06_Reimann_Sumations.ipynb

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    "**Riemann Sums**\n",
    "\n",
    "A Riemann Sum is a certain kind of approximation of an integrail by a finite sum. <br> <br>\n",
    "<font color=\"green\"> Deffinition </font> \n",
    "Let $ f:[a,b] \\rightarrow R $ be defined on a closed interval  $ [a,b] \\in R $ and $ P =  [ [x_0,x_1],[x_1,x_2],...,[x_{n-1},x_n] ]$ be a partition of an Interval  where $ a =x_0 < x_1< x_2 < .... < x_{n-1} <x_n = b $ ( x is monotonically increasing) <br>\n",
    "\n",
    "We define a Reimann sum ( $S$ ) of $ f(x) $ over the interval $ I $  with partion  $ P $  as : <br>\n",
    "$$ S = \\sum_{i=1}^n f(x_i^*)  \\Delta x_i$$\n",
    "Where: \n",
    "<br>\n",
    "$ x_i = a + i*\\Delta x  $ ................................. $ x_i  $ . incrementes in $ *\\Delta x  $ steps.\n",
    "<br>\n",
    "$ x_i^* \\in [x_{i-1},x_i] $.................................... $ x_i^* $ can be any point in $[x_{i-1},x_i]$.\n",
    "<br>\n",
    "$\\Delta x =  x_i - x_{i-1}  = \\frac{ b-a}{n}$.......................$\\Delta x $ depends on $ Inf(f(X)) $ , $ Sup(f(x)) $ and the number of points $ n $.\n",
    "<br>\n",
    "\n",
    "It does not matter which point $ x_i^* $ in the interval $ [x_{i-1},x_i] $ is taken,  when  the differnece of width of the summands $  \\Delta x $ approaches zero, so does the difference between any two points in the interval.\n",
    "\n",
    "$$\\lim_{\\Delta x \\rightarrow 0} \\rightarrow  x_{i-1} - x_i = 0$$\n",
    "\n",
    "<br>\n",
    "\n",
    "**Riemann Sums:**\n",
    "<ol>\n",
    "    <li> Left Riemann Sum: $ x_i^* = x_{i-1}$</li>\n",
    "    <li> Right Riemann Sum: $ x_i^* = x_{i}$</li>\n",
    "    <li> Middle Riemann Sum: $ x_i^* = \\frac{x_{i}+x_{i-1}}{2} $ </li>\n",
    "   \n",
    "  \n",
    "</ol>\n",
    "\n",
    "**Deffinition of integral by Right Riemann Sum**\n",
    "$$\\lim_{n \\rightarrow \\infty} \\sum_{i=1}^n f(x_i^*) \\Delta x = \\int_a^b f(x) dx = \\lim_{\\Delta x \\to 0}  \\sum_{i=1}^n f(x_i^*) \\Delta x $$\n",
    "\n",
    "If we make the substitions for $ x_i $ and $ \\Delta x $:\n",
    "\n",
    "$$ \\int_a^b f(x) dx = \\lim_{\\Delta x \\to 0}  \\sum_{i=1}^n f( a + (\\frac{b-a}{n})i )*\n",
    "( \\frac{b-a}{n} )$$\n",
    "\n",
    "**Deffinition of integral by Middle Riemann Sum**\n",
    "\n",
    "$$ \\int_a^b f(x) dx = \\lim_{\\Delta x \\to 0}  \\sum_{i=0}^n f( \\frac{a + (\\frac{b-a}{n})i + a + (\\frac{b-a}{n})(i+1)}{2})*\n",
    "( \\frac{b-a}{n} )$$\n",
    "\n",
    "**Example:**\n",
    "Obtain the Riemann sumation (integral approximation) of the function $ f(x) = e^{2x} $ for the close interval $[2,5]$. <br>Analitical solution: $ F(x) = 2e^{2x} |_2^5= 2e^{(2*5)} - 2e^{(2*2)} = 282.048..$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       ".output_png {\n",
       "    display: table-cell;\n",
       "    text-align: center;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n"
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     "execution_count": 1,
     "metadata": {},
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   ],
   "source": [
    "# Import Libraries\n",
    "import numpy as np\n",
    "from math import *\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.core.display import HTML\n",
    "HTML(\"\"\"\n",
    "<style>\n",
    ".output_png {\n",
    "    display: table-cell;\n",
    "    text-align: center;\n",
    "    vertical-align: middle;\n",
    "}\n",
    "</style>\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Interval: 2 to 5, number of points: 20, dx: 0.15, Analytical Solution: 282.0482060072919\n"
     ]
    }
   ],
   "source": [
    "# Intialization for Right Riemann Sum (ONLY FOR PLOT )\n",
    "F_analytical= 2*(np.exp(5) - np.exp(2)) # Analytical Solution\n",
    "a = 2        # inf[f(x)]  Lower boundary\n",
    "b = 5        # sup[f(x)]  Upper boundary\n",
    "n = 20       # Number of points\n",
    "dx = (b-a)/n # dx by definition\n",
    "#SUMMARY\n",
