Lista8.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ae3119ea",
   "metadata": {},
   "source": [
    "# Lista 8 de Cálculo Numérico"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 359,
   "id": "6b08a5de",
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ff0f7d8",
   "metadata": {},
   "source": [
    "## Questão 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9411c6c6",
   "metadata": {},
   "source": [
    "### a. Compressão"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1194b91d",
   "metadata": {},
   "source": [
    "### Compressão de Dados em Matrizes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdeddfa3",
   "metadata": {},
   "source": [
    "A ideia da compressão de dados é encontrarmos vetores base dentro da matriz tal que forneçam uma base para a composição de todos os dados. Porém, na vida real precisamos encontrar aproximações, visto que o uso de números reais acabam dificultando uma base L.I de poucos vetores representarem muitos dados.\n",
    "\n",
    "Na ideia simples, transformamos a matriz em uma multiplicação de duas matrizes, tal que a primeira seja os vetores base e a segunda os coeficientes das combinações lineares.\n",
    "\n",
    "Com a preocupação de se adequar aos números reais, começamos a analisar isso pelo erro da matriz original com a nova multiplicação das duas matrizes. Para quantificar o erro, usamos a ideia de norma de matriz, que é dada pela soma das normas de cada vetor (colunas) da matriz.\n",
    "\n",
    "Então, temos que dado a matriz A e o vetor b, precisamos determinar c tal que \n",
    "\n",
    "$$min ||A - [b][c^T]|| \\longrightarrow min ||[(a_1)(a_2)...(a_n)] - [c_1(b) c_2(b) ... c_n(b)]|| \\longrightarrow ||(a_1 - c_1b)(a_2-c_2b)...(a_n-c_nb)||$$\n",
    "\n",
    "Visto que $e_n = a_n - c_nb \\space \\bot \\space c_nb$, pelo produto interno temos que $e_n^Tc_nb = 0 \\longrightarrow (a_n - c_nb)^Tc_nb = 0$. Aplicando a transposta tal que $(M.Q)^T = Q^T.M^T$ e $(M - Q)^T = M^T - Q^T$ obtemos:\n",
    "\n",
    "$$(a_n - c_nb)^Tc_nb = 0 \\longrightarrow (a_n^T - (c_nb)^T)c_nb = 0 \\longrightarrow a_n^Tc_nb - c_n^2b^Tb = 0 \\longrightarrow a_n^Tc_nb = c_n^2b^Tb $$\n",
    "\n",
    "Desconsiderando o caso em que $c_n = 0$ pois é um caso muito trivial, podemos dividir a equação em ambos os lados por $c_n$, resultando em:\n",
    "\n",
    "$$a_n^Tb = c_nb^Tb \\longrightarrow c_n = \\frac{a_n^Tb}{b^Tb}$$\n",
    "\n",
    "A partir da análise feita acima podemos escrever um código em Julia para achar o vetor c:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 360,
   "id": "b1768801",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "melhores_coords (generic function with 1 method)"
      ]
     },
     "execution_count": 360,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function melhores_coords(A,b) #retorna c tal que norm(A-b*c') é mínima\n",
    "    m, n = size(A)\n",
    "    A = 1.0 * A # multiplica-se por 1.0 para evitar erros de operações matriciais entre inteiros e float's.\n",
    "    c = zeros(n)    \n",
    "    c = A'*(b/(b'*b)) # Operação para achar o vetor c que minimiza a distância perpendicular à reta.\n",
    "    return c\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 361,
   "id": "96330ca7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " 1.0001  10.001  3.0\n",
       " 2.01    20.3    6.0\n",
       " 3.003   30.0    9.0"
      ]
     },
     "execution_count": 361,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=[1.0001 10.001 3;2.01 20.3 6;3.003 30 9]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 362,
   "id": "a7b0e9e7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Int64,1}:\n",
       " 1\n",
       " 2\n",
       " 3"
      ]
     },
     "execution_count": 362,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b=[1;2;3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 363,
   "id": "be5224b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       "  1.0020785714285714\n",
       " 10.04292857142857\n",
       "  3.0"
      ]
     },
     "execution_count": 363,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c=melhores_coords(A,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 364,
   "id": "bec671c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2534747564213406"
      ]
     },
     "execution_count": 364,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(A-b*c') # O erro ainda é bem grande"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7cf5da20",
   "metadata": {},
   "source": [
    "Podemos aplicar o método de melhores coordenadas mais vezes, até convergir e encontrarmos a melhor coordenada possível para a base dada. Para aplicar mais de uma vez, alteramos entre bases que são equivalentes como o algoritmo abaixo denota:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 365,
   "id": "614b3e3a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "acha_melhor_minimizacao (generic function with 1 method)"
      ]
     },
     "execution_count": 365,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function acha_melhor_minimizacao(A)\n",
    "    # Pega o tamanho da matriz A\n",
    "    n,m = size(A)\n",
    "    \n",
    "    # Inicializa os vetores que serão usados: b para as bases e para os coeficientes.\n",
    "    bv = randn(n)\n",
    "    cv = randn(m)\n",
    "    \n",
    "    bw = zeros(m)\n",
    "    cw = zeros(n)\n",
    "    \n",
    "    # Aplicamos o método de melhores_coords 100 vezes para maior garantia que o método convergiu\n",
    "    for i in 1:50\n",
    "        cv=melhores_coords(A, bv) \n",
    "        bw=cv\n",
    "        cw=melhores_coords(A', bw)\n",
    "        bv=cw\n",
    "    end\n",
    "    \n",
    "    return bv,cv\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34756adf",
   "metadata": {},
   "source": [
    "Como o método mais acima é usado somente com uma base e queremos usar um número qualquer, precisamos generalizar. Chamamos a generalização de método da compressão:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 366,
   "id": "9fd45a09",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "compressao (generic function with 1 method)"
      ]
     },
     "execution_count": 366,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function compressao(A,nível)\n",
    "    # Pega tamanho e inicializa os vetores\n",
    "    m,n=size(A)\n",
    "    B=zeros(m,nível)\n",
    "    C=zeros(n,nível)\n",
    "    \n",
    "    for i=1:nível\n",
    "        B[:,i], C[:,i] = acha_melhor_minimizacao(A) # aplica melhor minimizacao para cada base e coeficiente\n",
    "        A = A - B[:,i]*C[:,i]' # Vai reduzindo o erro\n",
    "    end\n",
    "    return B,C\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 367,
   "id": "2dc9f581",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×4 Array{Float64,2}:\n",
       "  1.2263    -0.323799    1.56699    -0.838926\n",
       " -0.629255   0.0554864  -1.89732    -0.283598\n",
       " -1.12734   -0.707122    0.0571774  -1.41415"
      ]
     },
     "execution_count": 367,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = randn(3,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 368,
   "id": "31e84207",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([0.16804435737749704 -0.493696198546017 -2.6354277246602993; -0.15702274014800294 -0.18225823672435376 -3.0238921872263136; -0.030617646701731713 -1.7749297719966164 1.043551095101874], [6.304694061484572 0.4406413582746449 -0.14585037574261348; -0.7704408542373948 0.4098923703218821 -0.003047714518885344; 10.39325635934108 -0.15443487141937703 0.09705650045563972; -0.9872756788694469 0.8682781675738647 0.09271896239604066])"
      ]
     },
     "execution_count": 368,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B,C = compressao(A,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 369,
   "id": "86aa3330",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0570533288019466e-16"
      ]
     },
     "execution_count": 369,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(A-B*C')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 370,
   "id": "56818e6a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1: 16.857043069575244\n",
      "2: 15.318083524544253\n",
      "3: 13.9022051519667\n",
      "4: 12.572813739146865\n",
      "5: 11.292520252372325\n",
      "6: 10.150171945878174\n",
