rHWxkuaJinsijFhkiakfD5DUlNjjMnAjAAAL1K/RH0Nu3eYvtj0hWZtn+U6DpCmXJa2YpL2X3b/QMrXrjrGWpso6bSk/BmSDgDglf7V6F+qdUctDZg3QIfPHHYdB0gzAQ6f+2orZvYWxlwcaMwASQMkKTAwUKGhobeXLhM4evSoChYs6DqGx2OeUo+5Sh3mKfVuda4uJF7QiaMndPcnd6t8vvLpkMyz8DOVOp46Tz/++ONCa23LG41zWdoOSCpx2f3ikg5dY8wBY0yApNySTlztway1n0r6VJKyZ89uo6Ki0jywrwkNDRXzdGPMU+oxV6nDPKXe7czVR+s/0lMLnlLf1n31RO0n0jiZZ+FnKnU8eJ5uWNgkty+P/iCpvDGmtDEmi6TOkuZeMWaupF4pn3eUtMxae9WVNgAALvdk7SfVomwL/f2bv2vn8Z2u4wC3zVlpS3mP2mBJiyRtlzTdWrvVGPOaMaZtyrDxkvIbY3ZLelbSXy4LAgDA1fgZP30e9rmCA4LVfVZ3JSQluI4E3Ban12mz1kZaaytYa8taa99M+dor1tq5KZ9fsNY+Yq0tZ62tY63dk5rHLVCgQHrG9hkDBgxwHcErME+px1ylDvOUerc7V0VzFtWYNmP0w6Ef9ObKN9MolefhZyp1vH2ejC++2hgaGsp72gAAl/Sc3VNTfp6i1X1Wq27xuq7jAFdK1eXM2MYKAODzRrYaqWK5iqn77O46G3/WdRzglnhtabvRFlhxcXHq1KmTypUrp7p162rv3r0OUnqGhQsX6q677lK5cuU0fPjwvxz/73//q0qVKikkJERNmzbV77//7iClezeap/+ZMWOGjDGeegZSukvNPE2fPl2VKlVS5cqV1bVr1wxO6DluNFf79u1T48aNVaNGDYWEhCgyMtJBSvf69OmjQoUKqUqVKlc9bq3VkCFDVK5cOYWEhGjDhg03/Ry5g3Nr4sMT9euJX/XcN8/dbmQnbjRPkydPVkhIiEJCQlS/fn1t3rw5gxN6jhvN1f/88MMP8vf314wZMzIo2W2y1nrdTZK/pF8llZGURdJmSZX+d7xWrVp21KhRduDAgdZaa6dOnWofffRRmxklJibaMmXK2F9//dXGxcXZkJAQu3Xr1j+NWbZsmT137py11tqPP/44U85VaubJWmtjYmJsw4YNbd26de0PP/zgIKlbqZmnnTt32urVq9sTJ05Ya639448/XER1LjVz1b9/f/vxxx9ba63dunWrLVmypIOk7n333Xf2xx9/tJUrV77q8fnz59uWLVva5ORku2bNGlunTp1bfq7nv3ne6lXZeTvm3fJjuHKjeVq9evWl37vIyMjbmidvd6O5svbi72jjxo1tq1at7FdffZWB6a4qVf3HW1fabrgFVkREhHr1uni1kI4dO2rp0qX/K3yZyvr161WuXDmVKVNGWbJkUefOnRUREfGnMY0bN1a2bNkkSfXq1dOBAwdcRHUqNfMkSS+//LKGDh2q4OBgByndS808jR07VoMGDVLevHklSYUKFXIR1bnUzJUxRjExFzc2P336tIoWLeoiqnP33Xef8uXLd83jERER6tmzp4wxqlevnk6dOqXDh29tp4PXG7+ukMIh6ju3r46eO3qrkZ240TzVr1//0u9dZv1b/j83mitJGjlypDp06OBVf6O8tbTdcAusgwcPqkSJi9fuDQgIUO7cuXX8+PGMS+ghLp8HSSpevLgOHjx4zfHjx49Xq1atMiKaR0nNPG3cuFH79+9XmzZtMjqex0jNPO3cuVM7d+5UgwYNVK9ePS1cuDCjY3qE1MzVq6++qkmTJql48eJq3bq1Ro4cmdExvcLN/h27nqCAIE1qN0mnLpxS/3n9ffYf85n1b3lqHTx4ULNnz9bjjz/uOspN8dbSdsPtra72i5gZ95q/mXmYNGmSoqKi9Pzzz6d3LI9zo3lKTk7WM888o3fffTcjY3mc1Pw8JSYmateuXVq+fLmmTp2qfv366dSpUxkV0WOkZq6mTp2qxx57TAcOHFBkZKR69Oih5OTkjIroNdL673nVwlX1VpO3FLEjQp9v+vx2onmkb7/9VuPHj9eIESNcR/FYTz/9tEaMGCF/f3/XUW6KV17ywxhzj6RXrbUPpNwfJknW2rclKWfOUBsYOF6lSpVSrly5Za3V999/rwYN6iuVZ9X6jJiY09q7d69CQqpJkvbtu3iSwZ13lvzTuJMnT2rXrl2qUaO6AgOzZHhO1240T4mJiVq3bt2lX/D4+HgFBgaoSpWqypkzp5vQDqTm52nnzp3KlSuXihQpIknavHmTypQpo5w5c2V84DS2XPeneuzpmBjt3btX1UJCJEm/79snSSp5552Xxqz/4QeFhIQoOChIkrR23TrVrFlTWQID0y60l7hw4YJ+/vln1a5d+y/HduzcqTx58qhwystY69avV/Xq1RWU5db/VllJm6M360x8jEKL1lbWAO94y8P15kmSzp47py1btigkJETZsmbN4HSe5XpztXbt2kufJyQkyM/fX3dVqODuOq/Ll/v0JT9uuAVW/vwFFB39h6SLG8TmzZtHma2wSVLOnLkUGxurCxdiZW2yjhw5ovz5//xDefbsGe3cuUNVq1bJlIVNuvE8BQQEXHq5r169esqVK1emK2xS6n6eChQooFOnTkq6+Mfw/PlYBQdnvv945MqZU7GxsYq9cEHJ1urIkSMqkD//n8YEBwXp1MmLc3Xu/HklJycrMBMWthspkD+//oiOlpUUExOjgICA2yps0sX/Gtxd4G4ZGW0/tl1W3reAcaULcXHaumWLKlasmOkL24387295vXr1VLBgQVUoX947Lsyf2jMWPO0mqbWknbp4FumLKV97TVLbWrVq2djYWNuxY0dbtmxZW7t2bfvrr7/ezlkdXm3+/Pm2fPnytkyZMvaNN96w1lr78ssv24iICGuttU2bNrWFChWy1apVs9WqVbMPPfSQy7jO3GieLteoUaNMefaotTeep+TkZPvMM8/YihUr2ipVqtipU6e6jOvUjeZq69attn79+jYkJMRWq1bNLlq0yGVcZzp37myLFCliAwICbLFixey4cePs6NGj7ejRo621F3+mnnzySVumTBlbpUqVNP3dm7R5ktWrsm+ueDPNHjO93Gie+vbta/PkyXPpb3mtWrUcJ3bnRnN1uV69ennN2aNe+fLojbAjAgAgNay16jKzi2Zun6m1fdeqVtFariMhc/Lpl0cBALhtxhh9/ODHKpy9sLrP7q7YhFjXkYBrorQBADK1fFnz6fOwz/XLsV/0wpIXXMcBronSBgDI9JqXba4hdYZo5PqR+ubXb1zHAa6K0gYAgKThzYarYoGK6h3RWydiT7iOA/wFpQ0AAElZA7NqUvtJOnLuiB7/+nGf3S0B3ovSBgBAipp31NRr97+mr7Z9pck/T3YdB/gTShsAAJcZ2mCoGpRooEGRg7Tv9D7XcYBLKG0AAFzG389f4e3ClWyT1WtOLyVb9oNF2vshZRu7CxcuyBiT3Riz1RhT5XrfE5BR4QDA291/v+sEyDilVezsTi0/tkPlPzioErlK3NR338w+tcicakuKOH1aB8uXl6T/SJpkrd1yve9hpQ0AgKsokqOI8mcroN9O7tG5+LOu48AHlSxVSicu7j8cqovF7bpYaQOAVFq+3HUCZCyjo+cKq+roZorLXkgr+69XcEBwKr93eXoGg484Eh2tzg0aSHv25JQULOnc9caz0gYAwDUUzF5Q49uO189HftbLy152HQc+ZsCAAXr99dclabKkETcaT2kDAOA6HqzwoB6v9bjeXfOulu9d7joOfMTEiRMVEBCgrl27StJwSbWNMU2u9z3GFy8eGBoaaqOiolzHAAD4iHPx51RjTA1dSLygn5/4WbmDc7uOBN9iUjOIlTYAAG4ge5bsCm8XrkNnDmnwgsGu4yCTorQBAJAKdYvX1Uv3vaRJP03S9K3TXcdBJkRpAwAglV5s+KLqFKujx79+XAdjDrqOg0yG0gYAQCoF+gcqvF244pLi1DuiN7slIENR2gAAuAkV8lfQuy3e1eI9izVq/SjXcZCJUNoAALhJA2sNVOvyrTV0yVBtO7rNdRxkEpQ2AABukjFG49uOV44sOdRjdg/FJ8W7joRMgG2sACCV2DD+rzLzxuhFJG07X1hbj2zQgVHlVSZv6f9/kD3PkA5YaQMA4BYVzFZARXIU0b6YfTodd9p1HPg4VtoAIJVYPLma5a4DOFcs7oxaf1JNxsRo08AVyhmU03Uk+ChW2gAAuA05g3JqYruJ2ntqr55Z9IzrOPBhlDYAAG7TvXfeqxcavKDxG8drzi9zXMeBj6K0AQCQBl69/1XVKFJD/ef1V/TZaNdx4IMobQAApIEs/lk0qf0knY0/q35z+8la6zoSfAylDQCANFKpYCWNaDZC83fN16c/fuo6DnwMpQ0AgDQ0uM5gNSvTTM9+86x2Hd/lOg58CKUNAIA05Gf89EXYFwryD1KP2T2UmJzoOhJ8BKUNAIA0VixXMY1+cLTWHVynt1a+5ToOfASlDQCAdNCpSid1q9pNr333mtYfXO86DnyA8cWzW0JDQ21UVJTrGADg8zxpP1aP2gc1ZfuMUxdOKWR0iIIDgrVx4EZlz5LdbS54KpOaQay0AQCQTvIE59GEhydo94nden7x867jwMux9ygA4JZ51n6sy10HuKrGpRvr2Xue1btr3lWbCm3Uunxr15HgpVhpAwAgnb3R5A1VLVRVfSL66Nj5Y67jwEtR2gAASGfBAcGa1H6STl44qQHzBrBbAm4JpQ0AgAwQUjhEbzR+Q7N/ma0Jmye4jgMvRGkDACCDPHvPs2pUspGGLBii307+5joOvAylDQCADOLv568JD0+QMUY95/RUUnKS60jwIpQ2AAAyUMk8JfVRq4+0at8qvfP9O67jwItQ2gAAyGDdQ7rrkUqP6JVvX9HGwxtdx4GXoLQBAJDBjDEa/eBoFchWQN1nd1dsQqzrSPAClDYAABzIny2/Pg/7XNuObtOwpcNcx4EXoLQBAODIA+Ue0ODag/XBug+0ZM8S13Hg4dgwHgDgE9Jq8/o02Xj+Jvb3Op9wXrU+raUzcWf00xM/KV/WfLf//PA2bBgPAICnyxaYTZPaTdIf5/7QoMhBruPAg7FhPADAJ6Td5vVp9kCpVqtoLb3a6FW99O1LeqjCQ+patWuGZ4DnY6UNAAAP8MK9L+ie4vfoyflPav/p/a7jwANR2gAA8AABfgEKbxeuJJukXnN6Kdkmu44ED0NpAwDAQ5TNV1bvP/C+vt37rd5f+77rOPAwlDYAADxInxp91Pauthq2dJi2HNniOg48iJPSZozJZ4xZbIzZlfIx7zXGJRljNqXc5mZ0TgAAMpoxRmMfGqs8wXnUbVY3xSXGuY4ED+Fqpe0fkpZaa8tLWppy/2pirbXVU25tMy4eAADuFMpeSOPbjtdPf/ykV759xXUceAhXpS1M0oSUzydIethRDgAAPFKbCm00oOYAvfP9O/pu73eu48ADuCptha21hyUp5WOha4wLNsZEGWPWGmModgCATOXdB95V2Xxl1XNOT52+cNp1HDiWbqXNGLPEGLPlKrewm3iYO621oZK6SnrfGFP2Os83IKXgRR09evS28wMA4FqOLDkU3i5cB2IOaMjCIa7jwLF0K23W2mbW2ipXuUVI+sMYc4ckpXw8co3HOJTycY8uXqK6xnWe71Nrbai1NrRgwYJp/v8HAAAX6hWvpxcbvqiJmydqxrYZruPAIScbxhtj3pF03Fo73BjzD0n5rLVDrxiTV9J5a22cMaaApDWSwqy12270+GwYDwCZT2o2jE