lista-4-solucoes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lista de Exercícios 4\n",
    "\n",
    "Solucionário matemático e computacional de exercícios selecionados da Lista de Exercícios 4."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sy\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Funções Implementadas\n",
    "\n",
    "# Solução de Sistemas\n",
    "\n",
    "def gaussjacobi(AB,ER,X0):\n",
    "    '''Realiza o cálculo de um sistema linear através do método de iterativo de Gauss-Jacobi.\n",
    "    \n",
    "    Sinopse:\n",
    "         X = gaussjacobi(AB,ER,X0)\n",
    "    \n",
    "    Entradas:\n",
    "         AB - Matriz aumentada (np.array) do sistema linear\n",
    "         ER - Erro relativo (forma decimal) entre a iteração i e a iteração i-1\n",
    "         X0 - Vetor estimativa inicial da solução\n",
    "     \n",
    "    Saídas:\n",
    "         X - Vetor solução do sistema linear\n",
    "         \n",
    "         @ney\n",
    "    '''\n",
    "    \n",
    "    A = AB[:,0:-1]\n",
    "    B = AB[:,-1]\n",
    "    X = X0\n",
    "    erro = 1\n",
    "    \n",
    "    while (erro > ER):\n",
    "        Xp = np.copy(X)\n",
    "        for i in range(len(A)):\n",
    "            s = 0\n",
    "            for j in range(len(A)):\n",
    "                if (i != j):\n",
    "                    s += A[i,j]*Xp[j]\n",
    "            X[i] = (B[i] - s)/A[i,i]\n",
    "        EA = np.absolute(X - Xp)\n",
    "        Emax = np.amax(EA)\n",
    "        idx = np.where(EA == Emax)\n",
    "        \n",
    "        if (len(idx[0]) > 1):\n",
    "            erro = np.absolute(EA[idx[0][0]]/X[idx[0][0]])\n",
    "        else:\n",
    "            erro = np.absolute(EA[idx[0]]/X[idx[0]])\n",
    "    \n",
    "    X = np.around(X, decimals=3)\n",
    "\n",
    "    return X\n",
    "\n",
    "def gaussseidel(AB,ER,X0):\n",
    "    ''' Função que realiza o cálculo de um sistema linear através do método de iterativo de Gauss-Seidel.\n",
    "    \n",
    "    Sinopse:\n",
    "         X = gaussseidel(AB,ER,X0)\n",
    "    \n",
    "    Entradas:\n",
    "         AB - Matriz aumentada (np.array) do sistema linear\n",
    "         ER - Erro relativo (forma decimal) entre a iteração i e a iteração i-1\n",
    "         X0 - Vetor estimativa inicial da solução\n",
    "     \n",
    "    Saídas:\n",
    "         X - Vetor solução do sistema linear\n",
    "         \n",
    "         @ney\n",
    "    '''\n",
    "    \n",
    "    A = AB[:,0:-1]\n",
    "    B = AB[:,-1]\n",
    "    X = X0\n",
    "    erro = 1\n",
    "    \n",
    "    while (erro > ER):\n",
    "        Xp = np.copy(X)\n",
    "        for i in range(len(A)):\n",
    "            s = 0\n",
    "            for j in range(len(A)):\n",
    "                if (i != j):\n",
    "                    s += A[i,j]*X[j]\n",
    "            X[i] = (B[i] - s)/A[i,i]\n",
    "        EA = np.absolute(X - Xp)\n",
    "        Emax = np.amax(EA)\n",
    "        idx = np.where(EA == Emax)\n",
    "        \n",
    "        if (len(idx[0]) > 1):\n",
    "            erro = np.absolute(EA[idx[0][0]]/X[idx[0][0]])\n",
    "        else:\n",
    "            erro = np.absolute(EA[idx[0]]/X[idx[0]])\n",
    "    \n",
    "    X = np.around(X, decimals=3)\n",
    "    \n",
    "    return X\n",
    "\n",
    "def jacobian(F,Xs):\n",
    "    '''Função que realiza o cálculo da matriz jacobiana.\n",
    "    \n",
    "    Sinopse:\n",
    "         J = jacobian(F,Xs)\n",
    "    \n",
    "    Entradas:\n",
    "         F - Vetor contendo as equações (simbólicas) do sistema não-linear\n",
    "         Xs - Vetor contendo as variáveis (simbólicas) do sistema não-linear\n",
    "    \n",
    "    Saídas:\n",
    "         J - Matriz jacobiana (simbólica)\n",
    "         \n",
    "         @ney\n",
    "    '''\n",
    "    J = sy.zeros(len(Xs), len(Xs))\n",
    "    \n",
    "    for i in range(len(Xs)):\n",
    "        for j in range(len(Xs)):\n",
    "            J[i,j] = sy.diff(F[i],Xs[j])\n",
    "    \n",
    "    return J\n",
    "\n",
    "def newtonnaolin(F,Xs,ER,X0):\n",
    "    '''Função que realiza o cálculo de um sistema não-linear através do método de iterativo de Newton.\n",
    "    \n",
    "    Sinopse:\n",
    "         X = newtonnaolin(F,Xs,ER,X0)\n",
    "    \n",
    "    Entradas:\n",
    "         F  - Vetor (sy.Matrix) contendo as equações (simbólicas) do sistema não-linear\n",
    "         Xs - Vetor (sy.Matrix) contendo as variáveis (simbólico) do sistema não linear\n",
    "         ER - Erro relativo (forma decimal) entre a iteração i e a iteração i-1\n",
    "         X0 - Vetor estimativa inicial da solução\n",
    "     \n",
    "    Saídas:\n",
    "         X - Vetor solução do sistema linear\n",
    "         \n",
    "         @ney\n",
    "    '''\n",
    "    \n",
    "    J = jacobian(F,Xs)\n",
    "    erro = 1\n",
    "    X = np.copy(X0)\n",
    "    \n",
    "    while (erro > ER):\n",
    "        Xp = np.copy(X)\n",
    "        A = J[:,:]\n",
    "        B = F[:,:]\n",
    "        subs = []\n",
    "        \n",
    "        for i in range(len(Xs)):\n",
    "            subs.append((Xs[i],X[i]))\n",
    "            \n",
    "        A = np.asarray(A.subs(subs), dtype=float)\n",
    "        B = np.asarray(B.subs(subs), dtype=float)\n",
    "        B *= -1\n",
    "        \n",
    "        S = np.linalg.solve(A,B)\n",
