Biblioteca de CN.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Trabalho 7 - Biblioteca de Cálculo Numérico"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Feito por: **Chris Ciafrino** - 117230691, **Luan Martins** - 119022028, **Thierry Pierre** - 119040432, **Vinicius Lettieri** - 119040830"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Questão 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Funções"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "import Pkg; \n",
    "Pkg.add(\"Polynomials\")\n",
    "\n",
    "using Polynomials\n",
    "using Plots\n",
    "using LinearAlgebra\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver Sistema de Equações Diagonal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolve_diagonal (generic function with 1 method)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O intuito da função resolve_diagonal é resolver equações Dx = b onde A é uma matriz diagonal NxN, \n",
    "X é uma matriz Nx1 e b também é uma matriz Nx1. Logo, a função recebe como entrada a Matriz Diagonal \n",
    "Quadrada NXN e o vetor B Nx1, e retorna o vetor x Nx1. \"\"\"\n",
    "\n",
    "function resolve_diagonal(D,b)     # D é diagonal e b é um vetor  \n",
    "    tamanho = length(b)            # Pega-se o tamanho tanto do vetor quanto da matriz (pressupõe que são de mesmo tamanho)\n",
    "    x=zeros(tamanho,1)             # Aloca o vetor x a ser retornado\n",
    "    for i in 1:tamanho \n",
    "        x[i] = b[i]/D[i,i]         # Resolve cada x[i] de acordo com o valor da diagonal e o valor de b de cada linha\n",
    "    end\n",
    "    \n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1.0; 0.3333333333333333; 1.0; -9.0]\n",
      "[-1.0, 0.3333333333333333, 1.0, -9.0]\n"
     ]
    }
   ],
   "source": [
    "T = [1 0 0 0 ; 0 3 0 0 ; 0 0 5 0 ; 0 0 0 -1]\n",
    "b = [-1 ; 1 ; 5 ; 9]\n",
    "\n",
    "println(resolve_triangular_superior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0; -1.0; 4.0]\n",
      "[1.0, -1.0, 4.0]\n"
     ]
    }
   ],
   "source": [
    "T = [1 0 0 ; 0 -3 0 ; 0 0 1]\n",
    "b = [1 ; 3 ; 4]\n",
    "\n",
    "println(resolve_triangular_superior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-5.0; -1.0; -1.0; 1.0]\n",
      "[-5.0, -1.0, -1.0, 1.0]\n"
     ]
    }
   ],
   "source": [
    "T = [1 0 0 0 ; 0 2 0 0 ; 0 0 -1 0 ; 0 0 0 1]\n",
    "b = [-5 ; -2 ; 1 ; 1]\n",
    "\n",
    "println(resolve_triangular_superior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver Sistema de Equações Triangular Superior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolve_triangular_superior (generic function with 1 method)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O intuito dessa função é resolver equações T.x = b tal que T seja uma matriz NxN e triangular superior,\n",
    "x e b serão uma matriz coluna Nx1. A função deve retornar a solução x para tal sistema de equações.\"\"\"\n",
    "\n",
    "function resolve_triangular_superior(T,b)     #T é triangular superior\n",
    "    tamanho = length(b)\n",
    "    x=zeros(tamanho,1)\n",
    "    \n",
    "    i = tamanho\n",
    "    while (i > 0)\n",
    "        x[i] = b[i]/T[i,i]\n",
    "        k = tamanho\n",
    "        for j in i:(tamanho - 1)\n",
    "            x[i] -= T[i, k] * x[k]/T[i,i]\n",
    "            k -= 1\n",
    "        end\n",
    "        i -= 1\n",
    "    end\n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[29.0; -22.7; 1.9; 4.5]\n",
      "[29.0, -22.7, 1.9, 4.5]\n"
     ]
    }
   ],
   "source": [
    "T = [1 2 1 3 ; 0 1 3 4 ; 0 0 5 -1 ; 0 0 0 2]\n",
    "b = [-1 ; 1 ; 5 ; 9]\n",
    "\n",
    "println(resolve_triangular_superior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-2.0; 1.0; 0.0]\n",
      "[-2.0, 1.0, 0.0]\n"
     ]
    }
   ],
   "source": [
    "T = [1 1 1 ; 0 1 0 ; 0 0 1]\n",
    "b = [-1 ; 1 ; 0]\n",
    "\n",
    "println(resolve_triangular_superior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-5.0; -4.25; 1.0; 0.25]\n",
      "[-5.0, -4.25, 1.0, 0.25]\n"
     ]
    }
   ],
   "source": [
    "T = [1 0 0 0 ; 0 1 2 1 ; 0 0 1 0 ; 0 0 0 4]\n",
    "b = [-5 ; -2 ; 1 ; 1]\n",
    "\n",
    "println(resolve_triangular_superior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver Sistema de Equações Triangular Inferior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolve_triangular_inferior (generic function with 1 method)"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O intuito dessa função é resolver equações T.x = b tal que T seja uma matriz NxN e triangular inferior,\n",
    "x e b serão uma matriz coluna Nx1. A função deve retornar a solução x para tal sistema de equações.\"\"\"\n",
    "\n",
    "function resolve_triangular_inferior(T,b)     #T é triangular inferior\n",
    "    tamanho = length(b)\n",
    "    x=zeros(tamanho,1)\n",
    "    \n",
    "    i = 1\n",
    "    while (i <= tamanho)\n",
    "        x[i] = b[i]/T[i,i]\n",
    "        for j in 1:(i-1)\n",
    "            x[i] -= T[i, j] * x[j]/T[i,i]\n",
    "        end\n",
    "        i += 1\n",
    "    end\n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0; 2.0; 0.8; -1.85]\n",
      "[1.0, 2.0, 0.8, -1.85]\n"
     ]
    }
   ],
   "source": [
    "T = [1 0 0 0 ; 1 1 0 0 ; -1 -2 5 0 ; 1 2 3 4]\n",
    "b = [1 ; 3 ; -1 ; 0]\n",
    "\n",
    "println(resolve_triangular_inferior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1.0; -1.5; 0.16666666666666669]\n",
      "[-1.0, -1.5, 0.16666666666666666]\n"
     ]
    }
   ],
   "source": [
    "T = [-1 0 0 ; 1 2 0 ; 0 1 3]\n",
    "b = [1 ; -4 ; -1]\n",
    "\n",
    "println(resolve_triangular_inferior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0; -1.0]\n",
      "[1.0, -1.0]\n"
     ]
    }
   ],
   "source": [
    "T = [1 0 ; 0 3]\n",
    "b = [1 ; -3]\n",
    "\n",
    "println(resolve_triangular_inferior(T,b))\n",
    "println(T\\b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Realizar Eliminação Gaussiana no Sistema de Equações"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "eliminacao_gaussiana (generic function with 1 method)"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O intuito dessa função é realizar o processo de eliminação gaussiana com o sistema A.x = b, \n",
    "onde são passados a matriz densa quadrada A de NxN e a matriz coluna b de Nx1.\n",
    "O retorno será o sistema escalonado, com A sendo uma matriz triangular superior.\"\"\"\n",
    "\n",
    "function eliminacao_gaussiana(A,b)      #A é uma matriz cheia (\"matriz densa\")\n",
    "    tamanho = length(b)\n",
    "    for i in 1:tamanho-1\n",
    "        for j in i+1:tamanho\n",
    "            multiplicador = A[j,i]/A[i,i]\n",
    "            A[j, :] = A[j, :] - multiplicador*A[i, :]\n",
    "            b[j] = b[j] - multiplicador*b[i]\n",
    "        end\n",
    "    end\n",
    "    return A, b\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "([1.0 1.0 1.0; 0.0 1.0 2.0; 0.0 0.0 0.0], [1.0, -5.0, 0.0])\n"
     ]
    }
   ],
   "source": [
    "A = [1 1 1 ; 2 3 4 ; 1 1 1]*1.0\n",
    "b = [1 ; -3 ; 1]*1.0\n",
    "\n",
    "println(eliminacao_gaussiana(A, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "([1.0 2.0 3.0 1.0; 0.0 -1.0 -10.0 -3.0; 0.0 0.0 17.0 5.0; 0.0 0.0 0.0 -2.0], [-1.0, 0.0, 4.0, 4.0])\n"
     ]
    }
   ],
   "source": [
    "A = [1 2 3 1 ; 2 3 -4 -1 ; 1 0 0 0 ; 0 0 0 -2]*1.0\n",
    "b = [-1 ; -2 ; 3 ; 4]*1.0\n",
    "\n",
    "println(eliminacao_gaussiana(A, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "([1.0 2.0 3.0 1.0; 0.0 -1.0 -10.0 -1.0; 0.0 0.0 21.0 1.0; 0.0 0.0 0.0 -3.571428571428571], [-1.0, 4.0, -4.0, 9.285714285714285])\n"
     ]
    }
   ],
   "source": [
    "A = [1 2 3 1 ; 2 3 -4 1 ; 1 0 4 0 ; 0 3 0 -2]*1.0\n",
    "b = [-1 ; 2 ; 3 ; 3]*1.0\n",
    "\n",
    "println(eliminacao_gaussiana(A, b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver Sistema de Equações de Matriz Cheia"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolve_cheia (generic function with 1 method)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O intuito dessa função é resolver o sistema de equações lineares no formato A.x = b, tal que\n",
    "A seja uma matriz quadrada densa. Essa implementação realiza a eliminação gaussiana do sistema e\n",
    "depois resolve como um sistema triangular superior.\"\"\"\n",
    "\n",
    "function resolve_cheia(A,b)\n",
    "    T,c= eliminacao_gaussiana(A,b)  \n",
    "    x=resolve_triangular_superior(T,c)  \n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.499780622986203e-14"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 20\n",
    "A = zeros(N, N)\n",
    "b = zeros(N)\n",
    "\n",
    "for i in 1:N\n",
    "    b[i] = rand() % N\n",
    "    for j in 1:N\n",
    "        A[i, j] = rand() % N\n",
    "    end\n",
    "end\n",
    "\n",
    "x = resolve_cheia(A, b)\n",
    "\n",
    "norm(A*x - b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.974235031731525e-12"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = [134 343 95.4 341 1003; 434 34 305 451 14; 35 36 37 38 39; 104 342 534 034 3259; 1930 234 20 453 954] * 1.0\n",
    "b = [54; 4530; 34; 1034; 100] * 1.0\n",
    "\n",
    "k = resolve_cheia(M,b)\n",
    "norm(M*k - b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = [1 1 -1 ; 0 1 0 ; 1 1 3]*1.0\n",