      "7: 8.961901629038135\n",
      "8: 7.908501058963985\n",
      "9: 6.863215550994088\n",
      "10: 5.9666529033628315\n",
      "11: 5.120219928056899\n",
      "12: 4.200002173831524\n",
      "13: 3.385085377710945\n",
      "14: 2.6660285257852956\n",
      "15: 1.9201267775308484\n",
      "16: 1.310381588885014\n",
      "17: 0.92204800303176\n",
      "18: 0.47281856575577685\n",
      "19: 0.16911633984131866\n",
      "20: 4.362764101475657e-15\n"
     ]
    }
   ],
   "source": [
    "# Chance pequena de existirem colunas Linearmente Dependentes\n",
    "N = 20\n",
    "A = randn(N, N)\n",
    "for i in 1:N\n",
    "    B,C = compressao(A, i)\n",
    "    print(i, \": \", norm(A-B*C'), \"\\n\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 371,
   "id": "33f14cd0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.1386160011060724e-14\n",
      "0.0\n",
      "NaN\n",
      "NaN\n",
      "NaN\n",
      "NaN\n",
      "NaN\n",
      "NaN\n",
      "NaN\n",
      "NaN\n"
     ]
    }
   ],
   "source": [
    "N = 20\n",
    "A = zeros(N, N)\n",
    "for i in 1:N\n",
    "    A[:, i] .= i\n",
    "end\n",
    "for i in 1:10\n",
    "    B,C = compressao(A, i)\n",
    "    print(norm(A-B*C'), \"\\n\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ff8dd1e",
   "metadata": {},
   "source": [
    "### b. Gauss-Jacobi"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3f2d4c9",
   "metadata": {},
   "source": [
    "O método utilizado será o do ponto fixo, em que iniciamos com o sistema linear:\n",
    "\n",
    "$$A.x=b$$\n",
    "\n",
    "Chamaremos $A=M-K$ para um $M$ facilmente invertível e um $K$ resto da matriz.\n",
    "\n",
    "$$M.x-K.x=b \\\\\n",
    "M.x=b+K.x \\\\\n",
    "x=M^{-1}.b+M^{-1}.K.x$$\n",
    "\n",
    "Chamaremos $M^{-1}.b$ de $c$ e o $M^{-1}.K$ de $R$:\n",
    "\n",
    "$$x=R.x+c$$\n",
    "\n",
    "Nosso método numérico a ser implementado em cima dessa equação é dar um chute inicial e ir substituindo nela:\n",
    "\n",
    "$$x^{(0)}=chute_{inicial}$$\n",
    "\n",
    "$$x^{(k+1)}=R.x^{(k)}+c$$\n",
    "\n",
    "No caso específico do **Gauss-Jacobi**, a **matriz $M$ facilmente inversível é uma diagonal.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 372,
   "id": "9b9f6d15",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "gauss_jacobi (generic function with 1 method)"
      ]
     },
     "execution_count": 372,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function gauss_jacobi(A,b,iteracoes)\n",
    "    M = Diagonal(A)\n",
    "    K = M-A\n",
    "    R = inv(M)*K\n",
    "    c = inv(M)*b\n",
    "    \n",
    "    lin = length(b)\n",
    "    x = zeros(lin,1)\n",
    "    for i = 1:iteracoes\n",
    "        x = R*x + c\n",
    "    end\n",
    "    \n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 373,
   "id": "21c4668e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Norma: 8.881784197001252e-16\n"
     ]
    }
   ],
   "source": [
    "A = [4 2 1 ; 1 5 1 ; 3 4 10]\n",
    "b = [7 ; 7 ; 17]\n",
    "\n",
    "x = gauss_jacobi(A,b,300)\n",
    "println(\"Norma: \", norm(A*x-b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 374,
   "id": "d8177cb6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Norma: 1.1102230246251565e-16\n"
     ]
    }
   ],
   "source": [
    "A = [5 2 1 -1 ; 1 3 -1 1 ; 2 0 4 1 ; 1 1 1 8]\n",
    "b = [1 ; -1 ; 1 ; -3]\n",
    "\n",
    "x = gauss_jacobi(A,b,300)\n",
    "println(\"Norma: \", norm(A*x-b))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66c9bfba",
   "metadata": {},
   "source": [
    "### c. Gauss-Seidel"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf383bba",
   "metadata": {},
   "source": [
    "O método de Gauss-Seidel é semelhante ao anterior, a diferença está no M, que ao invés de utilizar a matriz diagonal que é fácil de inverter, utiliza a triangular superior, que também possui uma inversão fácil."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 375,
   "id": "b0f4e5da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "gauss_seidel (generic function with 1 method)"
      ]
     },
     "execution_count": 375,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function gauss_seidel(A,b,iteracoes)\n",
    "    M = UpperTriangular(A)\n",
    "    K = M-A\n",
    "    R = inv(M)*K\n",
    "    c = inv(M)*b\n",
    "    \n",
    "    lin = length(b)\n",
    "    x = zeros(lin,1)\n",
    "    for i = 1:iteracoes\n",
    "        x = R*x + c\n",
    "    end\n",
    "    \n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 376,
   "id": "2e7a67d4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Norma: 8.881784197001252e-16\n"
     ]
    }
   ],
   "source": [
    "A = [4 2 1 ; 1 5 1 ; 3 4 10]\n",
    "b = [7 ; 7 ; 17]\n",
    "\n",
    "x = gauss_seidel(A,b,300)\n",
    "println(\"Norma: \", norm(A*x-b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 377,
   "id": "b4a361aa",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Norma: 2.220446049250313e-16\n"
     ]
    }
   ],
   "source": [
    "A = [5 2 1 -1 ; 1 3 -1 1 ; 2 0 4 1 ; 1 1 1 8]\n",
    "b = [1 ; -1 ; 1 ; -3]\n",
    "\n",
    "x = gauss_seidel(A,b,300)\n",
    "println(\"Norma: \", norm(A*x-b))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa48e799",
   "metadata": {},
   "source": [
    "## Questão 2."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8bf39fca",
   "metadata": {},
   "source": [
    "Faça o gráfico do sistema linear\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\begin{cases}\n",
    "      2x_1 + x_2 = 2\\\\\n",
    "      -x_1 + 2x_2 = 2\n",
    "    \\end{cases}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "e resolva-os pelo método de Gauss-Jacobi e Gauss-Seidel com n passos\n",
    "tal que a **tolerância**=$||x^{(n)} − x^{(n-1)}||$ seja menor que $10^{-3}$. Marque as iterações no gráfico.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c5ae686",
   "metadata": {},
   "source": [
    "Temos o sistema de equações:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\begin{cases}\n",
    "      2x_1 + x_2 = 2\\\\\n",
    "      -x_1 + 2x_2 = 2\n",
    "    \\end{cases}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "que pode ser simplificado deixando x_1 em evidência:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\begin{cases}\n",
    "      x_1 = 1 - \\frac{x_2}{2}\\\\\n",
    "      x_1 = 2x_2 - 2\n",
    "    \\end{cases}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "E então, mudando apenas os nomes das variáveis para ficar mais explícito $(x_1 = y$ e $x_2 = x)$:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\begin{cases}\n",
    "      y = 1 - \\frac{x}{2}\\\\\n",
    "      y = 2x - 2\n",
    "    \\end{cases}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Temos duas retas e queremos encontrar para qual valor de x e y elas se interceptam. Fazendo o gráfico do sistema podemos observar melhor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 378,
   "id": "296c8fdd",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 378,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = 1 - x/2 # primeira equação\n",
    "g(x) = 2x - 2 # segunda equação\n",
    "\n",
    "P = plot([f,g], 1, 1.5) # intervalo arbitrário que contenha o ponto de interseção"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22bc4c95",
   "metadata": {},
   "source": [
    "Para resolver o sistema pelos métodos de Gauss-Jacobi e Gauss-Seidel precisamos montar a equação com matrizes $Ax=b$. Logo, a partir do sistema linear\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\begin{cases}\n",
    "      2x_1 + x_2 = 2\\\\\n",
    "      -x_1 + 2x_2 = 2\n",
    "    \\end{cases}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Temos:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} \n",
    "    2 & 1 \\\\\n",
    "    -1 & 2 \\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} \n",
    "    x_1 \\\\\n",
    "    x_2 \\\\\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix} \n",
    "    2 \\\\\n",
    "    2 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Ou seja,\n",
    "\n",
    "$$A = \\begin{bmatrix} \n",
    "    2 & 1 \\\\\n",
    "    -1 & 2 \\\\\n",
    "\\end{bmatrix}\n",
    "\\space e \\space b = \\begin{bmatrix} \n",
    "    2 \\\\\n",
    "    2 \\\\\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Agora podemos aplicar os dois métodos, pois já temos a Matriz A e o vetor B, a fim de encontrar uma aproximação do vetor x:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 379,