    "print(f\"Interval: {a} to {b}, number of points: {n}, dx: {dx}, Analytical Solution: {F_analytical}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.   2.15 2.3  2.45 2.6  2.75 2.9  3.05 3.2  3.35 3.5  3.65 3.8  3.95\n",
      " 4.1  4.25 4.4  4.55 4.7  4.85 5.  ]\n"
     ]
    }
   ],
   "source": [
    "# Initialization of xi (this is only used in order to plot the function)\n",
    "# by definition xi = a + dxi\n",
    "x = []\n",
    "xi = a\n",
    "for i in range(0,20+1):\n",
    "    xi = a + dx*i\n",
    "    x.append(xi)\n",
    "x = np.array(x) # convertion to numpy array (for better performance)\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot function\n",
    "#f_x = np.exp(x)             # Definition of f(x)\n",
    "plt.figure(figsize=(10,5))  # Adjust Fig size\n",
    "plt.plot(x,f_x)             # plot function x,f(x)\n",
    "plt.plot(a,np.exp(a),\"*r\")  # inf[f(x)]  Lower boundary\n",
    "plt.plot(b,np.exp(b),\"*g\")  # sup[f(x)]  Upper boundary\n",
    "\n",
    "# Cutomize output \n",
    "plt.title(\"Expontential from 2 to 5\")\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Right Reimman Sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "enter the number of points n=100\n",
      "Analytical Solution: 282.0482 Reimman Sumation with 100 points: 286.7434, Absolute Error: 4.69521975419832\n"
     ]
    }
   ],
   "source": [
    "# Right Riemman Sum from scratch \n",
    "# Right Reimman Sum\n",
    "F=0 # Init Integral \n",
    "a = 2   # inf[f(x)]  Lower boundary\n",
    "b = 5   # sup[f(x)]  Upper boundary\n",
    "n=int(input(\"enter the number of points n=\")) # Number of points\n",
    "\n",
    "for i in range(0,n+1): \n",
    "    fx = 2*np.exp(  a + i*(b-a)/n  ) # Evaluate function in x_i \n",
    "    F+= fx*( (b-a)/n )               # Multiply by dx (Area of rectangle)\n",
    "\n",
    "print(f\"Analytical Solution: {round(F_analytical,4)} Reimman Sumation with {n} points: {round(F,4)}, Absolute Error: {abs(F_analytical -F)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Middle Riemman Sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "enter the number of points n=100\n",
      "Analytical Solution: 282.0482 Reimman Sumation with 100 points: 279.217, Absolute Error: 2.8311654181069343\n"
     ]
    }
   ],
   "source": [
    "# Reimman Sum\n",
    "a = 2   # inf[f(x)]  Lower boundary\n",
    "b = 5   # sup[f(x)]  Upper boundary\n",
    "F=0     # Init Integral\n",
    "n=int(input(\"enter the number of points n=\")) # Number of points\n",
    "x = np.linspace(a,b,n) # Create vector from a to b with n points (a, a+dx, a+2dx, ..., b-dx, b)\n",
    "for i in range(0,len(x)):\n",
    "    xi_prev = x[i]\n",
    "    try:\n",
    "        xi = x[i+1]\n",
    "    except:\n",
    "        continue\n",
    "    middle_point = (xi_prev +xi)/2 # Obtain middle point \n",
    "    fx = 2*np.exp(middle_point)    # Evaluate function in middle point\n",
    "    F+= fx*( (b-a)/n )             # Multiply by dx (Area of rectangle)\n",
    "\n",
    "print(f\"Analytical Solution: {round(F_analytical,4)} Reimman Sumation with {n} points: {round(F,4)}, Absolute Error: {abs(F_analytical -F)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Reimman Sum number of points analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,5))  # Adjust Fig size\n",
    "values = [100,200,500,1000,2000,3000,4000]\n",
    "j=0\n",
    "for v in values:\n",
    "    F =0\n",
    "    n=v\n",
    "    for i in range(0+1,n+1):\n",
    "         F+= (2*np.exp(( a + i*(b-a)/n )))*((b-a)/n)\n",
    "    \n",
    "    j = j + 1\n",
    "    plt.plot(j,F,\"*\",label=f\" n = {v}\")\n",
    "    \n",
    "plt.axhline(y=F_analytical, color='green', linestyle='--', label=\"Real Integral\")\n",
    "\n",
    "# Cutomize output \n",
    "plt.legend()    # Add legends\n",
    "plt.grid()      # Turn on grid\n",
    "plt.title(\"Reimann Sumations\") # Set Title\n",
    "plt.ylim([280,287])\n",
    "plt.show()      # Print \n"
   ]
  }
 ],
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