    "        S = np.transpose(S)\n",
    "        \n",
    "        X += S[0]\n",
    "        \n",
    "        EA = np.absolute(X - Xp)\n",
    "        Emax = np.amax(EA)\n",
    "        idx = np.where(EA == Emax)\n",
    "        \n",
    "        if (len(idx[0]) > 1):\n",
    "            erro = np.absolute(EA[idx[0][0]]/X[idx[0][0]])\n",
    "        else:\n",
    "            erro = np.absolute(EA[idx[0]]/X[idx[0]])\n",
    "    \n",
    "    X = np.around(X, decimals=3)\n",
    "    \n",
    "    return X\n",
    "\n",
    "# Interpolação\n",
    "\n",
    "def int_newton(X, Y, x0):\n",
    "    '''Função que realiza a interpolação de grau n de n+1 pontos dados, utilizando o método das diferenças divididas\n",
    "    de Newton.\n",
    "    \n",
    "    Sinopse:\n",
    "         y0 = int_newton(X, Y, x0)\n",
    "    \n",
    "    Entradas:\n",
    "         X - Array com as coordenadas em x para os pontos dados\n",
    "         Y - Array com as coordenadas em y para os pontos dados\n",
    "         x0 - Ponto ao qual se deseja a estimativa f(x0)\n",
    "     \n",
    "    Saídas:\n",
    "         y0 - Estimativa desejada f(x0)\n",
    "         \n",
    "         @ney\n",
    "    '''\n",
    "    \n",
    "    n = len(X)\n",
    "    x = sy.symbols(\"x\")\n",
    "    \n",
    "    dx = []\n",
    "    f = []\n",
    "    aux = np.diff(Y)\n",
    "\n",
    "    for i in range(1,n):\n",
    "        dx.append(X[i:] - X[:-i])\n",
    "        f.append(aux/dx[i-1]) # Diferenças Divididas\n",
    "        aux = np.diff(f[i-1])\n",
    "    \n",
    "    b = []\n",
    "    \n",
    "    for i in f:\n",
    "        b.append(i[0]) # Coeficientes do polinômio\n",
    "    \n",
    "    f = Y[0]\n",
    "    aux = 1\n",
    "\n",
    "    for i in range(n-1):\n",
    "        aux *= x - X[i]\n",
    "        f += b[i]*aux # Polinômio\n",
    "        \n",
    "    y0 = f.subs(x,x0) # Estimativa desejada\n",
    "    \n",
    "    return float(y0)\n",
    "\n",
    "def int_lagrange(X, Y, x0):\n",
    "    '''Função que realiza a interpolação de grau n de n+1 pontos dados, utilizando o método de Lagrange.\n",
    "    \n",
    "    Sinopse:\n",
    "         y0 = int_lagrange(X, Y, x0)\n",
    "    \n",
    "    Entradas:\n",
    "         X - Vetor com as coordenadas em x para pontos dados\n",
    "         Y - Vetor com as coordenadas em y para pontos dados\n",
    "         x0 - Ponto ao qual se deseja a estimativa f(x0)\n",
    "     \n",
    "    Saídas:\n",
    "         y0 - Estimativa desejada f(x0)\n",
    "         \n",
    "         @ney\n",
    "    '''     \n",
    "    \n",
    "    n = len(X)\n",
    "    x = sy.symbols(\"x\")\n",
    "    f = 0\n",
    "    \n",
    "    for i in range(n):\n",
    "        L = 1\n",
    "        for j in range(n):\n",
    "            if (i != j):\n",
    "                L *= (x - X[j])/(X[i] - X[j])\n",
    "        f += L*Y[i] # Polinômio\n",
    "    \n",
    "    y0 = f.subs(x,x0) # Estimativa desejada\n",
    "    \n",
    "    return float(y0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Métodos iterativos para sistemas lineares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 1\n",
    "\n",
    "Verifique o condicionamento e o critério das linhas para os sistemas abaixo e em seguida aplique o método de Gauss-Jacobi para determinar uma solução aproximada, com erro absoluto inferior a $10^{-2}$, tomando a aproximação inicial $x^{(0)} = 0$.\n",
    "\n",
    "(a) \\begin{cases}\n",
    "2x_1 − x_2 = 1 \\\\\n",
    "x_1 + 2x_2 = 3\n",
    "\\end{cases}\n",
    "\n",
    "(b) \\begin{cases}\n",
    "x_1 − 0.25x_2 − 0.25x_3 = 0 \\\\\n",
    "−0.25x_1 + x_2 − 0.25x_4 = 0 \\\\\n",
    "−0.25x_1 + x_3 − 0.25x_4 = 0.25 \\\\\n",
    "−0.25x_2 + x_4 = 0.25\n",
    "\\end{cases}\n",
    "\n",
    "**Obs.** Tome como número de condicionamento o valor $C = det(Ã)$, em que $Ã$ e obtida de $A$ fazendo com que o maior elemento em valor absoluto em cada linha de $Ã$ seja igual a $1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(a)\n",
      "O número C de condicionamento é: 5.0\n",
      "\n",
      "A solução do problema é:\n",
      " [[0.998]\n",
      " [1.002]]\n",
      "\n",
      "(b)\n",
      "O número C de condicionamento é: 0.812\n",
      "\n",
      "A solução do problema é:\n",
      " [[0.107]\n",
      " [0.093]\n",
      " [0.343]\n",
      " [0.272]]\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "# (a)\n",
    "A = np.array([[2., -1.], [1., 2.]]) # Matriz de coeficientes constantes\n",
    "B = np.array([[1.], [3.]]) # Vetor de constantes\n",
    "X = np.array([[0.], [0.]]) # Estimativa inicial (X0)\n",
    "EA = 10**(-2)\n",
    "erro = 1\n",
    "\n",
    "while (erro > EA):\n",
    "    Xp = np.copy(X)\n",
    "    for i in range(len(A)):\n",
    "        s = 0\n",
    "        for j in range(len(A)):\n",
    "            if (i != j):\n",
    "                s += A[i,j]*Xp[j]\n",
    "        X[i] = (B[i] - s)/A[i,i]\n",
    "    erro = np.amax(np.absolute(X - Xp))\n",
    "\n",
    "X = np.around(X, decimals=3)    \n",
    "    \n",
    "print(\"(a)\\nO número C de condicionamento é:\", np.around(np.linalg.det(A), decimals=3))\n",
    "print(\"\\nA solução do problema é:\\n\", X)\n",
    "\n",
    "# (b)\n",
    "A = np.array([[1., -0.25, -0.25, 0.], [-0.25, 1., 0., -0.25], [-0.25, 0., 1., -0.25], [0., -0.25, 0., 1.]]) # Coeficientes\n",