   "id": "a41a1f01",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0; 1.0]\n",
      "[0.5; 1.5]\n",
      "[0.25; 1.25]\n",
      "[0.375; 1.125]\n",
      "[0.4375; 1.1875]\n",
      "[0.40625; 1.21875]\n",
      "[0.390625; 1.203125]\n",
      "[0.3984375; 1.1953125]\n",
      "[0.40234375; 1.19921875]\n",
      "[0.400390625; 1.201171875]\n",
      "[0.3994140625; 1.2001953125]\n",
      "[0.39990234375; 1.19970703125]\n",
      "\n",
      "Iterações: 13\n",
      "Tolerância do Sistema: 0.0006905339660024879\n",
      "Tolerância do passo: 0.0006905339660024879"
     ]
    }
   ],
   "source": [
    "\"\"\" Aplicando o método de Gauss-Jacobi primeiro\"\"\"\n",
    "A = [2 1; -1 2]\n",
    "b = [2; 2]\n",
    "\n",
    "# Para printar as iterações no gráfico\n",
    "array_x = []\n",
    "array_y = []\n",
    "\n",
    "iterador = 1 # número de passos até ecnontrar erro menor que 10^(-3)\n",
    "x_anterior = randn(2) # chute inicial qualquer do vetor x_0\n",
    "x_atual = randn(2)  # chute inicial qualquer do vetor x_1\n",
    "while (norm(x_atual - x_anterior) >= 1e-3) # enquanto || x^(n) - x^(n-1) || menor que 10^(-3)\n",
    "    x_anterior = x_atual # atualiza x^(n-1)\n",
    "    x_atual = gauss_jacobi(A, b, iterador) # atualiza x^n\n",
    "    push!(array_x, x_atual[2]) # array_x armazena o valor de x\n",
    "    push!(array_y, x_atual[1]) # array_y armazena o valor de y\n",
    "    print(x_atual, \"\\n\") # printa x^n\n",
    "    iterador += 1 # número de passos aumenta em 1\n",
    "end\n",
    "print(\"\\n\", \"Iterações: \", iterador, \"\\n\") # número de passos total\n",
    "print(\"Tolerância do Sistema: \", norm(A*x_atual - b), \"\\n\") # tolerância de acordo com o sistema\n",
    "print(\"Tolerância do passo: \", norm(x_atual - x_anterior)) # tolerância de acordo com o passo seguinte e o passo anterior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 380,
   "id": "b7925f81",
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     "execution_count": 380,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter!(P, array_x, array_y, alpha=0.5) # Faz o Plot com os pontos das iterações"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 381,
   "id": "16f32b15",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.5; 1.0]\n",
      "[0.375; 1.25]\n",
      "[0.40625; 1.1875]\n",
      "[0.3984375; 1.203125]\n",
      "[0.400390625; 1.19921875]\n",
      "[0.39990234375; 1.2001953125]\n",
      "[0.4000244140625; 1.199951171875]\n",
      "\n",
      "Iterações: 8\n",
      "Tolerância do Sistema: 0.0001220703125\n",
      "Tolerância do passo: 0.0002729575167846423"
     ]
    }
   ],
   "source": [
    "\"\"\" Aplicando o método de Gauss-Seidel \"\"\"\n",
    "\n",
    "A = [2 1; -1 2]\n",
    "b = [2; 2]\n",
    "\n",
    "# Para printar as iterações no gráfico\n",
    "array_x = []\n",
    "array_y = []\n",
    "\n",
    "iterador = 1 # número de passos até ecnontrar erro menor que 10^(-3)\n",
    "x_anterior = randn(2) # chute inicial qualquer do vetor x_0\n",
    "x_atual = randn(2)  # chute inicial qualquer do vetor x_1\n",
    "\n",
    "while (norm(x_atual - x_anterior) >= 1e-3) # enquanto || x^(n) - x^(n-1) || menor que 10^(-3)\n",
    "    x_anterior = x_atual # atualiza x^(n-1)\n",
    "    x_atual = gauss_seidel(A, b, iterador) # atualiza x^n\n",
    "    push!(array_x, x_atual[2]) # array_x armazena o valor de x\n",
    "    push!(array_y, x_atual[1]) # array_y armazena o valor de y\n",
    "    print(x_atual, \"\\n\") # printa x^n\n",
    "    iterador += 1 # número de passos aumenta em 1\n",
    "end\n",
    "\n",
    "print(\"\\nIterações: \", iterador, \"\\n\") # número de passos total\n",
    "print(\"Tolerância do Sistema: \", norm(A*x_atual - b), \"\\n\") # tolerância de acordo com o sistema\n",
    "print(\"Tolerância do passo: \", norm(x_atual - x_anterior)) # tolerância de acordo com o passo seguinte e o passo anterior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 382,
   "id": "4de7b99a",
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      ]
     },
     "execution_count": 382,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = 1-x/2\n",
    "g(x) = 2x - 2\n",
    "\n",
    "G = plot([f,g], 1, 1.5)\n",
    "\n",
    "scatter!(G, array_x, array_y, alpha=0.5) # Faz o Plot com os pontos das iterações"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09dbc1a0",
   "metadata": {},
   "source": [
    "## Questão 3."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90f02057",
   "metadata": {},
   "source": [
    "**(A)** Podemos modelar o problema levando em consideração as quatro direções e suas respectivas temperaturas, montando o seguinte sistema"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e400959",
   "metadata": {},
   "source": [
    "\\begin{align*}\n",
    "x_1 &= \\frac{1}{4} (5 + 15 + x_2 + x_3) &\\implies 4x_1 - x_2 - x_3 &= 20\\\\\n",
    "x_2 &= \\frac{1}{4} (x_1 + 15 + 35 + x_4) &\\implies 4x_2 - x_1 - x_4 &= 50\\\\\n",
    "x_3 &= \\frac{1}{4} (5 + x_1 + x_4 + 10) &\\implies 4x_3 - x_1 - x_4 &= 15\\\\\n",
    "x_4 &= \\frac{1}{4} (x_3 + x_2 + 35 + 10) &\\implies 4x_4 - x_2 - x_3 &= 45\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebe721b6",
   "metadata": {},
   "source": [
    "Assim chegando no seguinte sistema linear"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "283d3dec",
   "metadata": {},
   "source": [
    "$$\n",
    "  \\begin{bmatrix}\n",
    "        4 & -1 & -1 & 0\\\\\n",
    "        -1 & 4 & 0 & -1\\\\\n",
    "        -1 & 0 & 4 & -1\\\\\n",
    "        0 & -1 & -1 & 4\\\\\n",
    "   \\end{bmatrix} \n",
    "   \\begin{bmatrix}\n",
    "      x_1\\\\\n",
    "      x_2\\\\\n",
    "      x_3\\\\\n",
    "      x_4\n",
    "   \\end{bmatrix}\n",
    "   = \\begin{bmatrix}\n",
    "       20\\\\\n",
    "       50\\\\\n",
    "       15\\\\\n",
    "       45\n",
    "   \\end{bmatrix}\n",
    "    $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "123dbbad",
   "metadata": {},
   "source": [
    "**(B)** Vendo linha a linha temos:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ead6c2d8",
   "metadata": {},
   "source": [
    "\\begin{align*}\n",
    "|A_{11}| = 4 \\geq \\sum_{i\\ne j} |A_{ij}| = 2\\\\\n",
    "|A_{22}| = 4 \\geq \\sum_{i\\ne j} |A_{ij}| = 2\\\\\n",
    "|A_{33}| = 4 \\geq \\sum_{i\\ne j} |A_{ij}| = 2\\\\\n",
    "|A_{44}| = 4 \\geq \\sum_{i\\ne j} |A_{ij}| = 2\\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57c1c459",
   "metadata": {},
   "source": [
    "Como a desigualdade é satisfeita para todas as linhas, temos que ela é\n",
    "diagonal dominante, consequentemente, temos que ambos os métodos de Gauss-Jacobi\n",
    "e Gauss-Seidel convergem."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92891ca8",
   "metadata": {},
   "source": [
    "**(C)** Montando o sistema em Julia temos, podemos aproximar a temperatura nos quatro vértices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 383,
   "id": "fddb9db4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×1 Array{Float64,2}:\n",
       " 13.124534606933594\n",
       " 20.624534606933594\n",
       " 11.874534606933594\n",
       " 19.374534606933594"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "4-element Array{Float64,1}:\n",
       " 13.124849319458008\n",
       " 20.624698638916016\n",
       " 11.874698638916016\n",
       " 19.37439727783203"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [4 -1 -1 0; -1 4 0 -1; -1 0 4 -1; 0 -1 -1 4] * 1.0\n",
    "b = [20; 50;  15; 45] * 1.0\n",
    "\n",
    "function gauss_seidel(A, b, EPS)\n",
    "    U = UpperTriangular(A)\n",
    "    L = U - A\n",
    "    \n",
    "    Uinv = inv(U)\n",
    "    \n",
    "    T = Uinv * L\n",
    "    C = Uinv * b\n",
    "\n",
    "    basis = ones(length(b))\n",
    "\n",
    "    error = 1.0\n",
    "\n",
    "    iter = 1\n",
    "\n",
    "    while true\n",
    "        basis = T * basis + C\n",
    "        error = norm(T * basis + C - basis)\n",
    "        if (error < EPS)\n",
    "            break\n",
    "        end\n",
    "        iter += 1\n",
    "    end\n",
    "\n",
    "    return basis\n",
    "end\n",
    "\n",
    "function jacobi(A, b, EPS) \n",
    "    D = Diagonal(A)\n",
    "    K = D - A\n",