    "b = [-1 ; 5 ; -1]*1.0\n",
    "\n",
    "x = resolve_cheia(T,b)\n",
    "norm(T\\b - x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Realizar Decomposição LU em uma Matriz Quadrada"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "decomposicao_LU (generic function with 1 method)"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O intuito dessa função é realizar a Decomposição LU da matriz A, que compôe em duas matrizes:\n",
    "A matriz L é uma matriz triangular inferior e a U é uma trinagular inferior.\n",
    "O input da questão será a matriz A quadrada e o retorno serão as matrizes L e U descritas.\"\"\"\n",
    "\n",
    "function decomposicao_LU(A)          # A é uma matriz cheia (\"matriz densa\")\n",
    "    tamanho = size(A)\n",
    "    tamanho = tamanho[1]\n",
    "    L=zeros(tamanho,tamanho)\n",
    "    \n",
    "    for i in 1:tamanho\n",
    "        L[i,i] = 1\n",
    "    end\n",
    "    \n",
    "    U = zeros(tamanho, tamanho)\n",
    "    \n",
    "    for i in 1:tamanho\n",
    "        for j in 1:tamanho\n",
    "            U[i, j] = A[i, j]\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    for i in 1:tamanho-1\n",
    "        for j in i+1:tamanho\n",
    "            L[j,i] = U[j,i]/U[i,i]\n",
    "            U[j, :] = U[j, :] -L[j,i]*U[i, :]\n",
    "        end\n",
    "    end\n",
    "\n",
    "    return L,U \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0 2.0 1.0 3.0; 0.0 1.0 3.0 4.0; 0.0 0.0 5.0 -1.0; 0.0 0.0 0.0 2.0]\n"
     ]
    }
   ],
   "source": [
    "A = [1 2 1 3 ; 0 1 3 4 ; 0 0 5 -1 ; 0 0 0 2]*1.0\n",
    "\n",
    "L,U = decomposicao_LU(A)\n",
    "\n",
    "println(L*U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0 1.0 1.0 1.0; -1.0 1.0 -1.0 2.0; 3.0 4.0 2.0 1.0; 0.0 0.0 3.0 0.0]\n"
     ]
    }
   ],
   "source": [
    "A = [1 1 1 1 ; -1 1 -1 2 ; 3 4 2 1 ; 0 0 3 0]*1.0\n",
    "\n",
    "L, U = decomposicao_LU(A)\n",
    "\n",
    "println(L*U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0 1.0 -1.0; 0.0 1.0 0.0; 1.0 1.0 3.0]\n"
     ]
    }
   ],
   "source": [
    "A = [1 1 -1 ; 0 1 0 ; 1 1 3]*1.0\n",
    "\n",
    "L, U = decomposicao_LU(A)\n",
    "\n",
    "println(L*U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver Sistema de Equações após Decomposição LU"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolver_pos_LU (generic function with 1 method)"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Essa função resolve um sistema de equações A.x = b que foi decomposto em L.U.x = b.\n",
    "Para isso, primeiro pensamos em U.x como sendo um y e resolvemos L.y = b.\n",
    "Em seguida, resolvemos U.x = y.\n",
    "Como entrada, devem ser passadas as matrizes quadradas L, U e a matriz coluna b.\n",
    "O retorno será a matriz coluna x, resultado do sistema de equações.\"\"\"\n",
    "\n",
    "function resolver_pos_LU(L,U,b)    \n",
    "    y=resolve_triangular_inferior(L,b)\n",
    "    x=resolve_triangular_superior(U,y)\n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[29.0; -22.7; 1.9; 4.5]\n",
      "[29.0, -22.7, 1.9, 4.5]\n"
     ]
    }
   ],
   "source": [
    "A = [1 2 1 3 ; 0 1 3 4 ; 0 0 5 -1 ; 0 0 0 2]*1.0\n",
    "b = [-1 ; 1 ; 5 ; 9]*1.0\n",
    "\n",
    "L,U = decomposicao_LU(A)\n",
    "\n",
    "println(resolver_pos_LU(L,U,b))\n",
    "println(A\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-3.1904761904761907; 3.5714285714285716; -0.33333333333333304; -1.0476190476190477]\n",
      "[-3.190476190476191, 3.571428571428572, -0.3333333333333333, -1.0476190476190474]\n"
     ]
    }
   ],
   "source": [
    "A = [1 1 1 1 ; -1 1 -1 2 ; 3 4 2 1 ; 0 0 3 0]*1.0\n",
    "b = [-1 ; 5 ; 3 ; -1]*1.0\n",
    "\n",
    "L, U = decomposicao_LU(A)\n",
    "\n",
    "println(resolver_pos_LU(L,U,b))\n",
    "println(A\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-6.0; 5.0; 0.0]\n",
      "[-6.0, 5.0, 0.0]\n"
     ]
    }
   ],
   "source": [
    "A = [1 1 -1 ; 0 1 0 ; 1 1 3]*1.0\n",
    "b = [-1 ; 5 ; -1]*1.0\n",
    "\n",
    "L, U = decomposicao_LU(A)\n",
    "\n",
    "println(resolver_pos_LU(L,U,b))\n",
    "println(A\\b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Questão 2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Método de Taylor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "taylor (generic function with 1 method)"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Essa função faz a aproximação de uma função no ponto 'desejado' utilizando o polinômio de taylor\n",
    "centrado no ponto 'referencia'. Para isso, deve-se passar a função, as múltiplas derivadas (quantas \n",
    "quiser - quanto mais melhor) e os pontos de referencia e desejado.\"\"\"\n",
    "\n",
    "function taylor(funcao, derivadas, referencia, desejado)  \n",
    "    grau = length(derivadas)\n",
    "    soma = funcao(referencia)\n",
    "    fat = 1\n",
    "    for i in 1:grau\n",
    "        fat = fat * i\n",
    "        soma = soma + (derivadas[i](referencia)*(desejado-referencia)^i)/fat\n",
    "    end\n",
    "    return soma\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.40580357142857143\n",
      "0.4054651081081644\n"
     ]
    }
   ],
   "source": [
    "f(x) = log(x)\n",
    "fd1(x) = 1/x; fd2(x) = -1/x^2; fd3(x) = 2/x^3; fd4(x) = -6/x^4; fd5(x) = 24/x^5; fd6(x) = -120/x^6; fd7(x) = 720/x^7;\n",
    "der = [fd1,fd2,fd3,fd4,fd5,fd6,fd7]\n",
    "\n",
    "println(taylor(f, der, 1, 1.5))\n",
    "println(log(1.5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9635581840200099\n",
      "0.9320390859672263\n"
     ]
    }
   ],
   "source": [
    "f(x) = sin(x)\n",
    "fd1(x) = cos(x); fd2(x) = -sin(x); fd3(x) = -cos(x); fd4(x) = sin(x); fd5(x) = cos(x); fd6(x) = -sin(x); fd7(x) = -cos(x);\n",
    "der = [fd1,fd2,fd3,fd4,fd5,fd6,fd7]\n",
    "\n",
    "println(taylor(f, der, 1, 1.3))\n",
    "println(sin(1.2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11.067330000000004\n",
      "11.067330000000002\n"
     ]
    }
   ],
   "source": [
    "f(x) = x^5+4x^4-3x^2+1\n",
    "fd1(x) = 5x^4+16x^3-6x; fd2(x) = 20x^3+48x^2-6; fd3(x) = 60x^2+96x; fd4(x) = 120x+96; fd5(x) = 120;\n",
    "der = [fd1,fd2,fd3,fd4,fd5]\n",
    "\n",
    "println(taylor(f, der, 1, 1.3))\n",
    "println(f(1.3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Achar o Zero da Função por Bisseção"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "bissecao (generic function with 2 methods)"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Na bisseção, queremos reduzir o intervalo ao máximo sempre garantindo que a função nos\n",
    "extremos tenha sinais distintos, visto que isso garante, pelo Teorema do Valor Intermediário,\n",
    "a existência de um zero de função nesse intervalo. Sendo assim, passamos a função, os extremos\n",
    "esquerdo e direito do intervalo e o erro máximo esperado.\n",
    "Retornamos a estimativa com um erro menor que o esperado.\"\"\"\n",
    "\n",
    "function bissecao(f, esq, dir, erro = 1e-2)\n",
    "    m=0\n",
    "    \n",
    "    @assert(f(esq)*f(dir)<=0, \"O intervalo passado não possui extremos com sinais distintos. Não podemos garantir um zero da função!\")\n",
    "    \n",
    "    while abs(dir-esq) > erro\n",
    "        m = (esq+dir)/2\n",
    "        res = f(m)\n",
    "        \n",
    "        if res == 0.0\n",
    "            break\n",
    "        elseif res*f(esq) > 0     # verifica se f(m) e f(esq) possuem o mesmo sinal\n",
    "            esq = m\n",
    "        else                      \n",
    "            dir = m\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return m\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "\u001b[91mAssertionError: O intervalo passado não possui extremos com sinais distintos. Não podemos garantir um zero da função!\u001b[39m",
     "output_type": "error",
     "traceback": [
      "\u001b[91mAssertionError: O intervalo passado não possui extremos com sinais distintos. Não podemos garantir um zero da função!\u001b[39m",
      "",
      "Stacktrace:",
      " [1] bissecao(::typeof(f), ::Int64, ::Int64, ::Float64) at .\\In[119]:10",
      " [2] bissecao(::Function, ::Int64, ::Int64) at .\\In[119]:8",
      " [3] top-level scope at In[120]:3",
      " [4] include_string(::Function, ::Module, ::String, ::String) at .\\loading.jl:1091"
     ]
    }
   ],
   "source": [
    "# Testando intervalo sem zero\n",
    "f(x) = x^5+4x^4-3x^2+1\n",
    "bissecao(f,-3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3.798828125"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = x^5+4x^4-3x^2+1\n",
    "bissecao(f,-10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.2822265625"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = sin(x/2)\n",
    "bissecao(f, 1, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Achar o Zero da Função por Ponto Fixo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ponto_fixo"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Usamos o método do ponto fixo para encontrar o zero de função, a ideia é usar substituições\n",
    "sucessivas de forma a encontrar uma interseção entre a função g(x) que definimos e a reta\n",
    "y = x. O truque para achar o zero está em definir g(x) de forma que\n",
    "a função tenha um ponto Z t.q. f(Z) = Z - g(Z) = 0.\n",
    "\n",