    "B = np.array([[0.], [0.], [0.25], [0.25]]) # Vetor de constantes\n",
    "X = np.array([[0.], [0.], [0.], [0.]]) # Estimativa inicial (X0)\n",
    "EA = 10**(-2)\n",
    "erro = 1\n",
    "\n",
    "while (erro > EA):\n",
    "    Xp = np.copy(X)\n",
    "    for i in range(len(A)):\n",
    "        s = 0\n",
    "        for j in range(len(A)):\n",
    "            if (i != j):\n",
    "                s += A[i,j]*Xp[j]\n",
    "        X[i] = (B[i] - s)/A[i,i]\n",
    "    erro = np.amax(np.absolute(X - Xp))\n",
    "\n",
    "X = np.around(X, decimals=3)    \n",
    "    \n",
    "print(\"\\n(b)\\nO número C de condicionamento é:\", np.around(np.linalg.det(A), decimals=3))\n",
    "print(\"\\nA solução do problema é:\\n\", X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 2\n",
    "\n",
    "Considere o sistema\n",
    "\n",
    "\\begin{cases}\n",
    "15c_1 − 2c_2 − c_3 = 3800 \\\\\n",
    "−3c_1 + 18c_2 − 6c_3 = 1200 \\\\\n",
    "−4c_1 − c_2 + 12c_3 = 2350\n",
    "\\end{cases}\n",
    "\n",
    "(a) Verifique o condicionamento do sistema.\n",
    "\n",
    "(b) Verifique o critério das linhas.\n",
    "\n",
    "(c) Verifique o critério de Sassenfeld.\n",
    "\n",
    "(d) Determine uma solução aproximada do sistema com erro relativo percentual abaixo de $5\\%$ usando o método iterativo de Gauss-Jacobi.\n",
    "\n",
    "(e) Repita o item (d) usando o método de Gauss-Seidel.\n",
    "\n",
    "(f) Compare o número de iterações nos itens (d) e (e)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(d) A solução de X é:\n",
      " [[301.184]\n",
      " [216.637]\n",
      " [311.689]]\n",
      "\n",
      "(e) A solução de X é:\n",
      " [[302.072]\n",
      " [220.055]\n",
      " [314.862]]\n",
      "\n",
      "(f) O número de iterações pelo método de Gauss-Jacobi foi: 4\n",
      "O número de iterações pelo método de Gauss-Seidel foi: 3\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "AB = np.array([[15., -2., -1., 3800.], [-3., 18., -6., 1200.], [-4., -1., 12., 2350.]])\n",
    "ER = 5/100\n",
    "X0 = np.array([[0.], [0.], [0.]])\n",
    "\n",
    "# (d) Gauss-Jacobi\n",
    "A = AB[:,0:-1]\n",
    "B = AB[:,-1]\n",
    "X = np.copy(X0)\n",
    "erro = 1\n",
    "k1 = 0\n",
    "    \n",
    "while (erro > ER):\n",
    "    Xp = np.copy(X)\n",
    "    for i in range(len(A)):\n",
    "        s = 0\n",
    "        for j in range(len(A)):\n",
    "            if (i != j):\n",
    "                s += A[i,j]*Xp[j]\n",
    "        X[i] = (B[i] - s)/A[i,i]\n",
    "    EA = np.absolute(X - Xp)\n",
    "    Emax = np.amax(EA)\n",
    "    idx = np.where(EA == Emax)\n",
    "    erro = np.absolute(EA[idx[0]]/X[idx[0]])\n",
    "    k1 += 1\n",
    "\n",
    "X = np.around(X, decimals=3)    \n",
    "    \n",
    "print(\"(d) A solução de X é:\\n\", X)\n",
    "\n",
    "# (e) Gauss-Seidel\n",
    "A = AB[:,0:-1]\n",
    "B = AB[:,-1]\n",
    "X = np.copy(X0)\n",
    "erro = 1\n",
    "k2 = 0\n",
    "    \n",
    "while (erro > ER):\n",
    "    Xp = np.copy(X)\n",
    "    for i in range(len(A)):\n",
    "        s = 0\n",
    "        for j in range(len(A)):\n",
    "            if (i != j):\n",
    "                s += A[i,j]*X[j]\n",
    "        X[i] = (B[i] - s)/A[i,i]\n",
    "    EA = np.absolute(X - Xp)\n",
    "    Emax = np.amax(EA)\n",
    "    idx = np.where(EA == Emax)\n",
    "    erro = np.absolute(EA[idx[0]]/X[idx[0]])\n",
    "    k2 += 1\n",
    "\n",
    "X = np.around(X, decimals=3)\n",
    "\n",
    "print(\"\\n(e) A solução de X é:\\n\", X)\n",
    "\n",
    "# (f)\n",
    "print(\"\\n(f) O número de iterações pelo método de Gauss-Jacobi foi:\", k1)\n",
    "print(\"O número de iterações pelo método de Gauss-Seidel foi:\", k2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 3\n",
    "\n",
    "Use o método iterativo de Gauss-Seidel para determinar uma solução aproximada do sistema com erro relativo percentual abaixo de $5\\%$ dos seguintes sistemas. Antes verifique o critério de Sassenfeld e caso algum dos sitemas não o satisfaça, reorganize o sistema permutando linhas e/ou colunas para garantir a convergência.\n",
    "\n",
    "(a) \\begin{cases}\n",
    "10x_1 + 2x_2 − x_3 = 27 \\\\\n",
    "−3x_1 − 6x_2 + 2x_3 = −61.5 \\\\\n",
    "x_1 + x_2 + 5x_3 = −21.5\n",
    "\\end{cases}\n",
    "\n",
    "(b) \\begin{cases}\n",
    "−3x_1 + x_2 + 12x_3 = 50 \\\\\n",
    "6x_1 − x_2 − x_3 = 3 \\\\\n",
    "6x_1 + 9x_2 + x_3 = 40\n",
    "\\end{cases}\n",
    "\n",
    "(c) \\begin{cases}\n",
    "2x_1 − 6x_2 − x_3 = −38 \\\\\n",
    "−3x_1 − x_2 + 7x_3 = −34 \\\\\n",
    "−8x_1 + x_2 − 2x_3 = −20\n",
    "\\end{cases}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(a)\n",
      "A solução do sistema linear é: \n",
      " [ 0.5  8.  -6. ]\n",
      "\n",
      "(b)\n",
      "A solução do sistema linear é: \n",
      " [1.697 2.829 4.355]\n",
      "\n",
      "(c)\n",
      "A solução do sistema linear é: \n",
      " [ 4.005  7.992 -1.999]\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "# (a)\n",
    "AB = np.array([[10., 2., -1., 27.], [-3., -6., 2., -61.5], [1., 1., 5., -21.5]])\n",
    "ER = 5/100\n",
    "X0 = np.array([0., 0., 0.])\n",
    "X = gaussseidel(AB,ER,X0)\n",
    "print(\"(a)\\nA solução do sistema linear é: \\n\", X)\n",
    "\n",
    "# (b)\n",
    "AB = np.array([[6., -1., -1., 3.], [6., 9., 1., 40.], [-3., 1., 12., 50.]])\n",
    "ER = 5/100\n",
    "X0 = np.array([0., 0., 0.])\n",
    "X = gaussseidel(AB,ER,X0)\n",