    "    iD = inv(D)\n",
    "\n",
    "    T = iD * K\n",
    "    C = iD * b\n",
    "\n",
    "    basis = ones(size(A)[1], 1)\n",
    "    iter = 1\n",
    "    error = 1.0\n",
    "\n",
    "    while true\n",
    "        error = norm(T * basis + C - basis)\n",
    "        basis = T * basis + C\n",
    "        if error < 10^(-3)\n",
    "            break\n",
    "        end\n",
    "\n",
    "        iter += 1\n",
    "    end\n",
    "    \n",
    "    return basis\n",
    "end\n",
    "\n",
    "display(jacobi(A, b, 10^(-3)))\n",
    "display(gauss_seidel(A, b, 10^(-3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ab47260",
   "metadata": {},
   "source": [
    "**(D)** Para este item vamos generalizar o método acima para que seja viável construir\n",
    "sistemas maiores de mesmo formato (temperatura somente nos quatro lados do grid), podemos alterar o tamanho do grid e "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 384,
   "id": "16f3e097",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "25×1 Array{Float64,2}:\n",
       " 1.8814487552\n",
       " 0.9136544768\n",
       " 0.8744296448000001\n",
       " 0.9049166848000001\n",
       " 1.6724022272\n",
       " 1.1223789568\n",
       " 0.08519946240000001\n",
       " 0.0417562624\n",
       " 0.0757750784\n",
       " 0.9049310208000001\n",
       " 1.0925677568\n",
       " 0.0516097024\n",
       " 0.0078792704\n",
       " 0.042146918400000004\n",
       " 0.8747952128\n",
       " 1.1398545408\n",
       " 0.1040482304\n",
       " 0.0606818304\n",
       " 0.0946238464\n",
       " 0.9224066048\n",
       " 2.2995418112\n",
       " 1.3485503488\n",
       " 1.3099747328\n",
       " 1.3398125567999999\n",
       " 2.0904952832"
      ]
     },
     "execution_count": 384,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# dimensão do grid (Testado com matrizes quadradas)\n",
    "function constroi_sistema(dim) # dim = tamanho do grid (dim * dim)\n",
    "    # quantidade de variaveis no sistema\n",
    "    num_var = dim * dim\n",
    "\n",
    "    A = zeros((num_var, num_var))\n",
    "    b = zeros(num_var)\n",
    "\n",
    "    # arrays direção para facilitar na iteração -> [esq, cima, direita, baixo]\n",
    "    dx = [-1 0 1 0]\n",
    "    dy = [0 1 0 -1]\n",
    "\n",
    "    # temperatura em cada lado do grid -> [esq, cima, dir, baixo]\n",
    "    temperature = [25 20 20 30]\n",
    "\n",
    "    for i in 1 : dim\n",
    "        # temperatura de cima\n",
    "        b[i] += temperature[2]\n",
    "\n",
    "        # temperatura da dir\n",
    "        b[i * dim] += temperature[3]\n",
    "\n",
    "        # temperatura da esq\n",
    "        b[(i - 1) * dim + 1] += temperature[1]\n",
    "\n",
    "        # temperatura de baixo\n",
    "        b[(dim - 1) * dim + i] += temperature[4]\n",
    "    end\n",
    "\n",
    "    # checa se o vértice está dentro do grid\n",
    "    function check(i, j, id)\n",
    "        return (i >= 1 && j >= 1 && i <= dim && j <= dim && id >= 1 && id <= num_var)\n",
    "    end\n",
    "\n",
    "    # constroi a nossa matriz do sistema levando em consideração\n",
    "    # as direções possíveis\n",
    "    for i in 1 : dim\n",
    "        for j in 1 : dim\n",
    "            new_var = (i - 1) * dim + j\n",
    "            A[new_var, new_var] = num_var\n",
    "\n",
    "            for k in 1 : 4\n",
    "                new_i = i + dx[k]\n",
    "                new_j = j + dy[k]\n",
    "\n",
    "                loc = (new_i - 1) * dim + new_j\n",
    "\n",
    "                if check(new_i, new_j, loc)\n",
    "                    A[new_var, loc] = -1\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return A, b\n",
    "end\n",
    "\n",
    "A, b = constroi_sistema(5)\n",
    "\n",
    "temperatura_estimada_gs = gauss_seidel(A, b, 10^(-3))\n",
    "temperatura_estimada_jacobi = jacobi(A, b, 10^(-3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21e1615d",
   "metadata": {},
   "source": [
    "Temos então que a temperatura nos vértices do interior do grid são aproximadamente"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 385,
   "id": "7c620b5d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "25-element Array{Float64,1}:\n",
       " 1.8814400535262978\n",
       " 0.9136378565452907\n",
       " 0.8743911277977362\n",
       " 0.9048503160706224\n",
       " 1.6724112917214005\n",
       " 1.122363481612154\n",
       " 0.08513731962737954\n",
       " 0.04156518274377449\n",
       " 0.07579847651858843\n",
       " 0.9050429008052224\n",
       " 1.092531754469316\n",
       " 0.05141865276188393\n",
       " 0.007892951215112194\n",
       " 0.0424006110806016\n",
       " 0.87510445146112\n",
       " 1.1397887499211952\n",
       " 0.10404115395090843\n",
       " 0.060904545032601606\n",
       " 0.09505903083520001\n",
       " 0.92283932672\n",
       " 2.2995353875329845\n",
       " 1.3486157621428223\n",
       " 1.31022976090112\n",
       " 1.34019919872\n",
       " 2.090856448"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(temperatura_estimada_gs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f3c286d",
   "metadata": {},
   "source": [
    "**(E)** vamos testar para 104 no caso do jacobi e 60 no caso do gauss seidel. Escolhemos estes valores pois é o máximo que a máquina usada aguenta rodando em menos de dois minutos. Se botar valores maiores há a possibilidade do sistema crashar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 386,
   "id": "ebb59e5f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10816×1 Array{Float64,2}:\n",
       " 0.004160887741383967\n",
       " 0.0018496681752099963\n",
       " 0.0018494544733719243\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " 0.0018494544544043758\n",
       " ⋮\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816677760594\n",
       " 0.0027741816827920353\n",
       " 0.002774352632565838\n",
       " 0.004623208599957113"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 11.459976 seconds (1.38 k allocations: 1.745 GiB, 21.19% gc time)\n"
     ]
    }
   ],
   "source": [
    "A, b = constroi_sistema(104)\n",
    "\n",
    "@time display(jacobi(A, b, 10^(-3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 387,
   "id": "502221fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3600-element Array{Float64,1}:\n",
       " 0.012503475117408616\n",
       " 0.005560573882602405\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " 0.005558721001515141\n",
       " ⋮\n",
       " 0.008338274392647484\n",
       " 0.008338274392647484\n",
       " 0.008338274392647484\n",
       " 0.008338274392647484\n",
       " 0.008338274392647484\n",
       " 0.008338274392647484\n",
       " 0.008338274392647484\n",
       " 0.008338274392647517\n",
       " 0.00833827439276664\n",
       " 0.008338274821495687\n",
       " 0.008339817817048657\n",
       " 0.013893057784636488"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.657289 seconds (1.36 k allocations: 297.051 MiB)\n"
     ]
    }
   ],
   "source": [
    "A, b = constroi_sistema(60)\n",
    "\n",
    "@time display(gauss_seidel(A, b, 10^(-3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "676d131e",
   "metadata": {},
   "source": [
    "Vamos construir a matriz A no formato CSR (Compressed storage row)\n",
    "para se aproveitar do fato que ela é esparsa e ver se isso implicará em melhoras significativas no runtime dos nossos métodos\n",
    "\n",
    "**referencia**: https://www.wikiwand.com/en/Sparse_matrix#/Compressed_sparse_row_(CSR,_CRS_or_Yale_format)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 388,
   "id": "02e57489",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "jacobi2 (generic function with 1 method)"
      ]
     },
     "execution_count": 388,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Parametro é a matriz que queremos comprimir em CSR\n",
    "function constroi_csr(A)\n",
    "    row, col = size(A)\n",
    "    val = zeros(0) # valor de células não nulas em A\n",
    "    row_ptr = zeros(0) #row_ptr[i] = posição de val onde começa a i-esima linha de A\n",
    "    col_idx = zeros(0) #col_idx[i] = em qual coluna de A o val[i] aparece\n",
    "\n",
    "    for i in 1 : row\n",
    "        flag = false # flag para avisar se estamos procurando\n",
    "                    # pelo primeiro valor de uma linha\n",