/VZvBptx9WmklISlCDzxro15O/6ucnflbRnEVdR0La8ugN44dLam6M2SWpecp9GWNCjTHjUsZUlBRljNks6VtJw1NT2AAA8DWB/oEKbxeu2IRY9YnoIxcLLnDPyUpbemOlDQDgiz7+4WMNihykj1p9pEF1BrmOg7Tj0SttAADgJj0R+oRalmup5xY/p1+O/eI6DjIYpQ0AAC9hjNFnbT9T9sDs6j6ruxKSElxHQgaitAEA4EXuyHmHPn3oU/14+Ee99t1rruMgA1HaAADwMu0rtlevar301qq3tGb/GtdxkEEobQAAeKEPW32oErlKqMfsHjobf9Z1HGQAShsAAF4oV1AuhbcL156Te/Tsomddx0EGoLQBAOClGpZsqKENhmrshrGau2Ou6zhIZ5Q2AAC82GuNX1O1wtXUb24//XH2D9dxkI4obQAAeLEs/lk0uf1kxcTFqP+8/uyW4MMCXAcAACAtXG3v0b/sNeqB+4qmhcqFKmt4s+F6ZtEzGrdhnPrX6u86EtIBK20AAPiAIXWHqGnppnpm0TPafWK36zhIB+w9CgCAjzgQc0BVR1fV3QXu1sreKxXgxwtqXoK9RwEAyEyK5yquj1t/rLUH1mr4quGu4yCNUdoAAPAhXap2UZcqXfTv7/6tqEO86uRLKG0AAPiYUa1HqUiOIuo+q7vOJ5x3HQdphNIGAICPyZs1r74I+0I7ju/Q0MVDXcdBGqG0AQDgg5qWaapn6j2jUT+M0sLdC13HQRqgtAEA4KPeavqWKhesrN4RvXX8/HHXcXCbKG0AAPio4IBgTWo/ScfPH9fArweyW4KXo7QBAODDqheprtcbv66Z22cq/Kdw13FwGyhtAAD4uOfqP6eGdzbU4MjB2ntqr+s4uEWUNgAAfJy/n78mtpsoSeo5u6eSkpMcJ8KtoLQBAHzW/fenbCR/6ZPMq1SeUhrZaqRW7lupd9e86zoObgGlDQCATKJntZ5qX7G9Xlr2kjZHb3YdBzeJDeMBAMhEjp0/pqqjqyp/1vyKGhCl4IBg15HAhvEAAOBKBbIV0Odhn2vr0a3659J/uo6Dm0BpAwAgk2lZrqWeDH1S7619T0v3LHUdB6lEaQMAIBN6p8U7qpC/gh6LeEwnY0+6joNUoLQBAJAJZQvMpkntJin6bLQGLxjsOg5SgdIGAEAmVbtYbb1y3yua8vMUTdsyzXUc3AClDQCATGxYw2GqV7yenpj/hA7EHHAdB9dBaQMAIBML8AtQeLtwxSfF67E5jynZJruOhGugtAEAkMmVy1dO7z3wnpb+tlQj1410HQfXQGkDAADqX7O/2lRooxeWvKCtR7a6joOroLQBAHzWpX1HcUPGGI17aJxyBeVS99ndFZ8U7zoSrkBpAwAAkqTCOQprXNtx2hS9Sf/69l+u4+AKlDYAgM9avvx//4PUantXW/Wr0U8jVo/Qyv/X3p0HV1keehz/PVmAsASByYBIWsRCBQYGJeSiQRRBQ2QTB6atiEoJCREDCpZcaKhI2SwRoRlPA0kpDXQG7VSUhAgRiqBQNMGyGlMBIWUpEG2MRAhZ3vvHpXf0ypLlnDxn+X5mmCFnznne35/fyTkn78n3bc/BtxBtAADgO5bHLtft7W7Xk289qfLKcttzcBXRBgAAvqNN8zZaN3adSr4q0YwtM2zPwVVEGwAA+J57I+/VnEFztHb/Wr1Z9KbtORDRBgAAruPF+19U/1v7KyEnQWe/Pmt7TsAj2gAAwDWFBodq3dh1qqiq0ORNk+U4ju1JAY1oAwAA19UzoqeWPbRM7xx9RxmFGbbnBDSiDQAA3NC0AdMUe0esZuXPUnFpse05AYtoAwAAN2SM0ZoxaxQWGqaJGyeqqqbK9qSARLQBAICb6tyms1aNXKWCMwVauGuh7TkBiWgDAAB1Mq7XOE3sO1GL3l+kvaf22p4TcIg2AIB/44bxbpUel67bwm/TxI0TdfHKRdtzAgrRBgAA6qxti7bKfjRbx748phfyX7A9J6AQbQAA/8YN493u/q736zntbEMAAA2/SURBVIV7X9CqfauU+49c23MCBtEGAADq7ddDfq2+Hftq8qbJOl9x3vacgEC0AQCAemse0lzrx65X2eUyJeQkcLeEJkC0AQCABunTsY+WDF2it4vf1pq/r7E9x+8RbQAAoMGeG/ichnQdohlbZujYl8dsz/FrRBsAAGiwIBOktY+uVUhQiJ5860lV11bbnuS3iDYAANAoP2j7A732yGva8889+s3u39ie47eINgAA0GiP93lcP+n9E7343ovad2af7Tl+iWgDAACNZoyRa4RLHVt11BMbn9Clqku2J/kdog0AALhF+7D2WvvoWn1a+qlStqXYnuN3iDYAAOA2w7oN04z/mqH0j9KVfyzf9hy/QrQBAAC3WjJ0iXpF9NLTbz2tL775wvYcv0G0AQAAtwoLDdP6setV+k2pkjYncbcENyHaAACA291161166YGX9OdP/qw/HfqT7Tl+gWgDAAAeMTtmtmIiYzQtb5pOlp20PcfnEW0AAMAjgoOCtW7sOtU6tXrqradU69TanuTTrESbMWa8MeaIMabWGBN1g+cNN8YUG2OOGmP+uyk3AgCAxru93e367fDfaufJnVr+t+W25/g0W79pOyzpMUm7rvcEY0ywpNckxUnqJelnxpheTTMPAAC4y9P9ntajdz6qX/71lzp47qDtOT7LSrQ5jlPkOE7xTZ4WLemo4zjHHce5ImmDpDGeXwcAANzJGKPVI1erXYt2euLNJ3S5+rLtST7Jmz/Tdpukf37r51NXH7smY0yCMabQGFN44cIFj48DAAB1F9EqQmvGrNGh84eU+tdU23N8kseizRizzRhz+Br/6vrbMnONx677h14cx1ntOE6U4zhRERERDRsNAAA85pHuj2hq/6la/rfl2vH5DttzfE6Ipw52HGdYI484JSnyWz93kXSmkWcCAACL0h5O0/bPt+upt57SwaSDuqXFLbYn+Qxvfnu0QFJ3Y8ztxphmkn4qaZPlTQAAoBFaNWul9Y+t15mvzyj5nWTbc3yKrT/5MdYYc0rSPZI2G2O2Xn28szEmT5Icx6mW9KykrZKKJL3hOM4RG3sBAID7RN8WrXmD52n9wfV648gbtuf4DOOP9wOLiopyCgsLbc8AAADXUVVTpUF/GKTPvvhMh5IO6bbw637XMBBc63P83+PNb48CAAA/FRocqnVj16myplKT3p7E3RLqgGgDAABW9OjQQ688/IrePf6uXAUu23O8HtEGAACsSeyfqMn9JuuVPa/o+L+P257j1Yg2AABgjTFG84fM1xeXvlBibqL88bP27kK0AQAAq7qEd9HSYUu17fg2ZR/Itj3HaxFtAADAuqlRUxUTGaPntz6vcxfP2Z7jlYg2AABgXZAJUuaoTFVUVei5rc/ZnuOViDYAAOAVekb0VOp9qdpweINy/5Fre47XIdoAAIDXSBmUot4RvZW0OUnlleW253gVog0AAHiNZsHNlDU6S6fLT2vu9rm253gVog0AAHiVgV0GKjk6Wa4Cl3aX7LY9x2sQbQAAwOssfHChIttGakrOFFVWV9qe4xWINgAA4HXaNG+jjBEZKiot0pIPltie4xWINgAA4JXiusfp8T6Pa/H7i3Xk/BHbc6wj2gAAgNdaEbtC4c3DFZ8Tr5raGttzrCLaAACA14poFaEVw1do76m9+l3h72zPsYpoAwAAXm1CnwmKvSNWc7bPUclXJbbnWEO0AQAAr2aMUcbIDNU6tUranCTHcWxPsoJoAwAAXq/rLV216MFFyvssTxsOb7A9xwqiDQAA+ITk6GQN6DxA07dMV+k3pbbnNDmiDQAA+ITgoGBljc5S2eUyzcqfZXtOkyPaAACAz+jbsa9SYlKUfSBb+cfybc9pUkQbAADwKamDU9WjQw8l5iaq4kqF7TlNhmgDAAA+pUVIC2WOytSJshP61Y5f2Z7TZIg2AADgcwb/cLCm9p+qFR+uUMHpAttzmgTRBgAAfNLSYUvVqXUnxefEq6qmyvYcjyPaAACAT2rboq1cj7h08NxBLduzzPYcjyPaAACAzxpz5xiN6zVOC3YuUHFpse05HkW0AQAAn5Yel66w0DAl5Cao1qm1PcdjiDYAAODTOrXupLSH0rTr5C5lfZxle47HEG0AAMDn/fyun2tI1yH6xbu/0Jmvz9ie4xFEGwAA8HnGGK0auUpXaq7o2bxnbc/xCKINAAD4he4dumv+/fO18dONerPoTdtz3I5oAwAAfmPmPTPVr1M/TcubprLLZbbnuBXRBgAA/EZocKh+P/r3ulBxQbPfnW17jlsRbQAAwK/cfevdmnnPTGV+nKn3Trxne47bEG0AAMDvzH9gvrq166YpOVN0qeqS7TluQbQBAAC/0zK0pVaPXK2jXx7Vgp0LbM9xC6INAAD4paHdhmpSv0latmeZ9v9rv+05jUa0AQAAv5X2cJo6tOyg+E3xqq6ttj2nUYg2AADgt9qHtVd6XLr2nd2nlXtX2p7TKEQbAADwa+N7jdeoHqM0b8c8Hf/3cdtzGoxoAwAAfs0YI9cIl0KCQpSYmyjHcWxPahCiDQAA+L0u4V308rCXte34NmUfyLY9p0GINgAAEBASoxIVExmj57c+r3MXz9meU29EGwAACAhBJkiZozJVUVWhGVtm2J5Tb0QbAAAIGD0jeir1vlS9fuR15RTn2J5TL0QbAAAIKCmDUtQ7oreeyXtG5ZXltufUGdEGAAACSrPgZsoanaXT5ac1d/tc23PqjGgDAAABZ2CXgUqOTparwKXdJbttz6kTog0AAASkhQ8uVGTbSE3JmaLK6krbc26KaAMAAAGpTfM2yhiRoaLSIi35YIntOTdFtAEAgIAV1z1OE/pM0OL3F+vI+SO259wQ0QYAAALaq7GvKrx5uOJz4lVTW2N7znURbQAAIKBFtIrQiuErtPfUXrkKXLbnXBfRBgAAAt6EPhMUe0es5myfo5KvSmzPuSaiDQAABDxjjDJGZsiRo6TNSXIcx/ak7yHaAAAAJHW9pasWPbhIeZ/lacPhDbbnfA/RBgAAcFVydLIGdB6g6Vumq/SbUttzvoNoAwAAuCo4KFhZo7NUdrlMs/Jn2Z7zHUQbAADAt/Tt2FcpMSnKPpCt/GP5tuf8H6INAADg/0kdnKofd/ixEnMTVXGlwvYcSUQbAADA97QIaaHMUZk6UXZC83bMsz1HEtEGAABwTff98D5N7T9VKz9cqY9Of2R7jp1oM8aMN8YcMcbUGmOibvC8E8aYQ8aY/caYwqbcCAAAsHTYUnVq3Unxm+JVVVNldYut37QdlvSYpF11eO4Qx3H6OY5z3bgDAADwhLYt2sr1iEuHzh/Ssj3LrG6xEm2O4xQ5jlNs49oAAAD1MebOMRrXa5wW7Fyg4lJ7+eLtn2lzJOUbY/YZYxJsjwEAAIEpPS5dYaFhSshNUK1Ta2WDx6LNGLPNGHP4Gv/G1OOYGMdx7pYUJ2maMWbwDa6XYIwpNMYUXrhwodH7AQAA/qNT605KeyhNu07uUtbHWVY2GJs3RDXGvCfpBcdxbvolA2PMfEkXHcdJu9lzo6KinMJCvrcAAADcx3EcDc0eqn1n96loWpE6t+nsrqNNXZ7ktW+PGmNaGWPa/Of/kh7W/36BAQAAoMkZY7R61GpdqbmiZ/OebfLr2/qTH2ONMack3SNpszFm69XHOxtj8q4+raOkD4wxByR9JGmz4zhbbOwFAACQpB+1/5FeeuAlbfx0o/7yyV+a9NpW3x71FN4eBQAAnlJdW63ozGidvXhWnzzzidqFtWvskb799igAAIA3CgkKUdboLF2ouKDZ785