    "print(\"\\n(b)\\nA solução do sistema linear é: \\n\", X)\n",
    "\n",
    "# (c)\n",
    "AB = np.array([[-8., 1., -2., -20.], [2., -6., -1., -38.], [-3., -1., 7., -34.]])\n",
    "ER = 5/100\n",
    "X0 = np.array([0., 0., 0.])\n",
    "X = gaussseidel(AB,ER,X0)\n",
    "print(\"\\n(c)\\nA solução do sistema linear é: \\n\", X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 4\n",
    "Dos seguintes sistemas abaixo identifique qual(is) deles nao pode(m) ser resolvido(s) usando o método iterativo de Gauss-Seidel.\n",
    "\n",
    "(a) \\begin{cases}\n",
    "9x + 3y + z = 13 \\\\\n",
    "−6x + 8z = 2 \\\\\n",
    "2x + 5y − z = 6\n",
    "\\end{cases}\n",
    "\n",
    "(b) \\begin{cases}\n",
    "x + y + 6z = 8 \\\\\n",
    "x + 5y − z = 5 \\\\\n",
    "4x + 2y − 2z = 4\n",
    "\\end{cases}\n",
    "\n",
    "(c) \\begin{cases}\n",
    "−3x + 4y + 5z = 6 \\\\\n",
    "−2x + 2y − 3z = −3 \\\\\n",
    "2y − z = 1\n",
    "\\end{cases}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solução\n",
    "\n",
    "(a) Convergente\n",
    "\n",
    "(b) Divergente \n",
    "\n",
    "(c) Divergente"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 5\n",
    "\n",
    "Uma companhia de eletrônica produz transistores, resistores e chips de computador. Cada transistor usa quatro unidades de cobre, uma unidade de zinco e duas unidades de vidro. Cada resisitor usa três, três e uma unidades de cada material, respectivamente, e cada chip de computador usa duas, uma e três unidades desses materiais, respectivamente. Colocando essas \n",
    "informações em uma tabela, tem-se:\n",
    "\n",
    "Componente | Cobre | Zinco | Vidro\n",
    "-|-|-|-\n",
    "Transistor | 4 | 1 | 2\n",
    "Resistor | 3 | 3 | 1\n",
    "Chip de Computador | 2 | 1 | 3\n",
    "\n",
    "O fornecimento desses materiais varia de semana para semana. Assim, a companhia precisa determinar uma meta de produção diferente para cada semana. Por exemplo, em uma semana a quantidade total de materiais disponíveis era 960 unidades de cobre, 510 unidades de zinco e 610 unidades de vidro. Determine o sistema de equações que modela essa meta de produção e determine a sua solução, pelo método de Gauss-Seidel, para determinar o número de transistores, resistores e chips de computador fabricados nessa semana."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[T,R,C] = [120. 100.  90.]\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "AB = np.array([[4., 3., 2., 960.], [1., 3., 1., 510.], [2., 1., 3., 610.]])\n",
    "ER = 1/100\n",
    "X0 = np.array([[0.], [0.], [0.]])\n",
    "\n",
    "X = gaussseidel(AB,ER,X0)\n",
    "X = np.transpose(np.rint(X))\n",
    "print(\"[T,R,C] =\", X[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sistemas não-lineares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 6\n",
    "\n",
    "Usando o Método de Newton, determinar uma raiz para cada sistema não-linear abaixo com precisão $\\epsilon= 10^{-3}$:\n",
    "\n",
    "(i) $$\\begin{cases} x^2 + y^2 = 2 \\\\ x^2 - y^2 = 1 \\end{cases} \\quad \n",
    "x^{(0)} = \\begin{bmatrix} 1.2 \\\\ 0.7 \\end{bmatrix}$$\n",
    "\n",
    "(ii) $$\\begin{cases} 3x^2 y - y^3 = 4 \\\\ x^2 - x y^3 = 9 \\end{cases} \\quad\n",
    "x^{(0)} = \\begin{bmatrix} -1 \\\\ -2 \\end{bmatrix}$$\n",
    "\n",
    "(i) $$\\begin{cases} (x-1)^2 + y^2 = 4 \\\\ x^2 + (y-1)^2 = 4 \\end{cases} \\quad\n",
    "x^{(0)} = \\begin{bmatrix} 2 \\\\ 1 \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i) A solução do problema é: [1.225 0.707]\n",
      "\n",
      "(ii) A solução do problema é: [3.002 0.148]\n",
      "\n",
      "(iii) A solução do problema é: [1.823 1.823]\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "x, y = sy.symbols(\"x, y\")\n",
    "Xs = sy.Matrix([x, y])\n",
    "ER = 1/100\n",
    "\n",
    "# (i)\n",
    "F = sy.Matrix([x**2 + y**2 - 2, x**2 - y**2 - 1])\n",
    "X0 = np.array([1.2, 0.7])\n",
    "X = newtonnaolin(F,Xs,ER,X0)\n",
    "print(\"(i) A solução do problema é:\", X)\n",
    "\n",
    "# (ii)\n",
    "F = sy.Matrix([3*(x**2)*y - y**3 - 4, x**2 - x*y**3 - 9])\n",
    "X0 = np.array([-1., -2.])\n",
    "X = newtonnaolin(F,Xs,ER,X0)\n",
    "print(\"\\n(ii) A solução do problema é:\", X)\n",
    "\n",
    "# (iii)\n",
    "F = sy.Matrix([(x-1)**2 + y**2 - 4, x**2 + (y-1)**2 - 4])\n",
    "X0 = np.array([2., 1.])\n",
    "X = newtonnaolin(F,Xs,ER,X0)\n",
    "print(\"\\n(iii) A solução do problema é:\", X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpolação Polinomial "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 7\n",
    "\n",
    "Use um polinômio interpolador de Lagrange de primeiro e segundo graus para calcular $ln(2)$, com base nos seguintes dados:\n",
    "\n",
    "$x_i$ | $f(x_i)$\n",
    "-|-\n",
    "1 | 0\n",
    "4 | 1.386294\n",
    "6 | 1.791759"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i)\n",
      "A estimativa de ln(2) para o polinômio de primeiro grau é: 0.46209799999999995\n",
      "\n",
      "(ii)\n",
      "A estimativa de ln(2) para o polinômio de segundo grau é: 0.5658441999999999\n"
     ]
    },
    {
     "data": {