    "        for j in 1 : col\n",
    "            if !flag && A[i, j] != 0.0 # achamos uma célula não nula\n",
    "                append!(val, A[i, j]) # e ela é a primeira de uma linha\n",
    "                append!(col_idx, j)\n",
    "                append!(row_ptr, length(val))\n",
    "                flag = true \n",
    "            elseif flag && A[i, j] != 0.0 # achamos uma célula não nula\n",
    "                append!(val, A[i, j]) # e ela não é a primeira de sua linha\n",
    "                append!(col_idx, j)\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # SPG vamos botar o tamanho de val na ultima posição de row_itr\n",
    "    # para o algoritmo funcionar\n",
    "    append!(row_ptr, length(val) + 1)\n",
    "\n",
    "    # retornamos todos elementos necessários para executar\n",
    "    # multiplicação de matriz por vetor\n",
    "    return val, col_idx, row_ptr, row, col\n",
    "end\n",
    "\n",
    "val, col_idx, row_ptr, row, col = constroi_csr(A)\n",
    "\n",
    "# multiplica uma matriz CSR(val, col_idx, row_ptr, row, col) por um vetor x\n",
    "function multiply(val, col_idx, row_ptr, x, row, col) \n",
    "    b = zeros(length(row_ptr) - 1)\n",
    "    for i in 1 : row\n",
    "        for j in Int64(row_ptr[i]) : (Int64(row_ptr[i + 1]) - 1)\n",
    "            b[i] += val[j] * x[Int64(col_idx[j])]\n",
    "        end\n",
    "    end\n",
    "    return b\n",
    "end\n",
    "\n",
    "# função que checa se a matriz A é diagonal dominante\n",
    "function convergence_check(A) \n",
    "    n, m = size(A)\n",
    "    for i in 1 : n\n",
    "        S = 0.0\n",
    "        for j in 1 : m\n",
    "            if i != j \n",
    "                S += A[i, j]\n",
    "            end\n",
    "        end\n",
    "        if (abs(S) > abs(A[i, i])) \n",
    "            return false\n",
    "        end\n",
    "    end\n",
    "    return true\n",
    "end\n",
    "\n",
    "# vamos adaptar o método de gauss-seidel para usar o formato CSR\n",
    "function gauss_seidel2(A, b, EPS)\n",
    "    U = UpperTriangular(A)\n",
    "    L = U - A\n",
    "    \n",
    "    Uinv = inv(U)\n",
    "    \n",
    "    T = Uinv * L\n",
    "    C = Uinv * b\n",
    "\n",
    "    basis = ones(length(b))\n",
    "\n",
    "    error = 1.0\n",
    "    \n",
    "    val, col_idx, row_ptr, row, col = constroi_csr(T)\n",
    "\n",
    "    iter = 1\n",
    "\n",
    "    while true\n",
    "        cur = multiply(val, col_idx, row_ptr, basis, row, col)\n",
    "        basis = cur + C\n",
    "        error = norm(cur + C - basis)\n",
    "        if (error < EPS)\n",
    "            break\n",
    "        end\n",
    "        iter += 1\n",
    "    end\n",
    "\n",
    "    return basis\n",
    "end\n",
    "\n",
    "# vamos adaptar o método de gauss-jacobi para usar o formato CSR\n",
    "function jacobi2(A, b, EPS) \n",
    "    D = Diagonal(A)\n",
    "    K = D - A\n",
    "    iD = inv(D)\n",
    "\n",
    "    T = iD * K\n",
    "    C = iD * b\n",
    "\n",
    "    basis = ones(size(A)[1], 1)\n",
    "    iter = 1\n",
    "    error = 1.0\n",
    "    \n",
    "    val, col_idx, row_ptr, row, col = constroi_csr(T)\n",
    "\n",
    "    while true\n",
    "        cur = multiply(val, col_idx, row_ptr, basis, row, col)\n",
    "        error = norm(cur + C - basis)\n",
    "        basis = cur + C\n",
    "        if error < 10^(-3)\n",
    "            break\n",
    "        end\n",
    "\n",
    "        iter += 1\n",
    "    end\n",
    "    \n",
    "    return basis\n",
    "end\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "643db190",
   "metadata": {},
   "source": [
    "Vamos agora testar se melhorou a complexidade dos métodos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 389,
   "id": "e00916c7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 30.833554 seconds (50.10 k allocations: 1.721 GiB, 20.16% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "8100-element Array{Float64,1}:\n",
       " 0.0055562720057490585\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " 0.0025929431997236225\n",
       " ⋮\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074700581\n",
       " 0.003951105074705225\n",
       " 0.003951105112326674\n",
       " 0.003951409846060052\n",
       " 0.0064197530864197536"
      ]
     },
     "execution_count": 389,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, b = constroi_sistema(90)\n",
    "\n",
    "@assert(convergence_check(A))\n",
    "\n",
    "@time gauss_seidel2(A, b, 10^(-3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 390,
   "id": "6ce8cee5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  4.873155 seconds (69.72 k allocations: 1.497 GiB, 1.90% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10000-element Array{Float64,1}:\n",
       " 0.004500450153\n",
       " 0.00200065016\n",
       " 0.002000400159\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " 0.002000400135\n",
       " ⋮\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600185\n",
       " 0.003000600204\n",
       " 0.00300080019\n",
       " 0.005000500168"
      ]
     },
     "execution_count": 390,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, b = constroi_sistema(100)\n",
    "\n",
    "@assert(convergence_check(A))\n",
    "\n",
    "@time jacobi2(A, b, 10^(-3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20ef1959",
   "metadata": {},
   "source": [
    "## Questão 4."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15ee9fa7",
   "metadata": {},
   "source": [
    "(a) Faça a modelagem sem o cano pontilhado x9 e resolva o sistema linear com o método de Gauss-Jacobi e Gauss-Seidel com tolerância de $10^{-3}$. Os métodos convergem?\n",
    "\n",
    "### Resposta:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba554eaa",
   "metadata": {},
   "source": [
    "Modelando o problema da distribuição de água temos as equações:\n",
    "\n",
    "$$x_8 = x_6 + x_7 + 500$$\n",
    "$$x_7 = x_5 + 3000$$\n",
    "$$x_6 = x_4 + x_2$$\n",
    "$$x_5 = x_3 + 3000$$\n",
    "$$x_4 = x_1 + 30000$$\n",
    "$$x_3 = 9000$$\n",
    "$$x_2 = 3500$$\n",
    "$$x_1 = 7000$$\n",
    "\n",
    "E portanto podemos montar o sistema linear\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} \n",
    "    0 & 0 & 0 & 0 & 0 & -1 & -1 & 1 \\\\\n",
    "    0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 \\\\\n",
    "    0 & -1 & 0 & -1 & 0 & 1 & 0 & 0 \\\\\n",
    "    0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "    -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} \n",
    "    x_1 \\\\\n",
    "    x_2 \\\\\n",
    "    x_3 \\\\\n",
    "    x_4 \\\\\n",
    "    x_5 \\\\\n",
    "    x_6 \\\\\n",
    "    x_7 \\\\\n",
    "    x_8 \\\\\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix} \n",
    "    500 \\\\\n",
    "    3000 \\\\\n",
    "    0 \\\\\n",
    "    3000 \\\\\n",
    "    30000 \\\\\n",
    "    9000 \\\\\n",
    "    3500 \\\\\n",
    "    7000 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Obtemos uma matriz A com a diagonal toda zerada e os métodos de Gauss-Seidel e Gauss-Jacobi não funcionarão. No entanto, podemos fazer operações em cima da nossa matriz a fim de torná-la funcional para nossos métodos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 391,
   "id": "f08e4e0c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8×8 Array{Float64,2}:\n",
       "  1.0   0.0   0.0   0.0   0.0   0.0   0.0  0.0\n",
       "  0.0   1.0   0.0   0.0   0.0   0.0   0.0  0.0\n",
       "  0.0   0.0   1.0   0.0   0.0   0.0   0.0  0.0\n",
       " -1.0   0.0   0.0   1.0   0.0   0.0   0.0  0.0\n",
       "  0.0   0.0  -1.0   0.0   1.0   0.0   0.0  0.0\n",
       "  0.0  -1.0   0.0  -1.0   0.0   1.0   0.0  0.0\n",
       "  0.0   0.0   0.0   0.0  -1.0   0.0   1.0  0.0\n",
       "  0.0   0.0   0.0   0.0   0.0  -1.0  -1.0  1.0"
      ]
     },
     "execution_count": 391,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define a matriz A e o vetor B\n",
    "A = [0 0 0 0 0 -1 -1 1; 0 0 0 0 -1 0 1 0; 0 -1 0 -1 0 1 0 0; 0 0 -1 0 1 0 0 0; -1 0 0 1 0 0 0 0; 0 0 1 0 0 0 0 0; 0 1 0 0 0 0 0 0; 1 0 0 0 0 0 0 0]\n",
    "b = [500; 3000; 0; 3000; 30000; 9000; 3500; 7000]\n",
    "\n",
    "# Tamanho da Matriz Quadrada\n",
    "N = 8\n",
    "R = zeros(N,N)\n",
    "bzinho = zeros(N)\n",
    "for i in 1:N\n",