    "@param: - função f recebe a função escolhida pelo usuário para achar o zero (no caso, a nossa função g)\n",
    "        - a margem de erro eps\n",
    "        - e o ponto inicial x0\n",
    "\n",
    "@return: - zero da função f(x) (ponto fixo de g(x))\n",
    "         - Retorna um erro de execução caso o método falhe em encontrar (diverge)\n",
    "\"\"\"\n",
    "function ponto_fixo(f, eps, x0)\n",
    "    # Inicializa com infinito para saber se encontramos\n",
    "    ponto_fixo = -Inf\n",
    "    \n",
    "    # quantidade de iterações dadas até encontrar\n",
    "    steps = 1\n",
    "    \n",
    "    # Por conveniência vamos assumir que o método diverge \n",
    "    # se levar pelo menos 10^3 passos\n",
    "    while steps < 10^3 \n",
    "        nx = f(x0)\n",
    "\n",
    "        if abs(nx - x0) < eps\n",
    "            # achamos uma aproximação que respeita a margem de erro \n",
    "            ponto_fixo = nx\n",
    "            break\n",
    "        end\n",
    "\n",
    "        steps += 1\n",
    "        x0 = nx\n",
    "    end\n",
    "    \n",
    "    # método falhou\n",
    "    @assert(ponto_fixo != -Inf, \"Método divergiu com o input dado\")\n",
    "    \n",
    "    # retorna o ponto_fixo da função\n",
    "    return ponto_fixo\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.14558862306442932\n"
     ]
    }
   ],
   "source": [
    "function f(x)\n",
    "    return x*cos(x) - x*x + 8*x - 1\n",
    "end\n",
    "\n",
    "function g(x)\n",
    "    # g(x) é uma função cujo ponto fixo é um zero de f(x)\n",
    "    return (x*cos(x) - x*x -1)/8.0\n",
    "end\n",
    "\n",
    "println(ponto_fixo(g, 0.001, 0.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2891641649379035\n"
     ]
    }
   ],
   "source": [
    "function f(x)\n",
    "    return x^3 - 7*x + 2\n",
    "end\n",
    "\n",
    "function g(x)\n",
    "    # g(x) é uma função cujo ponto fixo é um zero de f(x)\n",
    "    return (x^3 + 2)/7.0\n",
    "end\n",
    "\n",
    "println(ponto_fixo(g, 1e-3, 0.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "\u001b[91mAssertionError: Método divergiu com o input dado\u001b[39m",
     "output_type": "error",
     "traceback": [
      "\u001b[91mAssertionError: Método divergiu com o input dado\u001b[39m",
      "",
      "Stacktrace:",
      " [1] ponto_fixo(::typeof(g), ::Float64, ::Float64) at .\\In[123]:37",
      " [2] top-level scope at In[126]:11",
      " [3] include_string(::Function, ::Module, ::String, ::String) at .\\loading.jl:1091"
     ]
    }
   ],
   "source": [
    "function f(x)\n",
    "    return x^3 - x - 1\n",
    "end\n",
    "\n",
    "# se pegarmos uma g simples nesse caso o método tem que divergir\n",
    "function g(x)\n",
    "    # g(x) é uma função cujo ponto fixo é um zero de f(x)\n",
    "    return (x^3 - 1)\n",
    "end\n",
    "\n",
    "println(ponto_fixo(g, 1e-6, 0.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3247179361449652\n"
     ]
    }
   ],
   "source": [
    "function f(x)\n",
    "    return x^3 - x - 1\n",
    "end\n",
    "\n",
    "# se pegarmos uma g esperta o método tem que convergir\n",
    "function g(x)\n",
    "    # g(x) é uma função cujo ponto fixo é um zero de f(x)\n",
    "    return cbrt(x + 1)\n",
    "end\n",
    "\n",
    "println(ponto_fixo(g, 1e-7, 0.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Achar o Zero da Função por Newton"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "newton (generic function with 2 methods)"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Para o método de Newton, queremos nos aproveitar do ângulo tangente da curva para nos\n",
    "aproximarmos precisamente do zero da função. Isso é feito encontrando a reta tangente a\n",
    "curva no ponto atual e indo para o ponto dessa reta que cruza o eixo x. Para isso, precisamos\n",
    "da função e sua derivada, o ponto inicial do método e o erro máximo esperado.\"\"\"\n",
    "\n",
    "function newton(f, fd, ponto, erro = 1e-8)\n",
    "    # Primeiro descobrimos quantos termos precisamos\n",
    "    termos = 1\n",
    "    while (termos+1)*(2^(termos+1)) < 1/erro\n",
    "        termos = termos+1\n",
    "    end\n",
    "\n",
    "    # Agora calculamos o taylor\n",
    "    for i in 1:termos\n",
    "        ponto = ponto - f(ponto)/fd(ponto)\n",
    "    end\n",
    "    \n",
    "    return ponto\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3.796694419165162"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = x^5+4x^4-3x^2+1\n",
    "fd(x) = 5x^4+16x^3-6x\n",
    "newton(f,fd,-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.283185307179586"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = sin(x/2)\n",
    "fd(x) = 1/2 * cos(x/2)\n",
    "newton(f, fd, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = log(x)\n",
    "fd(x) = 1/x\n",
    "newton(f, fd, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Método de Euler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "metodo_euler (generic function with 1 method)"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" A função consiste em aproximar o valor em y de uma função em um ponto x_ini utilizando\n",
    "apenas a derivada dessa função. Consiste em andar pequenos passos h estimando a curva pela\n",
    "tangente (derivada) da curva no dado ponto.\n",
    "Será necessário passar a derivada da curva, o x e y iniciais, o x final (ao qual desejamos\n",
    "o y relativo a ele) e o h, que é o intervalo andado em x a cada etapa.\n",
    "A função retorna o valor estimado de y no ponto x_final. \"\"\"\n",
    "\n",
    "function metodo_euler(derivada, x_ini, y_ini, x_final, h)\n",
    "    # passos iniciais\n",
    "    xi = x_ini\n",
    "    yi = y_ini\n",
    "\n",
    "    while xi + h <= x_final\n",
    "        # pegamos o próximo valor de y andando pela reta tangente ao \n",
    "        # ponto x_i com um passo de tamanho H\n",
    "        yi += derivada(xi, yi) * h\n",
    "        xi += h\n",
    "    end\n",
    "    \n",
    "    return yi # retornamos o valor de y no estado final\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "53.524117208294335\n"
     ]
    }
   ],
   "source": [
    "df(x, y) = y\n",
    "\n",
    "# queremos aproximar y(4) usando método de euler\n",
    "# sendo y'(x) = y(x)\n",
    "\n",
    "println(metodo_euler(df, 0.0, 1.0, 4, 0.01))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "67.15386504769752\n"
     ]
    }
   ],
   "source": [
    "df(x, y) = y + 3*x\n",
    "\n",
    "# queremos aproximar y(3) usando método de euler\n",
    "# sendo y'(x) = y + 3x\n",
    "\n",
    "println(metodo_euler(df, 0.0, 1.0, 3, 0.01))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8650971211771716\n"
     ]
    }
   ],
   "source": [
    "df(x, y) = exp(-x)\n",
    "\n",
    "# queremos aproximar y(5) usando método de euler\n",
    "# sendo y'(x) = x^2 + 3x\n",
    "\n",
    "println(metodo_euler(df, 0.0, 0.0, 2, 0.001))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver EDO com Problema de Valor Inicial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolver_PVI"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Usamos o método de euler para resolver o problema de valor inicial. O método é usado para\n",
    "aproximar o valor de EDOs, ele faz isso usando aproximações de retas tangentes para descrever\n",
    "a curva repetidas vezes.\n",
    "\n",
    "@param: - derivada de f(x, y(x))\n",
    "        - x inicial\n",
    "        - y inicial (isso é o valor inicial conhecido no nosso problema)\n",
    "        - x_final denota o ponto x em que queremos uma aproximação para f\n",
    "        - n denota em quantos subintervalos vamos dividir nosso intervalo\n",
    "\n",
    "@return: - aproximação para f no ponto x_final\n",
    "\"\"\"\n",
    "function resolver_PVI(derivada, x_inicial, y_inicial, x_final, n)\n",
    "    # tamanho do intervalo que usaremos dada a discretização desejada\n",
    "    h = (x_final - x_inicial) / n\n",
    "    \n",
    "    yi = metodo_euler(derivada, x_inicial, y_inicial, x_final, h)\n",
    "    \n",
    "    return yi # retornamos o valor de y no estado final\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41.144777789250874"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df(x, y) = y\n",
    "\n",
    "# queremos aproximar y(4) usando método de euler\n",
    "# sendo y'(x) = y(x), y(0) = 1 e dividindo o intervalo em 40 partes\n",
    "\n",
    "resolver_PVI(df, 0.0, 1.0, 4, 40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20.0"
      ]
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df(x, y) = y + 3*x\n",
    "\n",
    "# queremos aproximar y(3) usando método de euler\n",
    "# sendo y'(x) = y + 3x, y(3) = 1 e dividindo o intervalo em 3 partes\n",
    "\n",
    "resolver_PVI(df, 0.0, 1.0, 3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0987701305162527"
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df(x, y) = exp(-x)\n",
    "\n",
    "# queremos aproximar y(5) usando método de euler\n",
    "# sendo y'(x) = x^2 + 3x, y(2) = 10 e dividindo o intervalo em 40 partes\n",
    "\n",
    "resolver_PVI(df, 0.0, 0.0, 2, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver um Sistema Ax=b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolver_um_sistema (generic function with 1 method)"