ususSbQAAAPV09613a+Y9M5X19yzt+HxHk1yTaAMAAGiA+Q/MV7d23ZSQm6BLVZc8fj2iDQAAoAFahrbU6pGrdfTLo1qwc4HHr0e0AQAANNDQbkM1qd8kLduzTPv/td+j1yLaAAAAGiHt4TR1aNlB8ZviVV1b7bHrEG0AAACN0D6svdLj0rXv7D6t3LvSY9ch2gAAABppfK/xGtVjlObtmKdjXx7zyDWINgAAgEYyxsg1wqWQoBAl5ibKEzcvINoAAADcoEt4F7087GVt/3y7/njgj24/n2gDAABwk8SoRMVExmjm1pk6d/GcW88m2gAAANwkyAQpc1SmKqoqNGPLDPee7dbTAAAAAlzPiJ5KvS9Vrx95XTnFOW47l2gDAABws5RBKeod0VvP5D2j8spyt5xJtAEAALhZs+BmyhqdpdPlpzV3+1y3nEm0AQAAeMDALgOVHJ0sV4FLu0t2N/o8og0AAMBDFg1dpMi2kZqSM0WV1ZWNOotoAwAA8JDWzVorY0SGikqLtPj9xY06i2gDAADwoLjucZrQZ4KWfLBEh88fbvA5RBsAAICHvRr7qsKbhyt+U7xqamsadAbRBgAA4GERrSK0YvgKfXj6Q7kKXA06g2gDAABoAhP6TFDsHbGas32OSr4qqffriTYAAIAmYIxRxsgMOXKUtDlJjuPU6/VEGwAAQBPpektXLXpwkfI+y9OGwxvq9VqiDQAAoAklRydrQOcBmr5lukq/Ka3z64g2AACAJhQcFKys0Vkqu1ymmVtn1vl1pr7vp/oCY8wWx3GG294BAADgLn4ZbQAAAP6Gt0cBAAB8ANEGAADgA4g2AAAAH0C0AQAA+ACiDQAAwAcQbQAAAD6AaAMAAPABRBsAAIAPINoAAAB8wP8A1diTSbAqepcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x879d780>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Método da Bissecção para determinar uma raiz de uma função em uma variável do tipo f(x) = 0\n",
    "\n",
    "#Utiliza a biblioteca sympy, disponível em: http://docs.sympy.org/latest/index.html\n",
    "#Utiliza a biblioteca matplotlib, disponível em: https://matplotlib.org/\n",
    "\n",
    "import sympy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def sgn(s): # função que testa o sinal de f(x)\n",
    "    if (s < 0.0):\n",
    "        return -1\n",
    "    elif (s == 0.0):\n",
    "        return 0\n",
    "    else:\n",
    "        return 1\n",
    "\n",
    "def f(a, x, fx): #Retorna o valor numérico da função digitada pelo usuário\n",
    "    return float(fx.subs(x, a))\n",
    "\n",
    "x = sp.symbols('x')\n",
    "fx = sp.sympify(input(\"Digite a função f(x): \"))\n",
    "\n",
    "TOL = abs(float(input(\"Tolerância para o tamanho do intervalo (b-a)/2 (digite 0 para desativar): \"))) # Tolerância para o tamanho do intervalo (b-a)/2\n",
    "e = abs(float(input(\"Tolerância para o erro relativo de p (ponto médio do intervalo) (digite 0 para desativar): \"))) # Tolerância para o erro relativo de p\n",
    "efp = abs(float(input(\"Tolerância para o valor de f(p) ~= 0 (digite 0 para desativar): \"))) # Tolerância para f(p) ~= 0\n",
    "N0 = abs(int(input(\"Número máximo de iterações: \"))) # Número máximo de iterações\n",
    "\n",
    "# Variáveis extras para o cálculo\n",
    "TOL_val=0.0\n",
    "e_val=0.0\n",
    "FP=0.0\n",
    "pn=0.0\n",
    "n=0\n",
    "i=0.0\n",
    "passo=0.0\n",
    "\n",
    "# Obtém a e b e testa se f(a) e f(b) possuem sinais opostos\n",
    "while (True):\n",
    "    a = float(input(\"Defina a: \"))\n",
    "    b = float(input(\"Defina b: \"))\n",
    "    if (f(a,x,fx)*f(b,x,fx) > 0.0):\n",
    "        print(\"f(a) e f(b) devem ter sinais opostos!\")\n",
    "        continue\n",
    "    elif (abs(f(a,x,fx)) <= efp):\n",