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aNEhvueUWVVV9/vnndcaMGaqqeuTIEW3evHn++z7T8uXL8/c7JS4uTjt37qzDhg3TnJyc8/xLOOuCfhdNiXC5XBq/94j+eeEG7fDSEm3y3GKNeHmpvvzFJt2UeNTp8hwBxGgRmeqdTTd7Cx9zvsjlF6Fz587548106NCB3bt306NHj9O2+fnnn9m8eTPdu3cHICsr67QBz84cmfHUUMfXXHMNx44dIzU1lR9++IF58+YBcN1115GSksLRo0dP2y8hIYHBgwezf/9+srKy8ocVXrZsWf5gayJC7dq1gbyxaBYsWADAvn372LFjBwcOHDht2OFhw4axYsUK+vfvn/86W7dupWnTpvl3lN5zzz35nwgudYjljh07nndqRmMSU0+yMD6ReXEJ7Eo+QZVKPvRpezkDwoPp2SyISr7e2Uhxqbwz6Bs3zmuuKWx5MTk1DDCcPgxuQarKjTfeWOQgZ2cOHnbmTRkiUmgzxpnbPfbYYzz99NPcfvvtREdHM27cuPzXP1N0dDTLli1j5cqVBAQE0KtXLzIyMjxuLinqxhEtxiGWjSkoLTOHrzb+xrzYBH7+NQVV6Ny0Dg9ecwU3tW9ATX8/p0ss87zzz9+ECXBmZ2hAQN7yElajRg2OHz8OQNeuXfnxxx/zh8VNT08/bVKPM506+/7hhx+oVasWtWrV4pprrmGmu28hOjqaoKAgatasedp+R48eJTg4b2TOadOm5S/v06cP77//PpAXxEeOHOHo0aPUrl2bgIAAtm7dys8//wzkDTv83XffcejQIXJzc5k1a9ZpM2gBtGrVil9//ZVffvkF4LQ/YCU9xLKpWHJdyvc7knnq07V0emUZf/hsHfuPnuSpG1rw/bO9mfNgNwZ3amwh7yHvPKM/dXWNA1fdREVFcdNNN9GgQQOWL1/O1KlTGTp0KJmZmQC88sortGjRotB9a9euzdVXX82xY8eYMmUKkDcpyP33309YWBgBAQGnBfkp48aNY9CgQQQHB9O1a1d+/fVXAF544QUeffRRGjduTEBAAH//+9/p168fkyZNIiwsjJYtW9K1a1cgb9jhV199ld69e6Oq3Hzzzdxxxx2nvY6/vz+TJ0/mlltuISgoiB49erBxY95Uwp4OsdyzZ0+2bt1KWloaISEhfPzxx/Tt2/cif9rG22w/cJx5cQksjE/kwLFMavpX4s7wYAaGBxPeuHa5H4rAKTaoWRnRq1cv3njjDSIjCx2T6JIkJSXxwQcf8NJLLxX7sSsib/9dLG2H0jJZtDaJ+fEJbEw8NTtTPQaEh3Bdq/r4+1Xgq2UugA1qVoFFR0fz0EMPMawUPs0Y46mM7Fy+2XKQ+XEJRG9PJteltA+uxYu3teG2qxoSVL3K+Q9iPGZBX0ZER0eXyHFPTexhjNNUldg9R5gXl8ji9Un5szON7nkFA8KDaVFGZ2fyBhb0xpgStTclnQXxicyPT2BPSjpV/f43O1O3K8vG7EzezoLeGFPsjmVk8+X6/cyPS2T17sOIQLcr6vLYdc3p1+5yqpej2Zm8gf20jTHFIifXxfc7DjEvLoGlmw+QmePiynrVeKZvS/p3DCa4DM/O5O08mTN2CnArcFALmRxcRHoBn5M3OTjAfFUd717XD/gn4At8pKoTi6luY0wZoKps3p83O9Pna5M4lJZJ7QA/hnRqxIDwEMJCyv/sTN7AkxumpgLnG0rwe1Xt4H6cCnlf4F3gJqANMFRE2lxKsU6zYYpPd6HDFK9du5Zu3brRtm1bwsLC8m8QAxumuLw5cCyDySt+4aZ/fs8tb//A9JW7iWxSm8nDI1j1pxt46Y52XNUo0EK+jDjvGb2qrhCR0Is4dmdgp6ruAhCR2cAdwOaLONYFm7lhJmO/Gcveo3tpXKsxE66fwLD2NkwxODdMcUBAANOnT6d58+YkJSURERFB3759CQwMtGGKy4GTWbl8vfk35sUl8sOOZFwKHRsH8nL/dtzavgG1q3nn7EzeoLiGQOgmIutE5L8i0ta9LBjYV2CbBPeyQolIlIjEiEhMcnLyJRUzc8NMor6IYs/RPSjKnqN7iPoiipkbbJhicG6Y4hYtWuQPiNawYUPq16/PqX9rG6a4bHK5lJ93pfDs3HV0mrCMJ2av5ZeDaTzauxnf/v5aFjzSneFdm1jIl3VFDWtZ8AGEAhuLWFcTqO7+/mZgh/v7QeS1y5/abjjwf5683qUOU9zkzSbKOM56NHmzicfHKIwNU1w8wxSrqq5atUpbtWqlubm5qmrDFJc1vxw8rm8s2apXv/qNNnlusbb583/1D3PW6k87D2lursvp8kwhKMlhilX1WIHvvxSR90QkiLwz+EYFNg0Bki719Tyx92jhwxEXtfxi2DDFFz9M8f79+xk+fDjTpk3DxyfvQ6UNU+y81PQsvli/n/lxCcTvzZudqYfNzuQVLjnoReRy4ICqqoh0Jq85KAVIBZqLSFMgERgC3H2pr+eJxrUas+fo2cMUN65lwxQ7PUzxsWPHuOWWW3jllVfyB1QzzsnKcRG97SDz4xL5ZusBsnOVlpfV4E83t+KODsFcVtPf6RJNMThvG72IzAJWAi1FJEFERorIQyLykHuTu4CNIrIOeBsY4v4kkQOMAZYAW4A5qrqpZN7G6SZcP4EAv9M7QwP8AphwvQ1T7OQwxVlZWdx5553ce++9DBo06Jw/R1NyVJV1+1J58fONdPnrMqJmxBKz5zD3dgvlP4/34KsnexJ1zZUW8l7Ek6tuhp5n/TvAO0Ws+xL48uJKu3inrq5x4qobG6a46GGK58yZw4oVK0hJSWHq1KlA3mWiHTp0uMiftrkQSakn84YiiEvgl+QTVK7kQ582lzEwPIQezYPws9mZvJYNU1xG2DDF5Ud5+l08cWp2prgEVu7Km52pU2htBoSHcHP7BtSqahN3eAsbprgCs2GKK55cl7LylxTmxyXw342/cTI7l8Z1Anji+ubc2TGYJnWrnf8gxqtY0JcRNkyxuVQ7DhxnXlwiC+MT+e1YBjX8K9G/Y97sTBFNbHamiqxcBb2q2i+rcVRZa+pMSctk0bok5sclsiHxKL4+Qq8W9fjzrW24vrXNzmTylJug9/f3JyUlhbp161rYG0eoKikpKfj7l97VKAvjE3l9yTaSUk/SMLAqz/RtSb92l/PtVvfsTNuSyXEp7YJr8pdb23B7B5udyZyt3HTGZmdnk5CQQEZGhkNVGZN3whESEoKfX8l3Yi6MT+T5+Rs4mZ2bv8zXR/DzgYwcpX6NKtzZMZgB4SG0vNxmZ6rovKIz1s/PL/+OT2MqgteXbDst5CGvo7Wyry/TH4ige7Mgm53JeKTcBL0xFU1S6slCl2dk53JNi3qlXI0pz+wOCWPKoJxcFwFVCu9IbWgzNZkLZEFvTBlzLCObB6bFcCIz96ymmap+vjzT99zjCRlzJmu6MaYM2ZNygpHTYth96ASvDmhPVT/fs6666d+xyGkdjCmUBb0xZcSqXSk89O9YXArTR3bm6iuDACzYzSWzoDemDJgTs4+xCzbQqE4AH4/oRNMgG6bAFB8LemMclOtS/vbVViav2EXP5kG8c3e4DTRmip0FvTEOScvM4cnZ8SzbcpDhXZvw4m1tqGRDBZsSYEFvjAMSjqQzaloMOw6mMf6OttzbLdTpkowXs6A3ppTF7jnCgzNiyMxx8a/7OtnNT6bEWdAbU4oWxCfw3NwNNAj0Z3ZUJ5rVr+50SaYCsKA3phS4XMrfl27j3eW/0KVpHSbdE0HtapWdLstUEOcNehGZAtwKHFTVdoWsHwY8536aBjysquvc63YDx4FcIKeokdWM8WbpWTk8/ek6vtr0G0M6NWL8He2oXMk6XU3p8eSMfip5k39PL2L9r8C1qnpERG4CJgNdCqzvraqHLqlKY8qp/UdPMnp6DJuSjvHCLa0Z2aOpzadgSt15g15VV4hI6DnW/1Tg6c9AyKWXZUz5t25fKqOnx3AiM4ePR0RyXavLnC7JVFDF/flxJPDfAs8V+FpEYkUk6lw7ikiUiMSISExycnIxl2VM6Vq8PonffbCSypV8mP9Idwt546hi64wVkd7kBX2PAou7q2qSiNQHlorIVlVdUdj+qjqZvGYfIiMjy960V8Z4QFV5+5udvLlsO5FNajNpeIRN7WccVyxBLyJhwEfATaqacmq5qia5vx4UkQVAZ6DQoDemvMvIzuWZuev5Yl0SA8KDeXVAe6pUssm5jfMuOehFpDEwHxiuqtsLLK8G+Kjqcff3fYDxl/p6xpRFB49lMHpGLOsTUnmuXyseuvYK63Q1ZYYnl1fOAnoBQSKSALwI+AGo6iTgL0Bd4D33L/apyygvAxa4l1UCPlHVr0rgPRjjqI2JRxk9PYbU9Gwm3RNB37aXO12SMafx5KqboedZPwoYVcjyXcBVF1+aMWXfkk2/8eTstQQG+DH34W60bVjL6ZKMOYvdGWvMRVBVJn23i9eWbCUsJJAPh0dQv6a/02UZUygLemMuUGZOLs/P38D8uERuu6ohr98Vhr+fdbqassuC3pgLkJKWyYMzYonZc4SnbmjB49c3s05XU+ZZ0BvjoW2/HWfktDUkH8/knbs7cmtYQ6dLMsYjFvTGeODbrQd47JN4qlWpxJwHu3FVo0CnSzLGYxb0xpyDqvLxD7/y1y+30LpBTT4aEUmDWlWdLsuYC2JBb0wRsnJcvLhoI7NW76Nf28v5x+CrCKhs/2VM+WO/tcYU4siJLB6eGcvPuw4zpncznr6xBT4+1ulqyicLemPOsPNgGiOnrWF/agZvDr6KOzvayNumfLOgN6aA73ck88jMOKpU8mFWVFcimtR2uiRjLpkFvTFu01fu5qUvNtO8fnU+GhFJSO0Ap0syplhY0JsKLyfXxUtfbGbGz3u4vlV9/jm0I9Wr2H8N4z3st9lUaEdPZjPmkzi+33GIB6+5gmf7tcLXOl2Nl7GgNxXW7kMneGDaGvYdTue1gWH8rlMjp0sypkRY0JsKaeUvKTz071h8BP49sgtdrqjrdEnGlBgLelPhzF69lxcWbiQ0qBofj4ikSd1qTpdkTImyoDcVRq5L+euXW/j4h1+5pkU93rm7IzX9/Zwuy5gSZ0FvKoTjGdk8Piue5duSue/qUF64pTWVfH2cLsuYUuHRb7qITBGRgyKysYj1IiJvi8hOEVkvIuEF1o0QkR3ux4jiKtwYT+07nM7A939ixY5DvNK/HeNub2shbyoUT8/opwLvANOLWH8T0Nz96AK8D3QRkTrkTSYeCSgQKyKLVPXIpRRtjKfW7D7MgzNiycl1Mf2BznRvFuR0ScaUOo9Oa1R1BXD4HJvcAUzXPD8DgSLSAOgLLFXVw+5wXwr0u9SijfHEvNgEhn24ilpV/Vj4aHcLeVNhFVcbfTCwr8DzBPeyopYbU2JcLuX1r7fxfvQvXH1lXd4fFkGtAOt0NRVXcQV9YbcS6jmWn30AkSggCqBx48bFVJapaE5k5vDUp2v5evMB7u7SmJdub4uftcebCq64/gckAAVvKwwBks6x/CyqOllVI1U1sl69esVUlqlIklJPcteklSzbcoBxt7VhQv92FvLGUHxBvwi41331TVfgqKruB5YAfUSktojUBvq4lxlTrOL3HuH2d34k4XA6U+7rxH3dmyJiY9YYAx423YjILKAXECQiCeRdSeMHoKqTgC+Bm4GdQDpwv3vdYRF5GVjjPtR4VT1Xp64xF+zztYk8M3c9l9f0Z9boLjS/rIbTJRlTpngU9Ko69DzrFXi0iHVTgCkXXpox5+ZyKW99s4