      ]
     },
     "execution_count": 152,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Essa função é um encapsulamento da resolução de sistema do formato 'A.x = b' \n",
    "utilizando decomposição LU. Para isso, é passado como parâmetro a matriz A e o \n",
    "vetor b, e o resultado será o vetor x.\"\"\"\n",
    "\n",
    "function resolver_um_sistema(A,b) \n",
    "    L, U = decomposicao_LU(A)\n",
    "    x = resolver_pos_LU(L,U,b)\n",
    "    return x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.1040532365396274; -5.474289171203875; -0.6170598911070786; 3.5075620084694505]\n",
      "[2.104053236539625, -5.474289171203873, -0.6170598911070775, 3.507562008469449]\n"
     ]
    }
   ],
   "source": [
    "A=[1.0 3.0 2.0 7.0 ; 10.0 3.0 10.0 7.0 ; 3.0 5.0 18.0 22.0 ; 5.0 2.0 1.0 6.0]\n",
    "b=[9.0 ; 23.0 ; 45.0 ; 20.0]\n",
    "\n",
    "println(resolver_um_sistema(A,b))\n",
    "println(A\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-8.307692307692305; 6.692307692307692; -1.3846153846153844]\n",
      "[-8.307692307692307, 6.692307692307692, -1.3846153846153846]\n"
     ]
    }
   ],
   "source": [
    "A=[1.0 3.0 2.0 ; -2.0 -1.0 5.0 ; 1.0 1.0 1.0]\n",
    "b=[9.0 ; 3.0 ; -3.0]\n",
    "\n",
    "println(resolver_um_sistema(A,b))\n",
    "println(A\\b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.4; 0.2]\n",
      "[2.4, 0.2]\n"
     ]
    }
   ],
   "source": [
    "A=[1.0 3.0 ; -1.0 2.0]\n",
    "b=[3.0 ; -2.0]\n",
    "\n",
    "println(resolver_um_sistema(A,b))\n",
    "println(A\\b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver EDO com Problema de Valor de Contorno"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolver_PVC (generic function with 1 method)"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Essa função utiliza da aproximação do valor de y por meio da sua segunda derivada. Para isso, além da segunda derviada\n",
    "    recebemos dois pontos, e uma quantidade n de pontos utilizados entre eles para aproximar a curva. \"\"\"\n",
    "\n",
    "function resolver_PVC(segunda_derivada, x_inicial, y_inicial, x_final, y_final, n)  #Não dá pra fazer genericamente\n",
    "    # Calcula o h\n",
    "    h = (x_final - x_inicial)/n\n",
    "\n",
    "    # Inicializa a matriz vazia\n",
    "    A = []\n",
    "    \n",
    "    # Vetor auxiliar para construir a matriz \n",
    "    v = [-2, 1]\n",
    "    \n",
    "    # Preenche a primeira linha com 0's\n",
    "    for i = 3:n-1\n",
    "        push!(v, 0)\n",
    "    end\n",
    "    \n",
    "    # Coloca o vetor auxiliar na matriz\n",
    "    push!(A, v)\n",
    "    \n",
    "    # Constrói a matriz por meio dos vetores auxiliares\n",
    "    for i = 2:n-2\n",
    "        v = []\n",
    "        for j = 1:i-2\n",
    "            push!(v, 0)\n",
    "        end\n",
    "        push!(v, 1)\n",
    "        push!(v, -2)\n",
    "        push!(v, 1)\n",
    "        for j = i+2:n-1\n",
    "            push!(v, 0)\n",
    "        end\n",
    "        v = vcat(v...)\n",
    "        push!(A, v)\n",
    "    end\n",
    "    \n",
    "    # Termina de construir a matriz\n",
    "    v = []\n",
    "    for i = 1:n-3\n",
    "        push!(v, 0)\n",
    "    end\n",
    "    push!(v, 1)\n",
    "    push!(v, -2)\n",
    "    push!(A, v)\n",
    "    \n",
    "    # Gera o vetor b de acordo com as aproximações da segunda derivada\n",
    "    b = Any[segunda_derivada(x_inicial)*h^2 - y_inicial]\n",
    "    for i = 2:n-2\n",
    "        push!(b, segunda_derivada((x_inicial + h*(i)))*h^2)\n",
    "    end\n",
    "    push!(b, segunda_derivada(x_final)*h^2 - y_final)\n",
    "    b = vcat(b...)\n",
    "    A = hcat(A...)\n",
    "\n",
    "    # Resolve o sistema e acha os pontos y\n",
    "    y = resolver_um_sistema(A,b)\n",
    "    return y\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
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     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# Teste 1\n",
    "\n",
    "g(x) = x^2 - 5x + 6\n",
    "gd(x) = 2x - 5\n",
    "gdd(x) = 2\n",
    "x_inicial = 0\n",
    "y_inicial = 6\n",
    "x_final = 5\n",
    "y_final = 6\n",
    "n = 30\n",
    "\n",
    "y = Any[y_inicial]\n",
    "for e in resolver_PVC(gdd, x_inicial, y_inicial, x_final, y_final, n)\n",
    "    push!(y, e)\n",
    "end\n",
    "push!(y, y_final)\n",
    "\n",
    "plot(g, x_inicial, x_final)\n",
    "\n",
    "h = (x_final - x_inicial)/(n)\n",
    "scatter!(x_inicial:h:x_final, y, leg=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
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     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "#Teste 2\n",
    "\n",
    "g(x) = sin(x)\n",
    "gd(x) = cos(x)\n",
    "gdd(x) = -sin(x)\n",
    "x_inicial = 0\n",
    "y_inicial = 0\n",
    "x_final = 2*pi\n",
    "y_final = 0\n",
    "n = 30\n",
    "\n",
    "y = Any[y_inicial]\n",
    "for e in resolver_PVC(gdd, x_inicial, y_inicial, x_final, y_final, n)\n",
    "    push!(y, e)\n",
    "end\n",
    "push!(y, y_final)\n",
    "\n",
    "plot(g, x_inicial, x_final)\n",
    "\n",
    "h = (x_final - x_inicial)/(n)\n",
    "scatter!(x_inicial:h:x_final, y, leg=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Teste 3\n",
    "\n",
    "g(x) = log(x)\n",
    "gd(x) = 1/x\n",
    "gdd(x) = -1/x^2\n",
    "x_inicial = 1\n",
    "y_inicial = 0\n",
    "x_final = exp(2)\n",
    "y_final = 2\n",
    "n = 30\n",
    "\n",
    "y = Any[y_inicial]\n",
    "for e in resolver_PVC(gdd, x_inicial, y_inicial, x_final, y_final, n)\n",
    "    push!(y, e)\n",
    "end\n",
    "push!(y, y_final)\n",
    "\n",
    "plot(g, x_inicial, x_final)\n",
    "\n",
    "h = (x_final - x_inicial)/(n)\n",
    "scatter!(x_inicial:h:x_final, y, leg=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gerar a Matriz de Vandermonde"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matriz_de_vandermonde (generic function with 1 method)"
      ]
     },
     "execution_count": 160,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Essa função recebe uma lista de pontos x e retorna uma matriz cujas colunas são \n",
    "dadas por essa lista de pontos elevado a uma potência. A primeira coluna possui apenas 1s, \n",
    "a segunda é potência de 1, a terceira de 2, e assim em diante até o grau especificado. \"\"\"\n",
    "\n",
    "function matriz_de_vandermonde(pontos_x, grau)\n",
    "    matriz = ones(length(pontos_x))\n",
    "    for i in 1:grau\n",
    "        matriz = hcat(matriz, pontos_x.^i)\n",
    "    end \n",
    "    return matriz\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×4 Array{Float64,2}:\n",
       " 1.0   1.0   1.0   1.0\n",
       " 1.0   2.0   4.0   8.0\n",
       " 1.0   4.0  16.0  64.0\n",
       " 1.0   2.0   4.0   8.0\n",
       " 1.0  -1.0   1.0  -1.0"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matriz_de_vandermonde([1,2,4,2,-1], 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×2 Array{Float64,2}:\n",
       " 1.0   1.0\n",
       " 1.0  -1.0\n",
       " 1.0   2.0\n",
       " 1.0   3.0\n",
       " 1.0  -5.0"
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matriz_de_vandermonde([1,-1,2,3,-5], 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×6 Array{Float64,2}:\n",
       " 1.0  2.0  4.0   8.0  16.0   32.0\n",
       " 1.0  3.0  9.0  27.0  81.0  243.0\n",
       " 1.0  1.0  1.0   1.0   1.0    1.0\n",
       " 1.0  2.0  4.0   8.0  16.0   32.0"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matriz_de_vandermonde([2,3,1,2], 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fazer Interpolação de Curva por Vandermonde"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "interpolacao_coeficientes (generic function with 1 method)"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Nessa função, realizamos o processo de interpolação de curvas utilizando a matriz de \n",
    "Vandermonde para o sistema. Consiste em passarmos os pontos de x e y como duas listas \n",
    "separadas e o grau do polinômio, e a função retorna os coeficientes\n",
    "do polinômio, na ordem do grau 0 até o grau especificado.\"\"\"\n",
    "\n",
    "function interpolacao_coeficientes(pontos_x, pontos_y, grau)\n",
    "    V = matriz_de_vandermonde(pontos_x,grau)\n",
    "    coeficientes = resolver_um_sistema(V,pontos_y)\n",
    "    return coeficientes\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×1 Array{Float64,2}:\n",
       "  9.500000000000004\n",
       " -8.000000000000004\n",
       "  1.0000000000000009\n",
       " -0.10000000000000007"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = 9.5 -8*x + x^2 - 0.1x^3\n",
    "interpolacao_coeficientes([1,5,6,9], [f(1),f(5),f(6),f(9)], 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×1 Array{Float64,2}:\n",
       " -3.0\n",
       " -4.999999999999997\n",
       " -2.999999999999999\n",
       "  0.39999999999999714\n",
       " -2.220446049250313e-16\n",
       "  1.0000000000000007"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = -3 - 5*x - 3*x^2 + 0.4*x^3 + x^5\n",
    "interpolacao_coeficientes([-1,0,1.5,1,-2,2], [f(-1),f(0),f(1.5),f(1),f(-2),f(2)], 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×1 Array{Float64,2}:\n",
       " 3.0\n",
       " 5.0\n",