    "        print(\"a já é uma raiz da função para a tolerância desejada!\")\n",
    "        N0 = -1 # Para finalizar o programa\n",
    "    elif (abs(f(b,x,fx)) <= efp):\n",
    "        print(\"b já é uma raiz da função para a tolerância desejada!\")\n",
    "        N0 = -1 # Para finalizar o programa\n",
    "    break\n",
    "\n",
    "# Plota o gráfico da função no intervalo [a, b]\n",
    "plt.rcParams['figure.figsize'] = 10, 10 # Tamanho da figura\n",
    "graf1 = sp.plotting.plot(fx,show=sp.false) # Plota o gráfico da função\n",
    "if (a<b): # Determina os limites do eixo x\n",
    "    graf1.xlim = [a, b]\n",
    "else:\n",
    "    graf1.xlim = [b, a]\n",
    "# Determina os limites do eixo y do gráfico e calcula o passo com que a sequência de subdivisões de [a, b] serão plotadas no eixo y\n",
    "if (f(a,x,fx) < 0):\n",
    "    graf1.ylim = [f(a,x,fx), f(b,x,fx)]\n",
    "    passo = abs(f(a,x,fx))/N0\n",
    "else:\n",
    "    graf1.ylim = [f(b,x,fx), f(a,x,fx)]\n",
    "    passo = abs(f(b,x,fx))/N0    \n",
    "graf1[0].line_color = 'green' # Cor da linha do gráfico da função\n",
    "graf1.legend = sp.false # Não mostrar legenda\n",
    "\n",
    "# Realiza os cálculos do método da biseccção\n",
    "print ('\\nn\\t a\\t         b\\t         p\\t         f(p)\\t         (b-a)/2\\t e. relat. p\\tMundança')\n",
    "for n in range (1, N0+1):\n",
    "    p = a + (b-a)/2\n",
    "    i -= passo # i vai descendo no eixo y e plotando as subdivisões do intervalo [a, b] uma abaixo da outra\n",
    "    graf1.append(sp.plotting.plot(i, (x, a, p), show=sp.false, line_color='blue')[0]) # Plota a próxima subdivisão do intervalo [a, p]\n",
    "    graf1.append(sp.plotting.plot(i, (x, p, b), show=sp.false, line_color='red')[0]) # Plota a próxima subdivisão do intervalo [p, b]\n",
    "    FP = f(p,x,fx)\n",
    "    TOL_val = abs((b-a)/2)\n",
    "    if(p != 0.0): \n",
    "        e_val = abs((pn-p)/p)\n",
    "    print (n, '\\t', \"{:.10f}\".format(a), '\\t', \"{:.10f}\".format(b), '\\t', \"{:.10f}\".format(p), '\\t', \"{:.10f}\".format(FP), '\\t', \"{:.10f}\".format(TOL_val), '\\t', \"{:.10f}\".format(e_val), end=\"\")\n",
    "    if (abs(FP) <= efp or TOL_val < TOL or  e_val < e): #Se F(p) ~= 0 ou (b-2)/2 < erro ou erro relativo de p < erro, encerra o programa\n",
    "        print (\"\\n\\nA raiz da função, dentro dos parâmetros de tolerância, é\", p)\n",
    "        break\n",
    "    if (sgn(f(a,x,fx)) * sgn(FP) > 0.0):\n",
    "        a = p\n",
    "        print ('   f(a)*f(p)>0.0, a=p')\n",
    "    else:\n",
    "        b = p\n",
    "        print ('   f(a)*f(p)<0.0, b=p')\n",
    "    pn = p\n",
    "\n",
    "print (\"\\nO método da Bissecção foi encerrado após\", n, \"iterações.\")\n",
    "graf1.show() # Mostra o gráfico ao final"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Iteração de Ponto Fixo ##\n",
    "\n",