O3v9lB59A6TBoeQZ1qlZ0uy5gyx+6MNeXSyaxc/vDZOv6zYT+DIkKYcGd7Kley9nhjCmNBb8qdA8cyGD09hg2JR/nTza0Y3fMKa4835hws6E25siHhKKOmryEtI4cPh0dyQ5vLnC7JmDLPgt6UG//dsJ+n5qylbrUqzH34alo3qOl0ScaUCxb0psxTVd5dvpM3vt5OeONAPhgeSb0aVZwuy5hyw4LelGkZ2bn8cd56Fq5Non+HhkwcGIa/n6/TZRlTrljQmzIr+XgmUTNiiN+byjN9W/JIryut09WYi2BBb8qkLfuPMWpaDCknMnl/WDg3tW/gdEnGlFsW9KbMWbr5AE/MjujlOR4AABOcSURBVKemvx9zH7qadsG1nC7JmHLNgt6UGarK5BW7mPjVVtoH1+LDeyO5rKa/02UZU+5Z0JsyISvHxdgFG/gsNoFb2jfgjUFXUbWydboaUxws6I3jDp/I4qEZsazefZgnrm/OE9c3x8fHOl2NKS4W9MZROw4c54FpazhwLJO3h3bk9qsaOl2SMV7Hgt44JnrbQR77JJ4qfr58GtWVjo1rO12SMV7Jgt6UOlVl6k+7eXnxZlpeXpOPR0TSMLCq02UZ47Us6E2pys518eKiTXyyai992lzGm4M7UK2K/RoaU5Lsf5gpNanpWTwyM46ffknh4V5X8kyfltbpakwpsKA3pWJXchojp8WQeOQkfx90FQMjQpwuyZgKw4LelLgfdx7i4X/H4ufrwyejuxAZWsfpkoypUDyae01E+onINhHZKSJ/LGT9myKy1v3YLiKpBdblFli3qDiLN2XfzFV7uHfKai6v5c/CR7tbyBvjgPOe0YuIL/AucCOQAKwRkUWquvnUNqr6VIHtHwM6FjjESVXtUHwlm/IgJ9fFK//ZwtSfdtO7ZT3eHtqRGv5+TpdlTIXkSdNNZ2Cnqu4CEJHZwB3A5iK2Hwq8WDzlmfLoWEY2Yz6JZ8X2ZEb1aMrzN7fG1zpdjXGMJ0EfDOwr8DwB6FLYhiLSBGgKfFtgsb+IxAA5wERVXVjEvlFAFEDjxo09KMuURXtSTjByWgy7D51g4oD2DOls/5bGOM2ToC/sVEyL2HYIMFdVcwssa6yqSSJyBfCtiGxQ1V/OOqDqZGAyQGRkZFHHN2XYql0pPPTvWBSYMbIL3a6s63RJxhg864xNABoVeB4CJBWx7RBgVsEFqprk/roLiOb09nvjJebE7OOej1dRp1plFj7S3ULemDLEk6BfAzQXkaYiUpm8MD/r6hkRaQnUBlYWWFZbRKq4vw8CulN0274ph3Jdyl+/3MKzc9fT9Yq6zH+kO6FB1ZwuyxhTwHmbblQ1R0TGAEsAX2CKqm4SkfFAjKqeCv2hwGxVLdjs0hr4QERc5P1RmVjwah1TvqVl5vDk7HiWbTnIvd2a8Jdb21DJ16Mrdo0xpUhOz+WyITIyUmNiYpwuw5xDwpF0Rk2LYcfBNF68rQ33dgt1uiRjKjQRiVXVyMLW2Z2x5oLF7jnCgzNiyMxxMfX+TvRsXs/pkowx52BBby7IgvgEnpu7gQaB/syO6kSz+tWdLskYcx4W9MYjLpfy96XbeHf5L3S9og7vD4ugdrXKTpdljPGABb05r/SsHJ7+dB1fbfqNIZ0aMf6OdlSuZJ2uxpQXFvTmnPYfPcno6TFsTjrGC7e0ZmSPpojYcAbGlCcW9KZI6/alMnp6DOlZuXw0IpLrWl3mdEnGmItgQW8KtXh9Er+fs456NaowY2QXWl5ew+mSjDEXyYLenEZVefubnby5bDuRTWozaXgEQdWrOF2WMeYSWNCbfBnZuTwzdz1frEtiQHgwrw5oT5VKvk6XZYy5RBb0BoCDxzIYPSOW9QmpPNevFQ9de4V1uhrjJSzoDRsTjzJ6egyp6dlMuieCvm0vd7okY0wxsouhy7GZG2YS+lYoPi/5EPpWKDM3zLzgYyzZ9BuDJuUNODr34W4W8sZ4ITujL6dmbphJ1BdRpGenA7Dn6B6ivogCYFj7YefdX1WZ9N0uXluylatCApl8bwT1a/iXaM3GGGfYGX05Nfabsfkhf0p6djpjvxl73n0zc3L5/Wfr+NtXW7ktrCGzo7payBvjxeyMvpzae3TvBS0/JSUtkwdnxBKz5whP39iCx65rZp2uxng5C/pyqnGtxuw5uqfQ5UXZ9ttxRk5bw6G0TN69O5xbwhqUZInGmDLCmm7KqQnXTyDAL+C0ZQF+AUy4fkKh23+79QAD3vuRrBwXcx7sZiFvTAViQV9ODWs/jMm3TaZJrSYIQpNaTZh82+SzOmJVlY++38WoaTE0rVeNRWN6EBYS6FDVxhgn2FSCXiwrx8WLizYya/U+bmp3OX//3VUEVLbWOmO80bmmEvTojF5E+onINhHZKSJ/LGT9fSKSLCJr3Y9RBdaNEJEd7seIi38b5kIcOZHFvVNWMWv1Psb0bsa7d4dbyBtTQZ33f76I+ALvAjcCCcAaEVmkqpvP2PRTVR1zxr51gBeBSECBWPe+R4qlelOonQfTGDltDfuPZvDW4A707xjsdEnGGAd5ckbfGdipqrtUNQuYDdzh4fH7AktV9bA73JcC/S6uVOOJFduTufO9HzmRmcOs0V0t5I0xHgV9MLCvwPME97IzDRSR9SIyV0QaXeC+iEiUiMSISExycrIHZZkzTV+5m/unriE4sCoLH+1ORJPaTpdkjCkDPAn6wu6mObMH9wsgVFXDgGXAtAvYN2+h6mRVjVTVyHr16nlQlmHmTAgNRX18OFK/ITF/fYfeLesx9+GrCakdcP79jTEVgidBnwA0KvA8BEgquIGqpqhqpvvph0CEp/uaizRzJkRFwZ49iCq1k/fzxtJ3mey7nepVrNPVGPM/ngT9GqC5iDQVkcrAEGBRwQ1EpODdN7cDW9zfLwH6iEhtEakN9HEvM5dq7FhIP32sm8qZGfi8cP6xbowxFct5T/1UNUdExpAX0L7AFFXdJCLjgRhVXQQ8LiK3AznAYeA+976HReRl8v5YAIxX1cMl8D4qHN27t9B2Mfaee6wbY0zFYzdMlUOzVu/lmhsjCT5WSKd1kyawe3ep12SMcdYl3zBlyoZcl/Ly4s08P38Di343Bg04o8M1IAAmFD7WjTGm4rKgLyeOZ2QzatoaPv7hV+7vHsroSS8gkyfnncGL5H2dPBmGnX/SEWNMxWKXZ5QD+w6nM3LaGnYln2DCne0Y1qVJ3ophwyzYjTHnZUFfxq3ZfZgHZ8SSk+ti+gOdubpZkNMlGWPKGQv6MmxubAJ/mr+B4NpV+XhEJFfUq+50ScaYcsiCvgxyuZTXlmxj0ne/0L1ZXd67O4JaAX5Ol2WMKacs6MuYE5k5PPXpWr7efIBhXRoz7va2+Plan7kx5uJZ0JchiaknGTUthm2/HWPcbW0YcXWoTdxtjLlkFvRlRPzeI4yeHktmdi5T7utEr5b1nS7JGOMlLOjLgM/XJvLM3PVcXtOfWaO70PyyGk6XZIzxIhb0DnK5lLe+2cHb3+ygc9M6TLongjrVKjtdljHGy1jQO+RkVi5/+Gwd/9mwn0ERIUy4sz2VK1mnqzGm+FnQO+DAsQxGT49hQ+JRxt7cmlE9m1qnqzGmxFjQl7INCUcZNX0NaRk5fDg8khvaXOZ0ScYYL2dBX4r+u2E/T81ZS91qVZj78NW0blDT6ZKMMRWABX0pUFXeXb6TN77eTnjjQD4YHkm9GlWcLssYU0FY0JewjOxc/jhvPQvXJtG/Q0MmDgzD38/X6bKMMRWIBX0JSj6eSdSMGOL3pvJM35Y80utK63Q1xpQ6j67nE5F+IrJNRHaKyB8LWf+0iGwWkfUi8o2INCmwLldE1rofi87c11tt2X+M/u/+yNb9x5l0TziP9m5mIW+MccR5z+hFxBd4F7gRSADWiMgiVd1cYLN4IFJV00XkYeA1YLB73UlV7VDMdZdpSzcf4InZ8dT09+Ozh7rRLriW0yUZYyowT87oOwM7VXWXqmYBs4E7Cm6gqstVNd399GcgpHjLLB9UlQ+++4WoGTE0q1+dz8d0t5A3xjjOk6APBvYVeJ7gXlaUkcB/Czz3F5EYEflZRPoXtZOIRLm3i0lOTvagrLIlK8fFs3PX8+p/t3Jz+wZ8GtWNy2r6O12WMcZ41BlbWMOyFrqhyD1AJHBtgcWNVTVJRK4AvhWRDar6y1kHVJ0MTAaIjIws9Phl1eETWTw0I5bVuw/zxPXNefKG5tYeb4wpMzwJ+gSgUYHnIUDSmRuJyA3AWOBaVc08tVxVk9xfd4lINNAROCvoy6sdB47zwLQ1HDiWydtDO3L7VQ2dLskYY07jSdPNGqC5iDQVkcrAEOC0q2dEpCPwAXC7qh4ssLy2iFRxfx8EdAcKduKWa9HbDjLgvZ/IyHbxaVRXC3ljTJl03jN6Vc0RkTHAEsAXmKKqm0RkPBCjqouA14HqwGfuJou9qno70Br4QERc5P1RmXjG1Trlkqoy9afdvLx4M60ur8lHIyJpGFjV6bKMMaZQolr2msMjIyM1JibG6TIKlZ3r4sVFm/hk1V76tLmMNwd3oFoVu+/MGOMsEYlV1cjC1llCXYDU9CwemRnHT7+k8EivK/lDn5b4+FinqzGmbLOg99Cu5DRGTosh8chJ/vG7qxgQXiFvFTDGlEMW9B74cechHv53LH6+PnwyuguRoXWcLskYYzxmQX8eM1ft4S+fb6JZvep8NCKSRnUCnC7JGGMuiAV9EXJyXbzyny1M/Wk317Wqzz+HdKCGv5/TZRljzAWzoC/EsYxsxnwSz4rtyYzq0ZTnb26Nr3W6GmPKKQv6M+xJOcHIaTHsPnSCiQPaM6RzY6dLMsaYS2JBX8CqXSk89O9YFJgxsgvdrqzrdEnGGHPJLOjd5sTsY+yCDTSuE8DHIzoRGlTN6ZKMMaZYVPigz3Upf/tqK5NX7KJn8yDeuTucWlWt09UY4z0qdNCnZebw5Ox4lm05yIhuTfjzrW2o5OvR7IrGGFNuVNigTziSzqhpMew4mMbLd7RleLdQp0syxpgSUSGDPnbPER6cEUNmjoup93eiZ/N6TpdkjDElpsIF/YL4BJ6bu4GGgf7MjupEs/rVnS7JGGNKVIUJepdL+fvSbby7/Be6XlGH94dFULtaZafLMsaYElchgj49K4enP13HV5t+Y2jnRrx0ezsqV7JOV2NMxeD1Qb//6ElGT49hc9Ix/nxrGx7oHmoTdxtjKhSvDvp1+1IZPT2G9KxcPh7Rid6t6jtdkjHGlDqvDfrF65P4/Zx11KtRhRkju9Dy8hpOl2SMMY7wKOhFpB/wT/ImB/9IVSeesb4KMB2IAFKAwaq6273ueWAkkAs8rqpLiq36AhbGJ/L6km0kpZ6kun8ljmfk0Cm0NpPuiaBu9Sol8ZLGGFMunLdHUkR8gXeBm4A2wFARaXPGZiOBI6raDHgT+Jt73zbAEKAt0A94z328YrUwPpHn528gMfUkChzPyMFXhMGRjSzkjTEVnieXnnQGdqrqLlXNAmYDd5yxzR3ANPf3c4HrJa/H8w5gtqpmquqvwE738YrV60u2cTI797Rluaq8uWxHcb+UMcaUO54EfTCwr8DzBPeyQrdR1RzgKFDXw30BEJEoEYkRkZjk5GTPqndLSj15QcuNMaYi8SToC7sWUT3cxpN98xaqTlbVSFWNrFfvwoYkaBhY9YKWG2NMReJJ0CcAjQo8DwGSitpGRCoBtYDDHu57yZ7p25Kqfqc3/Vf18+WZvi2L+6WMMabc8STo1wDNRaSpiFQmr3N10RnbLAJGuL+/C/hWVdW9fIiIVBGRpkBzYHXxlP4//TsG8+qA9gQHVkWA4MCqvDqgPf07FtpKZIwxFcp5L69U1RwRGQMsIe/yyimquklExgMxqroI+BiYISI7yTuTH+Led5OIzAE2AznAo6qaW+gLXaL+HYMt2I0xphCSd+JdtkRGRmpMTIzTZRhjTLkhIrGqGlnYOhvZyxhjvJwFvTHGeDkLemOM8XIW9MYY4+XKZGesiCQDey5y9yDgUDGWUx7Ye/Z+Fe39gr3nC9VEVQu927RMBv2lEJGYonqevZW9Z+9X0d4v2HsuTtZ0Y4wxXs6C3hhjvJw3Bv1kpwtwgL1n71fR3i/Yey42XtdGb4wx5nTeeEZvjDGmAAt6Y4zxcl4T9CIyRUQOishGp2spDSLSSESWi8gWEdkkIk84XVNJExF/EVktIuvc7/klp2sqLSLiKyLxIrLY6VpKg4jsFpENIrJWRCrECIciEigic0Vkq/v/dbdiO7a3tNGLyDVAGjBdVds5XU9JE5EGQANVjRORGkAs0F9VNztcWolxz0NcTVXTRMQP+AF4QlV/dri0EiciTwORQE1VvdXpekqaiOwGIlW1wtwwJSLTgO9V9SP33B8BqppaHMf2mjN6VV1B3lj4FYKq7lfVOPf3x4EtFDEfr7fQPGnup37uh3ecqZyDiIQAtwAfOV2LKRkiUhO4hry5PVDVrOIKefCioK/IRCQU6AiscraSkuduwlgLHASWqqrXv2fgLeBZwOV0IaVIga9FJFZEopwuphRcASQD/3I30X0kItWK6+AW9OWciFQH5gFPquoxp+spaaqaq6odyJt/uLOIeHUznYjcChxU1Vinayll3VU1HLgJeNTdNOvNKgHhwPuq2hE4AfyxuA5uQV+Oudup5wEzVXW+0/WUJvfH2mign8OllLTuwO3uNuvZwHUi8m9nSyp5qprk/noQWAB0draiEpcAJBT4hDqXvOAvFhb05ZS7Y/JjYIuq/sPpekqDiNQTkUD391WBG4CtzlZVslT1eVUNUdVQ8uZi/lZV73G4rBIlItXcFxjgbr7oA3j11XSq+huwT0RauhddT95c28XivJODlxciMgvoBQSJSALwoqp+7GxVJao7MBzY4G6zBviTqn7pYE0lrQEwTUR8yTtJmaOqFeJywwrmMmBB3rkMlYBPVPUrZ0sqFY8BM91X3OwC7i+uA3vN5ZXGGGMKZ003xhjj5SzojTHGy1nQG2OMl7OgN8YYL2dBb4wxXs6C3hhjvJwFvTHGeLn/B/jG923gwZHAAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "X = [1, 4, 6]\n",
    "Y = [0, 1.386294, 1.791759]\n",
    "\n",
    "# (i) Polinômio de Primeiro Grau\n",
    "y1 = int_lagrange(X[0:2],Y[0:2],2)\n",
    "print(\"(i)\\nA estimativa de ln(2) para o polinômio de primeiro grau é:\", y1)\n",
    "\n",
    "# (i) Polinômio de Segundo Grau\n",
    "y2 = int_lagrange(X,Y,2)\n",
    "print(\"\\n(ii)\\nA estimativa de ln(2) para o polinômio de segundo grau é:\", y2)\n",
    "\n",
    "plt.plot(X,Y, 'o-')\n",
    "plt.plot(2,y1, 'or', 2,y2, 'og')\n",
    "plt.legend(['Spline Linear de y = ln(x)', 'Interpolação de 1°', 'Interpolação de 2°']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 8\n",
    "\n",
    "Faça uma estimativa de $log(10)$ usando interpolação linear\n",
    "\n",
    "(i) entre $log(8) = 0.9030900$ e $log(12) = 1.0791812$\n",
    "\n",
    "(ii) entre $log(9) = 0.9542425$ e $log(11) = 1.0413927$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i)\n",
      "A estimativa de ln(10) é: 0.9911356000000002\n",
      "\n",
      "(ii)\n",
      "A estimativa de ln(10) é: 0.9978176\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "# (i)\n",
    "X = [8, 12]\n",
    "Y = [0.9030900, 1.0791812]\n",
    "\n",
    "y1 = int_lagrange(X,Y,10)\n",
    "print(\"(i)\\nA estimativa de ln(10) é:\", y1)\n",
    "\n",
    "# (i)\n",
    "X = [9, 11]\n",
    "Y = [0.9542425, 1.0413927]\n",
    "\n",
    "y2 = int_lagrange(X,Y,10)\n",
    "print(\"\\n(ii)\\nA estimativa de ln(10) é:\", y2)\n",
    "\n",
    "X = [8, 9, 11, 12]\n",
    "Y = [0.9030900, 0.9542425, 1.0413927, 1.0791812]\n",
    "\n",
    "plt.plot(X,Y, 'o-')\n",
    "plt.plot(10,y1, 'or', 10,y2, 'og')\n",
    "plt.legend(['Spline Linear de y = log(x)', 'Interpolação (i)', 'Interpolação (ii)']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 9\n",
    "\n",
    "Considere os dados e faça o que se pede:\n",
    "\n",
    "x | y\n",
    "-|-\n",
    "1 | 3\n",
    "2 | 6\n",
    "3 | 19\n",
    "5 | 99\n",
    "7 | 291\n",
    "8 | 444\n",
    "\n",
    "(i) Calcule f(4) usando polinomios interpoladores de Lagrange de primeiro a terceiro graus.\n",
    "\n",
    "(ii) Calcule f(4) usando polinomios interpoladores de Newton de primeiro a quarto graus."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i)\n",
      "A estimativa de f(4) de primeiro grau é: 59.0\n",
      "A estimativa de f(4) de segundo grau é: 50.0\n",
      "A estimativa de f(4) de terceiro grau é: 48.0\n",
      "\n",
      "(ii)\n",
      "A estimativa de f(4) de primeiro grau é: 59.0\n",
      "A estimativa de f(4) de segundo grau é: 50.0\n",
      "A estimativa de f(4) de terceiro grau é: 48.0\n",
      "A estimativa de f(4) de quarto grau é: 48.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "X = np.array([1, 2, 3, 5, 7, 8])\n",
    "Y = np.array([3, 6, 19, 99, 291, 444])\n",
    "\n",
    "# (i)\n",
    "y1 = int_lagrange(X[2:4],Y[2:4],4)\n",
    "print(\"(i)\\nA estimativa de f(4) de primeiro grau é:\", y1)\n",
    "y2 = int_lagrange(X[1:4],Y[1:4],4)\n",
    "print(\"A estimativa de f(4) de segundo grau é:\", y2)\n",
    "y3 = int_lagrange(X[:4],Y[:4],4)\n",
    "print(\"A estimativa de f(4) de terceiro grau é:\", y3)\n",
    "\n",
    "# (i)\n",
    "y4 = int_newton(X[2:4],Y[2:4],4)\n",
    "print(\"\\n(ii)\\nA estimativa de f(4) de primeiro grau é:\", y4)\n",
    "y5 = int_newton(X[1:4],Y[1:4],4)\n",
    "print(\"A estimativa de f(4) de segundo grau é:\", y5)\n",
    "y6 = int_newton(X[:4],Y[:4],4)\n",
    "print(\"A estimativa de f(4) de terceiro grau é:\", y6)\n",
    "y7 = int_newton(X[:-1],Y[:-1],4)\n",
    "print(\"A estimativa de f(4) de quarto grau é:\", y7)\n",
    "\n",
    "plt.plot(X,Y, 'o-')\n",
    "plt.plot(4,y1, 'o', 4,y2, 'o', 4,y3, 'o', 4,y4, 'o', 4,y5, 'o', 4,y6, 'o', 4,y7, 'o')\n",
    "plt.legend(['Spline Linear de y = log(x)']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 10\n",