       " 0.0\n",
       " 0.0"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = 3 + 5*x\n",
    "interpolacao_coeficientes([0,1,-1,2], [f(0),f(1),f(-1),f(2)], 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Obter Polinômio por Lagrange"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "lagrange (generic function with 1 method)"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Polinômio de Lagrange dado um conjunto de pontos e o (grau - 1) do polinômio desejado\n",
    "\n",
    "@param: - coordenadas x\n",
    "        - coordenadas y\n",
    "        - (grau - 1) desejado para o polinômio de lagrange (se grau = 3, teremos P_2(x))\n",
    "\n",
    "@return: - aproximação para f no ponto x_final\n",
    "\"\"\"\n",
    "\n",
    "function lagrange(pontos_x, pontos_y, grau)\n",
    "    \n",
    "    # Vamos usar a classe Polynomials para criar um polinômio de uma variável e\n",
    "    # facilitar a nossa vida na hora de fazer as operações nele\n",
    "    poly = Polynomial(0.0)\n",
    "\n",
    "    for i in 1 : grau\n",
    "        # no primeiro instante dizemos que o coeficiente é y_i por conveniência mesmo,\n",
    "        # temos então que o coeficiente pode ser computado da seguinte forma:\n",
    "        # coeff_i = prod_{i=1}^grau (x - x_i) / (x_i - x_j)\n",
    "        \n",
    "        coeff = Polynomial(pontos_y[i])\n",
    "        \n",
    "        for j in 1 : grau\n",
    "            if i == j\n",
    "                # caso especial, não consideramos os\n",
    "                # mesmos índices\n",
    "                continue\n",
    "            end\n",
    "\n",
    "            # adicionamos esse coeficiente no polinômio\n",
    "            # ele vai ter como numerador o polinômio (x - x_i) e denominador (x_i - x_j)\n",
    "            coeff = coeff * Polynomial([-pontos_x[j], 1.0]) / (pontos_x[i] - pontos_x[j])\n",
    "        end\n",
    "\n",
    "        # adicionamos o coeficiente no polinômio\n",
    "        # nesse momento o polinômio já tem todos coeficientes até grau i computados\n",
    "        poly += coeff\n",
    "        \n",
    "    end\n",
    "\n",
    "    # retornamos polinômio na forma [1, x, x^2, ..., x^(grau-1)]\n",
    "    return poly\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1.733333333333333, 4.1, -0.36666666666666675]\n"
     ]
    }
   ],
   "source": [
    "x = [1.0, 2.0, 7.0]\n",
    "y = [2.0, 5.0, 9.0]\n",
    "\n",
    "println(coeffs(lagrange(x, y, 3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.5, 0.41666666666666674, -0.08333333333333331]\n"
     ]
    }
   ],
   "source": [
    "x = [-1.0, 2.0, 3.0, 5.0]\n",
    "y = [0.0, 1.0, 1.0, 2.0]\n",
    "\n",
    "println(coeffs(lagrange(x, y, 3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-2.0, 1.5]\n"
     ]
    }
   ],
   "source": [
    "x = [0.0, 2.0, 4.0]\n",
    "y = [-2.0, 1.0, 4.0]\n",
    "\n",
    "println(coeffs(lagrange(x, y, 2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fazer Interpolação de Curva por Lagrange"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "interpolacao_funcao (generic function with 1 method)"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Essa função faz a interpolação dos pontos x e y dados e retorna uma função com o grau passado.\n",
    "Vale notar que a função é apenas um encapsulamento da função que gera o lagrange.\"\"\"\n",
    "\n",
    "function interpolacao_funcao(pontos_x,pontos_y, grau)\n",
    "    polinomio =lagrange(pontos_x,pontos_y,grau)\n",
    "    return polinomio\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;1.733333333333333 &#43; 4.1&#8729;x &#45; 0.36666666666666675&#8729;x<sup>2</sup>"
      ],
      "text/latex": [
       "$-1.733333333333333 + 4.1\\cdot x - 0.36666666666666675\\cdot x^{2}$"
      ],
      "text/plain": [
       "Polynomial(-1.733333333333333 + 4.1*x - 0.36666666666666675*x^2)"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [1.0, 2.0, 7.0]\n",
    "y = [2.0, 5.0, 9.0]\n",
    "\n",
    "interpolacao_funcao(x,y,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.5 &#43; 0.41666666666666674&#8729;x &#45; 0.08333333333333331&#8729;x<sup>2</sup>"
      ],
      "text/latex": [
       "$0.5 + 0.41666666666666674\\cdot x - 0.08333333333333331\\cdot x^{2}$"
      ],
      "text/plain": [
       "Polynomial(0.5 + 0.41666666666666674*x - 0.08333333333333331*x^2)"
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [-1.0, 2.0, 3.0, 5.0]\n",
    "y = [0.0, 1.0, 1.0, 2.0]\n",
    "\n",
    "interpolacao_funcao(x, y, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "&#45;2.0 &#43; 1.5&#8729;x"
      ],
      "text/latex": [
       "$-2.0 + 1.5\\cdot x$"
      ],
      "text/plain": [
       "Polynomial(-2.0 + 1.5*x)"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [0.0, 2.0, 4.0]\n",
    "y = [-2.0, 1.0, 4.0]\n",
    "\n",
    "interpolacao_funcao(x, y, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aplicação de um x em uma função"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "aplica (generic function with 1 method)"
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Função auxiliar que transforma uma lista de coeficientes num polinômio centrado em x0 que também é dado como entrada.\n",
    "\n",
    "Ex de entrada: [ c, b, a ]\n",
    "Saída: ax^2 + bx + c\n",
    "\"\"\"\n",
    "\n",
    "function aplica(coeficientes, valor_de_x) # Recebe a lista de coeficientes e o valor x0 \n",
    "    grau_x = 1                             # Começamos com x^0 no polinômio, ou seja, o termo de coeficientes constante\n",
    "    valor_do_polinomio_no_x = 0            # Polinômio começa com valor igual a 0\n",
    "    for i = 1:length(coeficientes)         \n",
    "        valor_do_polinomio_no_x += coeficientes[i] * grau_x # x0^(i-1) * C_i\n",
    "        grau_x *= valor_de_x # x0^(i-1) = x0^i\n",
    "    end\n",
    "    return valor_do_polinomio_no_x # retorna o valor do polinômio em x0\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "f(x) = 1+3x-x^3\n",
    "func = [1,3,0,-1]\n",
    "\n",
    "println(aplica(func, 1))\n",
    "println(f(1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n",
      "4\n"
     ]
    }
   ],
   "source": [
    "f(x) = 4+x-x^2+x^3\n",
    "func = [4,1,-1,1]\n",
    "\n",
    "println(aplica(func, 0))\n",
    "println(f(0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "f(x) = -x+x^3+x^4\n",
    "func = [0,-1,0,1,1]\n",
    "\n",
    "println(aplica(func, 1))\n",
    "println(f(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolver um Sistema por Mínimos Quadrados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "resolver_um_sistema_por_minimos_quadrados (generic function with 1 method)"
      ]
     },
     "execution_count": 190,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Resolvemos um sistema por mínimos quadrados aproveitando a função já implementada que resolve sistemas da forma\n",
    "    Ax = b. Como a fórmula para mínimos quadrados é A'Ax = A'b, passamos como entrada apenas A e b, e para a função aninhada\n",
    "    resolver um sistema passamos A'A e A'b. Como saída temos o vetor de x solução do sistema. \"\"\"\n",
    "\n",
    "function resolver_um_sistema_por_minimos_quadrados(A,b) \n",
    "    x=resolver_um_sistema(A'*A,A'*b) # Resolve o Sistema A'Ax = A'b uma vez que sabemos resolver o sistema Ax=b\n",
    "    return x                         # Retorna a solução do sistema, ou seja, o vetor x\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regressão Linear de Funções Lineares"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "regressão_coeficientes_lineares (generic function with 2 methods)"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Com coeficientes lineares usamos polinômios para fazer as regressões. Dessa forma, precisamos da matriz de Vandermonde\n",
    "    que representa o grau dos valores de x. Em suma, teremos uma matriz alta de Vandermonde e o vetor de b que contêm \n",
    "    os valores de y dados e retornaremos os coeficientes do polinômio que minimiza a soma dos quadrados das distâncias. \"\"\"\n",
    "\n",
    "\n",
    "function regressão_coeficientes_lineares(pontos_x,pontos_y, grau) \n",
    "    V=matriz_de_vandermonde(pontos_x,grau)  #Matriz alta de Vandermonde -> muitas linhas e poucas colunas e o grau do polinômio.\n",
    "    coeficientes= resolver_um_sistema_por_minimos_quadrados(V,pontos_y) # retorna os coeficientes do polinômio por mínimos quadrados.\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      ]
     },
     "execution_count": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 21\n",
    "x = range(-10, 10, length=n)\n",
    "y = zeros(n)\n",
    "y[11] = 1;\n",
    "\n",
    "a = regressão_coeficientes_lineares(x, y, 1) # recebe os coeficientes com polinômio de grau 1: c0 + c1*x\n",
    "println(a)  # printa os coeficientes\n",
    "\n",
    "# Plota os pontos junto com a curva que conseguimos que minimiza a soma dos quadrados das distâncias\n",
    "scatter(x,y,leg=false)\n",
    "plot!(x -> a[1] + a[2] * x, extrema(x)..., c=:red, lw=2)\n",
    "title!(\"Mínimos Quadrados\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1035.3475650780576; 54.54998093497881]\n"
     ]
    },
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   "source": [