    "    R[i, :] .= A[(N+1)-i, :]\n",
    "    bzinho[i] = b[(N+1)-i]\n",
    "end\n",
    "\n",
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 392,
   "id": "07bdbbfc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{Float64,1}:\n",
       "  7000.0\n",
       "  3500.0\n",
       "  9000.0\n",
       " 30000.0\n",
       "  3000.0\n",
       "     0.0\n",
       "  3000.0\n",
       "   500.0"
      ]
     },
     "execution_count": 392,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bzinho"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c73d37b5",
   "metadata": {},
   "source": [
    "Aplicando então os dois métodos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 393,
   "id": "bfec2657",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "35640.566774393475\n",
      "38239.377609997784\n",
      "16000.0\n",
      "0.0\n",
      "0.0\n",
      "\n",
      "[7000.0; 3500.0; 9000.0; 37000.0; 12000.0; 40500.0; 15000.0; 56000.0]"
     ]
    }
   ],
   "source": [
    "for i in 1:5\n",
    "    x = gauss_jacobi(R,bzinho, i)\n",
    "    print(norm(R*x - bzinho), \"\\n\") # erro entre as iterações\n",
    "end\n",
    "print(\"\\n\", x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 394,
   "id": "a5ab1483",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n",
      "0.0\n",
      "0.0\n",
      "0.0\n",
      "0.0\n",
      "\n",
      "[7000.0, 3500.0, 9000.0, 37000.0, 12000.0, 40500.0, 15000.0, 56000.0]"
     ]
    }
   ],
   "source": [
    "for i in 1:5\n",
    "    x = gauss_seidel(R,bzinho, i)\n",
    "    print(norm(R*x - bzinho), \"\\n\") # erro entre as iterações\n",
    "end\n",
    "print(\"\\n\", x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f2c9d16",
   "metadata": {},
   "source": [
    "Os dois métodos convergem e encontram o mesmo vetor x como solução"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 395,
   "id": "5ea8b49d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 395,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(R*x - bzinho) # erro no sistema alterado"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 396,
   "id": "7585d957",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 396,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(A*x - b) # erro no sistema original"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa905099",
   "metadata": {},
   "source": [
    "(b) (Desafio) Faça a modelagem com o cano pontilhado x9 e tente resolver o sistema linear com Julia e com o método de Gauss-Jacobi e Gauss-Seidel com tolerância de $10^{-3}$. O que aconteceu?\n",
    "\n",
    "### Resposta:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ee1c6c8",
   "metadata": {},
   "source": [
    "A modelagem com o cano pontilhado $x_9$ possui esta variável a mais porém nenhuma equação adicional:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} \n",
    "    0 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & 0 \\\\\n",
    "    0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 \\\\\n",
    "    0 & -1 & 0 & -1 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "    0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n",
    "    -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 \\\\\n",
    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix} \n",
    "    x_1 \\\\\n",
    "    x_2 \\\\\n",
    "    x_3 \\\\\n",
    "    x_4 \\\\\n",
    "    x_5 \\\\\n",
    "    x_6 \\\\\n",
    "    x_7 \\\\\n",
    "    x_8 \\\\\n",
    "    x_9\\\\\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix} \n",
    "    500 \\\\\n",
    "    3000 \\\\\n",
    "    0 \\\\\n",
    "    3000 \\\\\n",
    "    30000 \\\\\n",
    "    9000 \\\\\n",
    "    3500 \\\\\n",
    "    7000 \\\\\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 397,
   "id": "83092331",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 397,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NewA = [0 0 0 0 0 -1 -1 1 0; 0 0 0 0 -1 0 1 0 0; 0 -1 0 -1 0 1 0 0 0; 0 0 -1 0 1 0 0 0 1; -1 0 0 1 0 0 0 0 -1; 0 0 1 0 0 0 0 0 0; 0 1 0 0 0 0 0 0 0; 1 0 0 0 0 0 0 0 0]\n",
    "\n",
    "rank(NewA) # Vemos que o número de colunas L.I ainda é 8"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "612671b2",
   "metadata": {},
   "source": [
    "Como o número de variáveis é 9 e o número de equações é apenas 8 sabemos que não é possível encontrar uma solução para o sistema proposto. Estamos no caso de infinitas soluções e portanto os métodos não convergem."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c4fe9ee",
   "metadata": {},
   "source": [
    "## Questão 5."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6504f11f",
   "metadata": {},
   "source": [
    "A."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 398,
   "id": "6c1d70d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×3 Array{Float64,2}:\n",
       " 1.50001  0.999972  1.49991\n",
       " 1.50009  1.00003   1.49999"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Erro: 3.000000000309557\n"
     ]
    }
   ],
   "source": [
    "A = [0 1 3; 3 1 0] * 1.0\n",
    "\n",
    "b, c = acha_melhor_minimizacao(A)\n",
    "\n",
    "println(\"Erro: \", norm(A - b * c'))\n",
    "\n",
    "display(b * c')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31bde7ca",
   "metadata": {},
   "source": [
    "Podemos plotar a solução, mostrando como os vetores originais são compactados na mesma reta, dada pelo vetor da base b."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 399,
   "id": "50119409",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 399,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = b*c'\n",
    "p = scatter(m[1,:], m[2,:], alpha=0.5)\n",
    "scatter!(p, A[1,:], A[2,:], alpha=0.5)\n",
    "\n",
    "f(x) = b[2]*x/b[1]\n",
    "plot!(p, f)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14d3c13c",
   "metadata": {},
   "source": [
    "B. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 400,
   "id": "70c24021",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×4 Array{Float64,2}:\n",
       " 1.0  2.0  2.0  1.0\n",
       " 1.0  2.0  2.0  1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Erro: 2.8284271247461903\n"
     ]
    }
   ],
   "source": [
    "A = [0 1 3 2; 2 3 1 0] * 1.0\n",
    "\n",
    "b, c = acha_melhor_minimizacao(A)\n",
    "\n",
    "println(\"Erro: \", norm(A - b * c'))\n",
    "\n",
    "display(b * c')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25e29c70",
   "metadata": {},
   "source": [
    "Podemos plotar a solução, mostrando como os vetores originais são compactados na mesma reta, dada pelo vetor da base b."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 401,
   "id": "33024536",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 401,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = b*c'\n",
    "p = scatter(m[1,:], m[2,:], alpha=0.5)\n",
    "scatter!(p, A[1,:], A[2,:], alpha=0.5)\n",
    "\n",
    "f(x) = b[2]*x/b[1]\n",
    "\n",
    "plot!(p, f)"
   ]
  },
  {
   "attachments": {
    "bandeiraGrecia.png": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "6eb1fbb2",
   "metadata": {},
   "source": [
    "## Questão 6.\n",
    "\n",
    "### Considere a bandeira da Grécia como uma imagem preta e branca e modele com uma matriz A.\n",
    "\n",
    "Para modelarmos a bandeira da grécia como uma matriz, consideraremos a seguinte imagem, que preparei com muito carinho:\n",
    "\n",
    "![bandeiraGrecia.png](attachment:bandeiraGrecia.png)\n",
    "\n",
    "Logo teremos \n",
    "\n",
    "$$A = \n",
    "\\begin{bmatrix} \n",
    "    1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "#### (a) Qual é o posto da bandeira da Grécia (tal que $A = BC^t$)? O que a matriz $B$ e matriz $C^t$ representam nesse caso? Explique com as suas próprias palavras.\n",
    "\n",
    "$A = \n",
    "\\begin{bmatrix} \n",
    "    1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "\\end{bmatrix} \n",
    "=\n",
    "\\begin{bmatrix}\n",
    "    1 & 0 & 1 \\\\\n",
    "    1 & 0 & 0 \\\\\n",
    "    0 & 0 & 1 \\\\\n",
    "    1 & 0 & 0 \\\\\n",
    "    1 & 0 & 1 \\\\\n",
    "    0 & 0 & 0 \\\\\n",
    "    1 & 1 & 1 \\\\\n",
    "    0 & 0 & 0 \\\\\n",
    "    1 & 1 & 1 \\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "A matriz $B$ representaria as diferentes \"colunas\" que a bandeira tem, que, nesse caso, são $3$, o nosso posto. $C^t$ por consequência representaria a ordem das \"colunas\" na bandeira.\n",