    "Um número $p$ é um **ponto fixo** de uma função $g$ se $g (p) = p$. \n",
    "\n",
    "### Teorema 2.3 ##\n",
    "**a.** Se $g ∈ C [a,b], e g(x) ∈ [a,b]$, pata todo $x ∈ [a,b], g$ terá um ponto fixo em $[a,b]$.\n",
    "\n",
    "**b.** Além disso, se $g'(x)$ existir em $(a,b)$ e existir uma constante positiva $k < 1$ com\n",
    "<p style=\"text-align: center;\"> $|g'(x)|≤ k$, para todo $x ∈ (a,b)$ <p>\n",
    "    \n",
    "então o ponto fixo em $[a,b]$ será único.\n",
    "\n",
    "#### Demostração####\n",
    "\n",
    "**a.** Se $g(a)=a$ ou $g(b)=b$, $g$ tem um ponto fixo em una extremidade. Caso contrário, $g(a) > a$ e $g(b) < b$. A função $h(x) = g(x) - x$ é contínua em $[a,b]$, com \n",
    "\n",
    "<p style=\"text-align: center;\"> $h(a) = g(a) - a > 0$ e $h(b) = g(b) - b < 0$.<p>\n",
    "\n",
    "O Teorema do Valor Intermediário implica a existência de $p ∈ (a,b)$ para o qual $h(p) = 0$. Esse número $p$ é um ponto fixo de $g$, uma vez que\n",
    "<p style=\"text-align: center;\"> $ 0 =h(p) = g(p) - p$ implica que $g(p) = p$ <p>\n",
    "\n",
    "**b.** Suponha também que $g'(x) ≤ k < 1$ e que $p$ e $q$ sejam pontos fixos em $[a,b]$. Se $p ≠ q$, o Teorema do Valor Médio impliqca a existência de um número ε entre $p$ e $q$ e, portanto, em $[a,b]$, com \n",
    "<p style=\"text-align: center;\"> $(g(p)-g(q))/(p-q) = g'(ε) $. \n",
    "\n",
    "PAra determinas a aproximação do ponto fixo de uma função $g$, escolhemos uma aproximação inicial $p_0$ e geramos a seqüência $[p_n]_{n=0}^{\\infty}$, definido $p_n = g(p_n-1)$, para cada $n ≥ 1.$ Se a seqüência converfir para $p$ e $q$ for cntínua, então\n",
    "<p style=\"text-align: center;\"> $p = lim_{n\\to\\infty} g(p_n-1) = g(lim_{n\\to\\infty} p_n-1) = g(p)$, <p>\n",
    "    \n",
    "e será obtida uma solução para $x=g(x).$ Essa técnica é denominada **iteração de ponto fixo** ou **iteração funcional**. \n",
    "\n",
    "### Teorema 2.4 (Teorema do Ponto Fixo)###\n",
    "\n",
    "Seja $g ∈ C[a,b]$ tal que $g(x) ∈ [a,b]$, para todo $x$ em $[a,b]$. Além disso, supoha que $g'$ exista em $(a,b)$ e que exista uma constante $0 < k < 1$ com \n",
    "<p style=\"text-align: center;\">  $|g'(x)|≤ k$, para todo $x ∈ (a,b)$. <p>\n",
    "    \n",
    "Então, para qualquer número $p_0$ em $[a,b]$, a seqüência definida por\n",
    "<p style=\"text-align: center;\"> $p_n = g(p_n-1), n≥1$ <p>\n",
    "\n",
    "converge para o único ponto fixo em $[a,b]$.\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaee6ba8>"
      ]
     },
     "metadata": {},
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    },
    {
     "name": "stdout",
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     "text": [
      "Ponto fixo de g(x) = pi+0.5*sin(x/2), para p0 = 0\n",
      "n\t p\t         |p - p0|\t |p - p0|/|p|\n",
      "1 \t 3.1415926536 \t 3.1415926536 \t 1.0000000000\n",