    "# Dados retirados da relação dos pesos de tipo um peixe com suas alturas\n",
    "x = [23.2, 24, 23.9, 26.3, 26.5, 26.8, 26.8, 27.6, 27.6, 28.5, 28.4, 28.7, 29.1, 29.5, 29.4, 29.4, 30.4, 30.4, 30.9, 31, 31.3, 31.4, 31.5, 31.8, 31.9, 31.8, 32, 32.7, 32.8, 33.5, 35, 35, 36.2, 37.4, 38]\n",
    "y = [242, 290, 340, 363, 430, 450, 500, 390, 450, 500, 475, 500, 500, 340, 600, 600, 700, 700, 610, 650, 575, 685, 620, 680, 700, 725, 720, 714, 850, 1000, 920, 955, 925, 975, 950]\n",
    "\n",
    "a = regressão_coeficientes_lineares(x, y, 1) # recebe os coeficientes\n",
    "println(a)  # printa os coeficientes\n",
    "\n",
    "# Plota os pontos junto com a curva que conseguimos que minimiza a soma dos quadrados das distâncias\n",
    "scatter(x,y,leg=false)\n",
    "plot!(x -> a[1] + a[2] * x, extrema(x)..., c=:red, lw=2)\n",
    "title!(\"Mínimos Quadrados\")"
   ]
  },
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     },
     "execution_count": 210,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Testando com uma função especificamente linear\n",
    "x = []\n",
    "y = []\n",
    "\n",
    "for i in 1:1000\n",
    "    push!(x, i)\n",
    "    push!(y, i)\n",
    "    \n",
    "end\n",
    "\n",
    "a = regressão_coeficientes_lineares(x, y, 1) # recebe os coeficientes\n",
    "println(a)  # printa os coeficientes\n",
    "\n",
    "# Plota os pontos junto com a curva que conseguimos que minimiza a soma dos quadrados das distâncias\n",
    "scatter(x,y,leg=false)\n",
    "plot!(x -> a[1] + a[2] * x, extrema(x)..., c=:red, lw=2)\n",
    "title!(\"Mínimos Quadrados\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regressão Linear de Funções Exponenciais"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "regressão_função_exponencial (generic function with 1 method)"
      ]
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" A função regressão_função_exponencial minimiza a soma dos quadrados das distâncias de acordo com a função c0*e^(x*c1).\n",
    "    Sendo assim, para calcular os coeficientes precisamos usar log natural como um dicionário, que \"linenizará\" a função.\n",
    "    Como entrada temos os valores de x, os valores de y correspondentes aos valores de x. E como saida o retorno dos\n",
    "    coeficientes c0 e c1. \"\"\"\n",
    "\n",
    "function regressão_função_exponencial(pontos_x,pontos_y) \n",
    "    #dicionário (\"linearização\") é feito manualmente ou por uma tabela\n",
    "    pontos_y_barra = zeros(length(pontos_y))\n",
    "    for i in 1:length(pontos_x)\n",
    "        pontos_y_barra[i] = log(pontos_y[i]) # aplicamos log nos valores de y e na direita da equação temos c0 + x*c1\n",
    "    end\n",
    "    pontos_x_barra = pontos_x # os valores de x se mantém os mesmos\n",
    "    \n",
    "    #resolver regresssão com coeficientes lineares no \"mundo barra\"\n",
    "    coeficientes_barra = regressão_coeficientes_lineares(pontos_x_barra,pontos_y_barra, 1)\n",
    "    return coeficientes_barra\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
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     "execution_count": 212,
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   ],
   "source": [
    "# Data Set aleatório\n",
    "pontos_x = [45, 99, 31, 57, 37, 85, 21, 64, 17, 41, 103]\n",
    "pontos_y = [33, 72, 19, 27, 23, 62, 24, 32, 18, 36, 76]\n",
    "\n",
    "# Recebe os coeficientes com a função exponencial\n",
    "Coeficientes = regressão_função_exponencial(pontos_x, pontos_y)\n",
    "polinomio(x) = (ℯ^Coeficientes[1])*ℯ^(x*Coeficientes[2]) # define o polinômio \"voltando para o mundo original\" aplicando e\n",
    "# Faz o plot da regressão junto os valores dados\n",
    "scatter(pontos_x,pontos_y,leg=false)\n",
    "plot!(pontos_x -> polinomio(pontos_x), c=:red, lw=2)\n",
    "title!(\"Regressão Exponencial\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 213,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Caso escolhamos uma função de potencia temos uma interpolação\n",
    "pontos_x = []\n",
    "pontos_y = []\n",
    "\n",
    "for i in 0:14\n",
    "    push!(pontos_x, i)\n",
    "    push!(pontos_y, 2^i)\n",
    "end    \n",
    "\n",
    "Coeficientes = regressão_função_exponencial(pontos_x, pontos_y)\n",
    "polinomio(x) = (ℯ^Coeficientes[1])*ℯ^(x*Coeficientes[2])\n",
    "scatter(pontos_x,pontos_y,leg=false)\n",
    "plot!(pontos_x -> polinomio(pontos_x), c=:red, lw=2)\n",
    "title!(\"Regressão Exponencial\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [
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     "execution_count": 214,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Data set que representa o número de casos de covid a partir de alguns dias\n",
    "pontos_x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];\n",
    "pontos_y = [78134, 84040, 92300, 100567, 130795, 139200, 165919, 199314, 221540, 243737, 250612, 254959, 266444, 278993,\n",
    "            300915, 342436, 353974, 408428, 501159];\n",
    "\n",
    "Coeficientes = regressão_função_exponencial(pontos_x, pontos_y)\n",
    "polinomio(x) = (ℯ^Coeficientes[1])*ℯ^(x*Coeficientes[2])\n",
    "scatter(pontos_x,pontos_y,leg=false)\n",
    "plot!(pontos_x -> polinomio(pontos_x), c=:red, lw=2)\n",
    "title!(\"Regressão Exponencial\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "regressão_função_potencia_x (generic function with 1 method)"
      ]
     },
     "execution_count": 215,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" A função regressão_função_potencia_x minimiza a soma dos quadrados das distâncias de acordo com a função c0*x^(c1).\n",
    "    Sendo assim, para calcular os coeficientes precisamos usar algum log como um dicionário, que \"linenizará\" a função.\n",
    "    Escolhi log2 mas talvez possa ser interessante passar o valor do log por parâmetro, por mais que o Julia não tenha\n",
    "    suporte para valores aleatórios de log.\n",
    "    Como entrada temos os valores de x, os valores de y correspondentes aos valores de x. E como saida o retorno dos\n",
    "    coeficientes c0 e c1. \"\"\"\n",
    "\n",
    "function regressão_função_potencia_x(pontos_x, pontos_y) #coeficientes não lineares\n",
    "    #dicionário (\"linearização\") é feito manualmente ou por uma tabela. É necessário usar o log tanto para os valores\n",
    "    # de x quanto para os valores de y visto que log(y) = log(c0) + log(x)*c1\n",
    "    pontos_y_barra = zeros(length(pontos_y))\n",
    "    pontos_x_barra = zeros(length(pontos_x))\n",
    "    for i in 1:length(pontos_x)\n",
    "        pontos_y_barra[i] = log2(pontos_y[i])\n",
    "        pontos_x_barra[i] = log2(pontos_x[i])\n",
    "    end\n",
    "\n",
    "    #resolver regresssão com coeficientes lineares no \"mundo barra\"\n",
    "    coeficientes_barra = regressão_coeficientes_lineares(pontos_x_barra,pontos_y_barra, 1)\n",
    "    return coeficientes_barra\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
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     "execution_count": 217,
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   "source": [
    "# Dados quaisquer\n",
    "x = [0.1, 0.2, 0.4, 0.8, 0.9]\n",
    "y = [22, 43, 84, 210, 320]\n",
    "\n",
    "# Pega os coeficientes de acordo com a função F(x) = c0*x^c1\n",
    "A = regressão_função_potencia_x(x, y)\n",
    "# Voltamos para \"o mundo normal\" ao elevar\n",
    "plot(x -> 2^A[1]*(x^A[2]), 0, 1, c=:red, lw=2)\n",
    "scatter!(x,y, leg=false, c=:cyan)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[14.999286479412877; 0.8189496652467084]\n"
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     "execution_count": 218,
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   ],
   "source": [
    "# DataSet casos de covid numa cidade\n",
    "pontos_x = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]\n",
    "pontos_y = [78134, 84040, 92300, 100567, 130795, 139200, 165919, 199314, 221540, 243737, 250612, 254959, 266444, 278993,\n",
    "            300915, 342436, 353974, 408428, 501159] \n",
    "\n",
    "B = regressão_função_potencia_x(pontos_x, pontos_y)\n",
    "println(B)\n",
    "# Plota \"voltando para o mundo normal\" aplicando log2\n",
    "scatter(pontos_x,pontos_y,leg=false)\n",
    "plot!(x -> 2^B[1]*(x^B[2]), c=:red, lw=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {},
   "outputs": [
    {
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     "output_type": "stream",
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 219,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#DataSet dos Peixes\n",
    "pontos_x = [23.2, 24, 23.9, 26.3, 26.5, 26.8, 26.8, 27.6, 27.6, 28.5, 28.4, 28.7, 29.1, 29.5, 29.4, 29.4, 30.4, 30.4, 30.9, 31, 31.3, 31.4, 31.5, 31.8, 31.9, 31.8, 32, 32.7, 32.8, 33.5, 35, 35, 36.2, 37.4, 38]\n",
    "pontos_y = [242, 290, 340, 363, 430, 450, 500, 390, 450, 500, 475, 500, 500, 340, 600, 600, 700, 700, 610, 650, 575, 685, 620, 680, 700, 725, 720, 714, 850, 1000, 920, 955, 925, 975, 950]\n",
    "D = regressão_função_potencia_x(pontos_x, pontos_y)\n",
    "println(D)\n",
    "scatter(pontos_x,pontos_y,leg=false)\n",