    "\n",
    "#### (b) Determine dois países tais quais sua bandeira tem posto = 1?\n",
    "\n",
    "Para posto = 1, podemos pensar em bandeiras horizontais, como Ucrânia, Rússia, Iémen, Indonésia, ...\n",
    "\n",
    "#### (c) Determine dois países tais quais sua bandeira tem posto = 2?\n",
    "\n",
    "Para posto = 2, podemos pensar em bandeiras horizontais, porém com algum detalhe simples vertical, como Dinamarca, Finlândia, Suécia, ...\n",
    "\n",
    "#### (c) Determine dois países tais quais sua bandeira tem posto = 3?\n",
    "\n",
    "Para posto = 3, podemos pensar em bandeiras verticais com três cores, como Bélgica, França, Chade, Guinea, ...\n",
    "\n",
    "##### Super interessante pensar nos postos para bandeiras complicadas como a do Brasil, Japão, Arábia Saudita, ..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed118e87",
   "metadata": {},
   "source": [
    "## Questão 7."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6279e1d0",
   "metadata": {},
   "source": [
    "Primeiro, fazemos a inserção da imagem:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 402,
   "id": "156ef4f3",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 402,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imagem = [0 1 0 1 0 ; 0 1 0 1 0 ; 0 0 0.5 0 0 ; 1 0 0 0 1 ; 0 1 1 1 0]\n",
    "plot(Gray.(1 .- imagem), size=(150,150), axis=nothing)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85eac0d7",
   "metadata": {},
   "source": [
    "Podemos descobrir o posto da imagem através da função rank:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 403,
   "id": "983f7608",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 403,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rank(imagem)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5cc3dd46",
   "metadata": {},
   "source": [
    "Para níveis de compressão abaixo do posto da imagem, a nitidez é prejudicada:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 404,
   "id": "c5a33d82",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 404,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B,c = compressao(imagem, 1)\n",
    "plot(Gray.(1 .- B*c'), size=(150,150), axis=nothing)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 405,
   "id": "97362bae",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 405,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B,c = compressao(imagem, 2)\n",
    "plot(Gray.(1 .- B*c'), size=(150,150), axis=nothing)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9315328d",
   "metadata": {},
   "source": [
    "A partir do nível de compressão igual ao posto da imagem a nitidez é mantida como a original."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 406,
   "id": "20389761",
   "metadata": {},
   "outputs": [
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     "execution_count": 406,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B,c = compressao(imagem, 3)\n",
    "plot(Gray.(1 .- B*c'), size=(150,150), axis=nothing)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 407,
   "id": "afbf6110",
   "metadata": {},
   "outputs": [
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       "ECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQ\n",
       "JvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAEPaamd0beGOttXsCwJ9pynn86AEgTOgBIEzoASBM\n",
       "6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzo\n",
       "ASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgB\n",
       "IEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEg\n",
       "TOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM\n",
       "6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzo\n",
       "ASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgB\n",
       "IEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEg7Bv8Dh9sZx/SMQAA\n",
       "AABJRU5ErkJggg==\n",
       "\" transform=\"translate(47, 47)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 407,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B,c = compressao(imagem, 4)\n",
    "plot(Gray.(1 .- B*c'), size=(150,150), axis=nothing)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 408,
   "id": "5cc36215",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"150\" height=\"150\" viewBox=\"0 0 600 600\">\n",
       "<defs>\n",
       "  <clipPath id=\"clip700\">\n",
       "    <rect x=\"0\" y=\"0\" width=\"600\" height=\"600\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<path clip-path=\"url(#clip700)\" d=\"\n",
       "M0 600 L600 600 L600 0 L0 0  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<defs>\n",
       "  <clipPath id=\"clip701\">\n",
       "    <rect x=\"120\" y=\"60\" width=\"421\" height=\"421\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<path clip-path=\"url(#clip700)\" d=\"\n",
       "M47.2441 552.756 L552.756 552.756 L552.756 47.2441 L47.2441 47.2441  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<defs>\n",
       "  <clipPath id=\"clip702\">\n",
       "    <rect x=\"47\" y=\"47\" width=\"507\" height=\"507\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<polyline clip-path=\"url(#clip700)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  47.2441,552.756 552.756,552.756 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip700)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  47.2441,47.2441 47.2441,552.756 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip702)\">\n",
       "<image width=\"506\" height=\"506\" xlink:href=\"data:image/png;base64,\n",
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       "ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQA\n",
       "ECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQ\n",
       "JvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm\n",
       "9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0\n",
       "ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQA\n",
       "ECb0ABAm9AAQtq7rmt0j+GnGswCfZ621ewIPfvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0\n",
       "ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQA\n",
       "ECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQ\n",
       "JvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAECb0ABAm\n",
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       "JvQAECb0ABAm9AAQJvQAECb0ABAm9AAQJvQAEPaamd0beGOttXsCwJ9pynn86AEgTOgBIEzoASBM\n",
       "6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzo\n",
       "ASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgB\n",
       "IEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEg\n",
       "TOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM\n",
       "6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzo\n",
       "ASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgB\n",
       "IEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEgTOgBIEzoASBM6AEg7Bv8Dh9sZx/SMQAA\n",
       "AABJRU5ErkJggg==\n",
       "\" transform=\"translate(47, 47)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 408,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B,c = compressao(imagem, 5)\n",
    "plot(Gray.(1 .- B*c'), size=(150,150), axis=nothing)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19517a6a",
   "metadata": {},
   "source": [
    "## Questão 8."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 409,
   "id": "ce67d9b6",
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra\n",
    "using Images\n",
    "using JLD"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7966f85e",
   "metadata": {},
   "source": [
    "### Encontre (ou desenhe) uma imagem A, com mais ou menos $200 × 200$ pixels (pode ser um pouco maior), que\n",
    "\n",
    "#### (a) exija um nível de compressão maior que 3 e menor que 6 para recuperar pelo menos 99% de sua norma total.\n",
    "\n",
    "Escolhemos o logo da Xiaomi levemente modificado para que fique sem curvas, o que justifica o posto 4~5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 410,
   "id": "99a99c54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "200×200 Array{Gray{N0f8},2} with eltype Gray{Normed{UInt8,8}}:\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)  …  Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)  …  Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)  …  Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " ⋮                                     ⋱  \n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)  …  Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)  …  Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)\n",