    "\n",
    "Conhecendo a seguinte tabela\n",
    "\n",
    "x | f(x)\n",
    "-|-\n",
    "-1 | 15\n",
    "0 | 8\n",
    "3 | -1\n",
    "\n",
    "determine:\n",
    "\n",
    "(i) O polinômio de interpolação para a função definida por este conjunto de pares de pontos. Ou seja, determine a solução do sistema\n",
    "\n",
    "$$ Xa = f $$\n",
    "\n",
    "para o vetor de $a$ dos coeficientes de $P_2(x)$.\n",
    "\n",
    "(ii) O polinomio de interpolação na forma de Newton."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "O polinômio é: P(x) = 1.0*x**2 - 6.0*x + 8.0\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "X = np.array([-1, 0, 3])\n",
    "Y = np.array([15, 8, -1])\n",
    "\n",
    "n = len(X)\n",
    "x = sy.symbols(\"x\")\n",
    "    \n",
    "dx = []\n",
    "f = []\n",
    "aux = np.diff(Y)\n",
    "\n",
    "for i in range(1,n):\n",
    "    dx.append(X[i:] - X[:-i])\n",
    "    f.append(aux/dx[i-1])\n",
    "    aux = np.diff(f[i-1])\n",
    "    \n",
    "b = []\n",
    "    \n",
    "for i in f:\n",
    "    b.append(i[0])\n",
    "    \n",
    "f = Y[0]\n",
    "aux = 1\n",
    "\n",
    "for i in range(n-1):\n",
    "    aux *= x - X[i]\n",
    "    f += b[i]*aux\n",
    "\n",
    "print(\"O polinômio é: P(x) =\", sy.simplify(f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 11\n",
    "\n",
    "Usando os dados da tabela do exercício anterior:\n",
    "\n",
    "(i) Determine $P_2(x)$ pela forma de Lagrange.\n",
    "\n",
    "(ii) Calcule uma aproximação para $f(1)$ usando o item (i)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i)\n",
      "P(x) = x**2 - 6*x + 8\n",
      "\n",
      "(ii)\n",
      "A aproximação para f(1) é: 3.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "X = [-1, 0, 3]\n",
    "Y = [15, 8, -1]\n",
    "\n",
    "# (i)\n",
    "n = len(X)\n",
    "x = sy.symbols(\"x\")\n",
    "f = 0\n",
    "    \n",
    "for i in range(n):\n",
    "    L = 1\n",
    "    for j in range(n):\n",
    "        if (i != j):\n",
    "            L *= (x - X[j])/(X[i] - X[j])\n",
    "    f += L*Y[i]\n",
    "\n",
    "print(\"(i)\\nP(x) =\", sy.simplify(f))\n",
    "\n",
    "# (ii)\n",
    "y = f.subs(x,1)\n",
    "\n",
    "print(\"\\n(ii)\\nA aproximação para f(1) é:\", float(y))\n",
    "\n",
    "plt.plot(X,Y, 'o-')\n",
    "plt.plot(1,y, 'o')\n",
    "plt.legend(['Spline Linear de y = f(x)', 'Valor Interpolado']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 12\n",
    "\n",
    "Dada a tabela:\n",
    "\n",
    "$x$ | $x e^{3x}$\n",
    "-|-\n",
    "0 | 1\n",
    "0.1 | 1.3499\n",
    "0.2 | 1.8221\n",
    "0.3 | 2.4596\n",
    "0.4 | 3.3201\n",
    "0.5 | 4.4817\n",
    "\n",
    "calcule $f(0.25)$, onde $f(x) = x e^{3x}$ usando polinômio de interpolação do 2° grau:\n",
    "\n",
    "(i) Usando $x_0 = 0.2, x_1 = 0.3, x_2 = 0.4$;\n",
    "\n",
    "(ii) Usando $x_0 = 0.1, x_1 = 0.2, x_2 = 0.3$;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i)\n",
      "A estimativa de f(0.25) é: 2.112975\n",
      "\n",
      "(ii)\n",
      "A estimativa de f(0.25) é: 2.1201875\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "# (i)\n",
    "X = [0.2, 0.3, 0.4]\n",
    "Y = [1.8221, 2.4596, 3.3201]\n",
    "\n",
    "y1 = int_lagrange(X,Y,0.25)\n",
    "print(\"(i)\\nA estimativa de f(0.25) é:\", y1)\n",
    "\n",
    "# (i)\n",
    "X = [0.1, 0.2, 0.3]\n",
    "Y = [1.3499, 1.8221, 2.4596]\n",
    "\n",
    "y2 = int_lagrange(X,Y,0.25)\n",
    "print(\"\\n(ii)\\nA estimativa de f(0.25) é:\", y2)\n",
    "\n",
    "X = [0, 0.1, 0.2, 0.3, 0.4, 0.5]\n",
    "Y = [1, 1.3499, 1.8221, 2.4596, 3.3201, 4.4817]\n",
    "\n",
    "plt.plot(X,Y, 'o-')\n",
    "plt.plot(0.25,y1, 'o', 0.25,y2, 'o')\n",
    "plt.legend(['Spline Linear de y = f(x)', 'Interpolação (i)', 'Interpolação (ii)']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Questão 13\n",
    "\n",
    "Para a seguinte função tabelada \n",
    "\n",
    "$x$ | $x e^{3x}$\n",
    "-|-\n",
    "0 | 1\n",
    "0.1 | 1.3499\n",
    "0.2 | 1.8221\n",
    "0.3 | 2.4596\n",
    "0.4 | 3.3201\n",
    "0.5 | 4.4817\n",
    "\n",
    "construir a tabela de diferenças divididas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " As 1 ªs diferenças divididas são:\n",
      " [ 3.499  4.722  6.375  8.605 11.616]\n",
      "\n",
      " As 2 ªs diferenças divididas são:\n",
      " [ 6.115  8.265 11.15  15.055]\n",
      "\n",
      " As 3 ªs diferenças divididas são:\n",
      " [ 7.16666667  9.61666667 13.01666667]\n",
      "\n",
      " As 4 ªs diferenças divididas são:\n",
      " [6.125 8.5  ]\n",
      "\n",
      " As 5 ªs diferenças divididas são:\n",
      " [4.75]\n"
     ]
    }
   ],
   "source": [
    "# Solução\n",
    "\n",
    "X = np.array([0, 0.1, 0.2, 0.3, 0.4, 0.5])\n",
    "Y = np.array([1, 1.3499, 1.8221, 2.4596, 3.3201, 4.4817])\n",
    "\n",
    "n = len(X)\n",
    "x = sy.symbols(\"x\")\n",
    "    \n",
    "dx = []\n",
    "f = []\n",
    "aux = np.diff(Y)\n",
    "\n",
    "for i in range(1,n):\n",
    "    dx.append(X[i:] - X[:-i])\n",
    "    f.append(aux/dx[i-1])\n",
    "    aux = np.diff(f[i-1])\n",
    "\n",
    "for i in range(len(f)):\n",
    "    print(\"\\n As\", i+1, \"ªs diferenças divididas são:\\n\", f[i])"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
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 },
 "nbformat": 4,
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