    "plot!(x -> 2^D[1]*(x^D[2]), c=:red, lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integração pela Regra do Retângulo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "regra_do_retangulo (generic function with 1 method)"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O objetivo dessa função é aproximar a integração da função no intervalo [a,b] utilizando \n",
    "o somatório de n retângulos dados pela base sendo o intervalo [x_k,x_k+1] e a altura como a \n",
    "média das funções nesses dois pontos. O retorno será a estimativa da integral.\"\"\"\n",
    "\n",
    "function regra_do_retangulo(funcao,a,b,n)\n",
    "    integral = 0.0\n",
    "    h = (b-a)/n \n",
    "    for i in 1:n\n",
    "        f = (funcao( a+h*i ) + funcao( a+h*(i-1) ))/2\n",
    "        integral = integral + f*h\n",
    "    end\n",
    "    return integral\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "70.83750000000002"
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = 2x^2-x\n",
    "regra_do_retangulo(f,0,5,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.047082902554811"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = log(x)\n",
    "regra_do_retangulo(f,1,5,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0408340855860843e-15"
      ]
     },
     "execution_count": 223,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = sin(x)\n",
    "regra_do_retangulo(f,-10,10,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integração pela Regra do Trapézio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "regra_do_trapezio (generic function with 1 method)"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O objetivo dessa função é aproximar a integração da função no intervalo [a,b] utilizando o \n",
    "somatório de n trapézios dados pelo intervalo [x_k,x_k+1] e as duas bases como as funções nesses \n",
    "dois pontos. O retorno será a estimativa da integral.\"\"\"\n",
    "\n",
    "function regra_do_trapezio(funcao,a,b,n)\n",
    "    soma = f(a)+f(b)\n",
    "    for i = 1:n-1\n",
    "        x = a + i*(b-a)/n\n",
    "        soma = soma + 2*f(x)\n",
    "    end\n",
    "    \n",
    "    return soma * (b-a) / (2*n)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "70.8375"
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = 2x^2-x\n",
    "regra_do_trapezio(f,0,5,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.047082902554812"
      ]
     },
     "execution_count": 226,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = log(x)\n",
    "regra_do_trapezio(f,1,5,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7763568394002506e-16"
      ]
     },
     "execution_count": 227,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = sin(x)\n",
    "regra_do_trapezio(f,-10,10,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integração pela Regra do Simpson"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "regra_de_simpson (generic function with 1 method)"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" O objetivo dessa função é aproximar a integração da função no intervalo [a,b] dividindo-o\n",
    "em n (sendo n necessariamente par) subintervalos e utilizando a regra de simpson em cada um deles.\n",
    "A regra de simpson em cada um desses subintervalos consiste em aproximar o valor da integral nesse\n",
    "subintervalo pegando 3 pontos (as duas extremidades do subintervalo e um médio), e logo em seguida\n",
    "aproximando a curva que os descreve usando um polinômio interpolador de grau dois.\n",
    "\n",
    "A função retorna uma aproximação para a interval definida dada pelo input\n",
    "\n",
    "ERRO: Essa aproximação tem erro inferior a f^4(x) * 1/180 * (b - a)[(b - a)/n]^4, sendo\n",
    "f^4(x) o valor máximo da quarta derivada de f(x) no intervalo [a, b]\n",
    "\n",
    "@param: - f, função f(x) que queremos integrar\n",
    "        - a, primeiro ponto do intervalo\n",
    "        - b, último ponto do intervalo\n",
    "        - n denota a metade da quantidade de subintervalos que pretendemos dividir nosso intervalo\n",
    "\n",
    "@return: - aproximação para integral de f no intervalo [a,b] dividido em 2n partes\n",
    "\"\"\"\n",
    "\n",
    "# regra de simpson composta\n",
    "function regra_de_simpson(f, a, b, n)\n",
    "    @assert(n % 2 == 0, \"você precisa escolher um n par!\")\n",
    "    \n",
    "    # h denota o tamanho do subintervalo\n",
    "    h = (b - a) / n / 2.0\n",
    "\n",
    "    # adicionamos de antemão a contribuição dos\n",
    "    # pontos nas extremidades\n",
    "    S = f(a) + f(b)\n",
    "\n",
    "    # vamos somar agora a imagem dos pontos médios\n",
    "    for i in 1 : (n << 1)-1\n",
    "        if (i & 1 == 1)\n",
    "            # termo ímpar tem peso 4\n",
    "            S += f(a + h * i) * 4\n",
    "        else \n",
    "            # termo par tem peso 2\n",
    "            S += f(a + h * i) * 2\n",
    "        end\n",
    "    end\n",
    "\n",
    "    return S / 3.0 * h\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-150.87986847952268\n"
     ]
    }
   ],
   "source": [
    "function f(x)\n",
    "    return x*x - 16*x - exp(x)\n",
    "end\n",
    "\n",
    "println(regra_de_simpson(f, 1.0, 4.0, 48))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8820813907624211\n"
     ]
    }
   ],
   "source": [
    "function f(x) \n",
    "    return exp(-x^2)\n",
    "end\n",
    "\n",
    "# nesse caso temos uma integral sem uma primitiva analítica\n",
    "println(regra_de_simpson(f, 0.0, 2.0, 2112));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0000000000000049\n"
     ]
    }
   ],
   "source": [
    "function f(x) \n",
    "    return cos(x)\n",
    "end\n",
    "\n",
    "println(regra_de_simpson(f, 0.0, acos(-1.0)/2.0, 5322))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Obter o Máximo de uma Função"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "maximar_uma_funcao (generic function with 4 methods)"
      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A estratégia é parecida com a dos pontos críticos, com exceção que checamos todos eles\n",
    "# Também checamos as laterais para saber se as laterais são os maiores valores\n",
    "# E por fim precisamos do denominador da função (caso haja) para verificar por assíntotas verticais\n",
    "function maximar_uma_funcao(funcao, derivada, segunda_derivada, esq, dir, denominador=0, qtd_iterações=1e2, erro=1e-8)\n",
    "    \n",
    "    # Inicializamos o maximo global\n",
    "    maximo_global = []\n",
    "    \n",
    "    # Se o denominador não for o padrão\n",
    "    if denominador != 0\n",
    "        \n",
    "        # Verificamos a bisseção para achar o 0\n",
    "        b = bissecao(denominador, esq, dir)\n",
    "        \n",
    "        # Se o retorno não for um erro (por não existir raízes no denominador)\n",
    "        # Retornamos o ponto da assíntota vertical\n",
    "        if typeof(b) != String && funcao(b) == Inf\n",
    "            return b\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    # Começamos uma lista de intervalos para o método da bisseção e selecionaremos os propícios\n",
    "    lista_de_intervalos = Any[[esq, dir]]\n",
    "    intervalos_propicios = Any[]\n",
    "    \n",
    "    # Escolhemos uma quantidade de iterações para diminuir o intervalo\n",
    "    for j in 1:qtd_iterações\n",
    "        \n",
    "        # Se a lista de intervalos estiver vazia, não temos mais o que procurar\n",
    "        if size(lista_de_intervalos, 1) == 0\n",
    "            break\n",
    "        end\n",
    "        \n",
    "        # Pega o primeiro intervalo da lista\n",
    "        intervalo = lista_de_intervalos[1]\n",
    "        \n",
    "        # Pega os pontos desse intervalo\n",
    "        a = intervalo[1]\n",
    "        b = intervalo[2]\n",
    "        \n",
    "        # Calcula o ponto médio do intervalo\n",
    "        metade_do_intervalo = (a+b)/2.0\n",
    "        \n",
    "        # Retira o intervalo da lista\n",
    "        deleteat!(lista_de_intervalos, 1)\n",
    "\n",
    "        # Sendo x a metade do intervalo, se um dos intervalos criados [a, x] ou [x, b] satisfizer as condições\n",
    "        # Adicionamos ele a uma outra lista contendo os intervalos bons\n",
    "        # Caso contrário adicionamos eles a nossa lista de intervalos\n",
    "        if derivada(a)*derivada(metade_do_intervalo) <= 0\n",
    "            push!(intervalos_propicios, [a, metade_do_intervalo])\n",
    "        else\n",
    "            push!(lista_de_intervalos, [a, metade_do_intervalo])\n",
    "        end\n",
    "        \n",
    "        if derivada(metade_do_intervalo)*derivada(b) <= 0\n",
    "            push!(intervalos_propicios, [metade_do_intervalo, b])\n",
    "        else\n",
    "            push!(lista_de_intervalos, [metade_do_intervalo, b])\n",
    "        end\n",
    "        \n",
    "    end\n",
    "    \n",
    "    maximos_locais = []\n",
    "    \n",
    "    # Checamos todos os intervalos\n",
    "    for i in 1:size(intervalos_propicios, 1)\n",
    "        \n",
    "        # Utilizando a bissecao para achar o ponto\n",
    "        x = bissecao(derivada, intervalos_propicios[i][1], intervalos_propicios[i][2], erro)\n",
    "    \n",
    "        # Se for maximo local, colocamos na lista para verificar se é máximo global\n",
    "        if segunda_derivada(x) <= 0\n",