       " Gray{N0f8}(0.541)  Gray{N0f8}(0.541)     Gray{N0f8}(0.541)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "img = load(\"xiaomi.png\");\n",
    "display(Gray.(img))\n",
    "raw = channelview(Gray.(img[:,:]));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 411,
   "id": "73416cc1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "200×200 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)  …  Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)  …  Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)  …  Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " ⋮                                                 ⋱  \n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)  …  Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)  …  Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)\n",
       " Gray{Float64}(0.540323)  Gray{Float64}(0.540323)     Gray{Float64}(0.540323)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B,C=compressao(raw,2)\n",
    "display(Gray.(B*C'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 412,
   "id": "5614486b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "200×200 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)  …  Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)  …  Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)  …  Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " ⋮                                                 ⋱  \n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)  …  Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)  …  Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)\n",
       " Gray{Float64}(0.541183)  Gray{Float64}(0.541183)     Gray{Float64}(0.541183)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B,C=compressao(raw,4)\n",
    "display(Gray.(B*C'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09290409",
   "metadata": {},
   "source": [
    "#### (b) exija um nível de compressão maior que 40 para recuperar pelo menos 99% de sua norma total.\n",
    "\n",
    "Escolhemos um qr code bem complicado para que necessite bastantes postos para restaurar a imagem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 413,
   "id": "7326822e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAADIAQAAAACFI5MzAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QAAd2KE6QAAASuSURBVFjD7dc9dq4qFAZgbGAK2sDUoIEpSMNPI1OQBqaGDU5BGzj7K0zuLQxnnTp2yZO4kI3v3qL+dqFf+ZV/l4aIT+w8nc0+9jMcuSA7ElvtfaYlVHWfMZ9+U20oXPlqk26qScSuqRmJxyJJLEs+CtzU97a0vxHlOKsIrZQ40pBEfyEWfiuNWZFopLC6hLMNBWH5/4ujkcB1XGozMyxEVXG36T/1eZGGuG0rRrqQ6KamyrzEoegeG9wm3rvyxw01oNwOpPu6RPj/86i2TsnG/LW2V2l86T0XXVmj0pYlIbQMBWF1X1NaLnIn2JeLY2lHwnI8N65CY0esKvSqr5H0sNmq4mY3ZCvLdYp37gNpU4xN5yTOHdnGdSPO2JFQqS5DYnfsSLrixWHVB9ILxXSBCty3j03kAnW0I6mc3Pt0d8d1JblixJ7neZeNsr5TUdm+wsFfknCzHUhbxXGtmOqifVsqXleOR9L9nQtrnyp4OFWwS6ENpZEQMix7I71iqkK+Qx8K5dyGw9+OqsqXpBwbSUOIc52QcPZwUDtxITKSDsWNuRhyURKOvulixFikJMVI5cS9caoc/Rzen6UhI/IlDbaXJBXpKg0bC5rXFaHJsViQUXdDYiS9x+4TJz5nN92Hh593O5IdrZycfYdDC4Xj0+fEDKSxHhKVU2WfR3ZLE/0eygIHwxHvnTr7Zdg1czEUyTZs6wzpNVV77rpYZwcC+ZU3znwI1cY7no6ap9qv0kgMO3QlqZKkWEAO9jwUiowus4aQObpvmFP5ZO+74HU6c108/AlGBos7tpH0XZ2xrlL7o+8697Cb57y9ClTBw/5DXFwY0k9X0eNI+r1R+zlcq/X9shsXRYykTbuUFM3YLJu6exJHedLlVTpEV+9JXVxcU8+VpPk5b6/SSIN4SLCAncKqZ6m/6/Mq1pF85OAbuyFgczw2NRSRsN7kDH2JwKZu4rufvsuKpoYl+nQZN0GX2dGz6nfhRuwsmSU4BE16t8WsdihyheSS0Pd8FRuJl3yq8CodBo0lV4iIO5wFAsYYaQfS8KfO6XMGs/t0GYroSHqM+9QjvB3OCEijZvPTmV4F3vodcTrrXR1panKFkBnKvDIYABJpS9hWSbn46s7vQoJPIsK5VdBo4BVL+HlLXgVKJ6fTw+0+rXNeGoT0UBxJZoUNKSxmZzfRcxhJoyzZs7KYdOjxTLqgZ7Z8le7LZzK7BERZ3lidcj77UCBWc4zJJiraPBvin/R/lx4u7cguEvPd9wz94ugDaVM+i1EVzws08zpz/vU8r9I3NBWSqHawLzDbsfRMtz/ItaKZ5fOMG6x62g0fS4NyzRgtoagj+xxivKsdySf5RUxyZg4bg+Atm0fSE4mfD6gLKWhkDDJtf/LtXZph4bhEmi6sYQiHb7GxOAEBQ+FeKjtEDcdIDQWy7wgVphpWVFGJwxePHQhM61RvBiLmaFIESLHvqeZV7AmDGTdmuaSCqXPj5tm3d4EUW+CUk6LPz5SGvlb9o0y+avhWm6FZwKrt17Txk8A5nKe+weCpYQ5waBqKrRRKsGOWVoThq0CaZSjwdaguDBOx9XexdSWXGsnL9Su/8q/yB1VWlXa0cXJwAAAAAElFTkSuQmCC",
      "text/plain": [
       "200×200 Array{Gray{N0f8},2} with eltype Gray{Normed{UInt8,8}}:\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)  …  Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)  …  Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)  …  Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " ⋮                                 ⋱                   \n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)  …  Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)  …  Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)\n",
       " Gray{N0f8}(1.0)  Gray{N0f8}(1.0)     Gray{N0f8}(1.0)  Gray{N0f8}(1.0)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "img = load(\"qrCode.png\");\n",
    "display(Gray.(img))\n",
    "raw = channelview(Gray.(img[:,:]));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 414,
   "id": "5c7b1b60",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "200×200 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)  …  Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)  …  Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)  …  Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " ⋮                                               ⋱  \n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)  …  Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)  …  Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)\n",
       " Gray{Float64}(1.11499)  Gray{Float64}(1.11499)     Gray{Float64}(1.11499)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B,C=compressao(raw,2)\n",
    "display(Gray.(B*C'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 415,
   "id": "657ceac1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "200×200 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)  …  Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)  …  Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)  …  Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " ⋮                                               ⋱  \n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)  …  Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)  …  Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)\n",
       " Gray{Float64}(1.00199)  Gray{Float64}(1.00199)     Gray{Float64}(1.00199)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B,C=compressao(raw,20)\n",
    "display(Gray.(B*C'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 416,
   "id": "167cf02d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "200×200 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)  …  Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)  …  Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)  …  Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " ⋮                                             ⋱  \n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)  …  Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)  …  Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)\n",
       " Gray{Float64}(1.0002)  Gray{Float64}(1.0002)     Gray{Float64}(1.0002)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B,C=compressao(raw,41)\n",
    "display(Gray.(B*C'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 417,
   "id": "619dc027",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "200×200 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)  …  Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)  …  Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)  …  Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " ⋮                                       ⋱  \n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)  …  Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)  …  Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)\n",
       " Gray{Float64}(1.0)  Gray{Float64}(1.0)     Gray{Float64}(1.0)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B,C=compressao(raw,100)\n",
    "display(Gray.(B*C'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d4b2cbba",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.5.4",
   "language": "julia",
   "name": "julia-1.5"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.5.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}