      "2 \t 3.6415926536 \t 0.5000000000 \t 0.1373025617\n",
      "3 \t 3.6260488644 \t 0.0155437891 \t 0.0042867015\n",
      "o ponto fixo da função g(x), dentro dos parâmetros de tolerância, é 3.6260488644451154\n",
      "O método foi encerrado após 3 iterações.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "import numpy\n",
    "\n",
    "#Método do Ponto Fixo - Com gráfico\n",
    "\n",
    "import sympy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def ponto_fixo (g, p0, TOL, TOL_rel, N0): # Função método do Ponto Fixo\n",
    "    #Argumentos: \n",
    "    #g: função g(x) a ser encontrado o seu ponto fixo: g(p) = p\n",
    "    # TOL: Tolerância para o erro absoluto |p - p0|\n",
    "    # TOL_rel: Tolerância para o erro relativo |p - p0|/|p|\n",
    "    # N0: número máximo de iterações\n",
    "    \n",
    "    x = sp.symbols('x') # variavel da função sympy\n",
    "    gx = sp.sympify(g) # Transforma a string em uma função sympy\n",
    "    \n",
    "    \n",
    "    # Plota o gráfico da função g(x) e o gráfico y = x\n",
    "    plt.rcParams['figure.figsize'] = 10, 10 # Tamanho da figura\n",
    "    graf1 = sp.plotting.plot(gx, show=sp.false, line_color='green', ylabel=\"g(x)\") # Plota o gráfico da função g(x)\n",
    "    graf1.append(sp.plotting.plot(x, show=sp.false, line_color='blue')[0]) # Plota o gráfico y = x\n",
    "    graf1.legend = sp.true # Mostrar legenda\n",
    "    graf1.show()\n",
    "  \n",
    "    print ('Ponto fixo de g(x) = ' + g + ', para p0 = ' + str(p0) + '\\nn\\t p\\t         |p - p0|\\t |p - p0|/|p|')\n",
    "    \n",
    "    for n in range (1, N0+1):\n",
    "        p = float(gx.subs(x, p0)) # Calcula pn\n",
    "        erro = abs(p - p0) # Calcula o erro absoluto |p - p0|\n",
    "        erro_rel = erro / abs(p) # Calcula o erro relativo |p - p0|/|p|\n",
    "        print (n, '\\t', \"{:.10f}\".format(p), '\\t', \"{:.10f}\".format(erro), '\\t', \"{:.10f}\".format(erro_rel))\n",
    "        if (erro < TOL or erro_rel < TOL_rel): # Testa os erros\n",
    "            print (\"o ponto fixo da função g(x), dentro dos parâmetros de tolerância, é\", p)\n",
    "            break\n",
    "        if (erro_rel > 1): # Se o erro relativo subir acima de 100%, o programa encerra por precaução\n",
    "            print (\"O método falhou!\")\n",
    "            break\n",
    "        p0 = p\n",
    "    print (\"O método foi encerrado após\", n, \"iterações.\\n\")\n",
    "\n",
    "\n",
    "# Exemplo de uso da função ponto_fixo (basta alterar aqui para usar):\n",
    "\n",
    "ponto_fixo('pi+0.5*sin(x/2)', 0, 0.01, 0.01, 30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Referências Bibliográficas**\n",
    "\n",
    "BURDEN, R.L.;FAIRES,D.J.;BURDEN, A.M. **Numerical Analysis**. 10 ed. Boston, MA: Cengage Learning, 2014, cap. 2, p.47-66. ISBN 978-1-305-25366-7"
   ]
  }
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