    "            push!(maximos_locais, x)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    # Achamos o maximo global\n",
    "    if size(maximos_locais, 1) > 0\n",
    "        \n",
    "        # Verificamos cada um dos maximos locais\n",
    "        for x in maximos_locais\n",
    "            if (size(maximo_global, 1) > 0)\n",
    "                if (maximo_global[1][2] < funcao(x))\n",
    "                    deleteat!(maximo_global, 1)\n",
    "                    push!(maximo_global, [x, funcao(x)])\n",
    "                end\n",
    "            else\n",
    "                push!(maximo_global, [x, funcao(x)])\n",
    "            end\n",
    "        end\n",
    "        \n",
    "    end\n",
    "    \n",
    "    # E verificamos as laterais\n",
    "    if (size(maximo_global, 1) > 0)\n",
    "        if maximo_global[1][2] < funcao(esq)\n",
    "            deleteat!(maximo_global, 1)\n",
    "            push!(maximo_global, [esq, funcao(esq)])\n",
    "        end\n",
    "    else\n",
    "        push!(maximo_global, [esq, funcao(esq)])\n",
    "    end\n",
    "    if maximo_global[1][2] < funcao(dir)\n",
    "        push!(maximo_global, [dir, funcao(dir)])\n",
    "    end\n",
    "        \n",
    "    return maximo_global[1][1]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Inf"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Teste 1\n",
    "\n",
    "f(x) = 1/x\n",
    "fd(x) = -1/x^2\n",
    "fdd(x) = 2/x^3\n",
    "esq = -1\n",
    "dir = 1\n",
    "d(x) = x\n",
    "p = maximar_uma_funcao(f, fd, fdd, esq, dir, d)\n",
    "f(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
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     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# Teste 2\n",
    "\n",
    "f(x) = x^2 * exp(-x)\n",
    "fd(x) = -exp(-x)*x*(x-2)\n",
    "fdd(x) = exp(-x)*(x^2 - 4*x + 2)\n",
    "esq = -1\n",
    "dir = 5\n",
    "p = maximar_uma_funcao(f, fd, fdd, esq, dir)\n",
    "\n",
    "plot(f, esq, dir, leg=false)\n",
    "scatter!(p:p, [f(p)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Teste 3\n",
    "\n",
    "f(x) = exp(-x^2)\n",
    "fd(x) = -2*exp(-x^2)*x\n",
    "fdd(x) = exp(-x^2)*(4*x^2 - 2)\n",
    "esq = -2\n",
    "dir = 2\n",
    "p = maximar_uma_funcao(f, fd, fdd, esq, dir)\n",
    "\n",
    "plot(f, esq, dir, leg=false)\n",
    "scatter!(p:p, [f(p)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Encontrar um ponto crítico de uma função"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "achar_um_ponto_critico_de_uma_funcao (generic function with 3 methods)"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" A ideia aqui é acharmos o ponto em que f'(x) = 0. Podemos usar bisseção, newton ou ponto fixo para isso. \"\"\"\n",
    "\n",
    "function achar_um_ponto_critico_de_uma_funcao(derivada, esq, dir, erro = 1e-2, qtd_iterações = 10)\n",
    "    \n",
    "    # Começamos uma lista de intervalos para o método da bisseção e selecionaremos os propícios\n",
    "    lista_de_intervalos = Any[[esq, dir]]\n",
    "    intervalos_propicios = Any[]\n",
    "    \n",
    "    # Escolhemos uma quantidade de iterações para diminuir o intervalo\n",
    "    for j in 1:qtd_iterações\n",
    "        \n",
    "        # Se a lista de intervalos estiver vazia, não temos mais o que procurar\n",
    "        if size(lista_de_intervalos, 1) == 0\n",
    "            break\n",
    "        end\n",
    "        \n",
    "        # Pega o primeiro intervalo da lista\n",
    "        intervalo = lista_de_intervalos[1]\n",
    "        \n",
    "        # Pega os pontos desse intervalo\n",
    "        a = intervalo[1]\n",
    "        b = intervalo[2]\n",
    "        \n",
    "        # Calcula o ponto médio do intervalo\n",
    "        metade_do_intervalo = (a+b)/2.0\n",
    "        \n",
    "        # Retira o intervalo da lista\n",
    "        deleteat!(lista_de_intervalos, 1)\n",
    "\n",
    "        # Sendo x a metade do intervalo, se um dos intervalos criados [a, x] ou [x, b] satisfizer as condições\n",
    "        # Adicionamos ele a uma outra lista contendo os intervalos bons\n",
    "        # Caso contrário adicionamos eles a nossa lista de intervalos\n",
    "        if derivada(a)*derivada(metade_do_intervalo) <= 0\n",
    "            push!(intervalos_propicios, [a, metade_do_intervalo])\n",
    "        else\n",
    "            push!(lista_de_intervalos, [a, metade_do_intervalo])\n",
    "        end\n",
    "        \n",
    "        if derivada(metade_do_intervalo)*derivada(b) <= 0\n",
    "            push!(intervalos_propicios, [metade_do_intervalo, b])\n",
    "        else\n",
    "            push!(lista_de_intervalos, [metade_do_intervalo, b])\n",
    "        end\n",
    "        \n",
    "    end\n",
    "    \n",
    "    println(intervalos_propicios)\n",
    "    \n",
    "    # Pega o primeiro ponto crítico da lista\n",
    "    ponto_crítico = bissecao(derivada, intervalos_propicios[1][1], intervalos_propicios[1][2], erro)\n",
    "        \n",
    "    return ponto_crítico\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Any[[0.0, 2.5], [2.5, 5.0]]\n"
     ]
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     "execution_count": 240,
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   ],
   "source": [
    "#Teste 1\n",
    "\n",
    "f(x) = x^2 - 5x + 6\n",
    "fd(x) = 2x - 5\n",
    "p = achar_um_ponto_critico_de_uma_funcao(fd, 0, 5)\n",
    "\n",
    "plot(f, 0, 5)\n",
    "scatter!(p:p, [f(p)], leg=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Any[[0.0, 2.5], [2.5, 5.0]]\n"
     ]
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     "execution_count": 241,
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   ],
   "source": [
    "#Teste 2\n",
    "\n",
    "f(x) = x^3/3 - 5x^2/2 + 6x\n",
    "fd(x) = x^2 - 5x + 6\n",
    "p = achar_um_ponto_critico_de_uma_funcao(fd, 0, 5)\n",
    "\n",
    "plot(f, 0, 5)\n",
    "scatter!(p:p, [f(p)], leg=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Any[[1.5707963267948966, 3.141592653589793]]\n"
     ]
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     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Teste 3\n",
    "\n",
    "f(x) = sin(x)\n",
    "fd(x) = cos(x)\n",
    "p = achar_um_ponto_critico_de_uma_funcao(fd, 0, pi)\n",
    "plot(f, 0, pi)\n",
    "scatter!(p:p, [f(p)], leg=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Encontrar um ponto crítico de uma função de 2 variáveis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "achar_um_ponto_critico_de_uma_funcao_de_2_variaveis (generic function with 1 method)"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\" Nessa função, queremos encontrar um ponto crítico em uma função de duas variáveis. Para isso, utilizamos o \n",
    "método do gradiente descendente, em que descemos a curva da função através do decremento do gradiente, visto que\n",
    "este aponta para a direção de maior crescimento. Para isso, é necessário passar o gradiente da função no formato \n",
    "[fdx, fdy], os intervalos de x e y no formato [a, b] e a quantidade de iterações a ser realizada pelo método. \"\"\"\n",
    "\n",
    "function achar_um_ponto_critico_de_uma_funcao_de_2_variaveis(gradiente_da_funcao, intervalox, intervaloy, iteracoes)\n",
    "    # Iniciamos aleatoriamente o ponto\n",
    "    px = rand(intervalox[1]:intervalox[2])\n",
    "    py = rand(intervaloy[1]:intervaloy[2])\n",
    "    taxa = 0.1\n",
    "    \n",
    "    for i in 1:iteracoes\n",
    "        # Obtemos o gradiente no ponto\n",
    "        grad = gradiente_da_funcao(px,py)\n",
    "        \n",
    "        # Checamos se esta dentro do intervalo pedido para descer o gradiente\n",
    "        if (intervalox[1]<=px-grad[1]*taxa<=intervalox[2]) && (intervaloy[1]<=py-grad[2]*taxa<=intervaloy[2])\n",
    "            px = px - grad[1]*taxa\n",
    "            py = py - grad[2]*taxa\n",
    "        else\n",
    "            break       # Senão acabamos a iteracao\n",
    "        end \n",
    "    end\n",
    "    \n",
    "    return [px,py]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " 1.4999999999999996\n",
       " 5.951071883248072e-111"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Testando a função x^2+2y^2-3x\n",
    "\n",
    "d(x,y) = [2x-3, 4y]\n",
    "achar_um_ponto_critico_de_uma_funcao_de_2_variaveis(d, [-10,10], [-10,10], 500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.7697453971177661\n",
       " -1.0740126016896432"
      ]
     },
     "execution_count": 248,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A função x^4 + x*y^3 - 2x^2*y + x - 5yx tem mínimo local em (-0.769, -1.074)\n",
    "\n",
    "d(x,y) = [4x^3+y^3-4y*x+1-5y,-2x^2+3x*y^2-5x]\n",
    "achar_um_ponto_critico_de_uma_funcao_de_2_variaveis(d, [-5,5], [-5,5], 500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -1.5707963267948957\n",
       "  0.28195272428243545"
      ]
     },
     "execution_count": 246,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Função sen(x)*cos(y/5) possui muitos pontos locais de minimo e maximo\n",
    "\n",
    "d(x,y) = [cos(y/5)*cos(x), -1/5*sin(x)*sin(y/5)]\n",
    "achar_um_ponto_critico_de_uma_funcao_de_2_variaveis(d, [-5,5], [-5,5], 500)"
   ]
  }
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