tutorial8.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "# Import necessary libraries\n",
    "import sympy as sym"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "outputs": [],
   "source": [
    "# Step 1: Define the symbolic variable x\n",
    "x = sym.symbols('x')"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "outputs": [
    {
     "data": {
      "text/plain": "x**2*sin(0.5*x)",
      "text/latex": "$\\displaystyle x^{2} \\sin{\\left(0.5 x \\right)}$"
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Step 2: Define the function f(x) = x^2 * sin(0.5x)\n",
    "f = (x ** 2) * sym.sin(0.5 * x)\n",
    "f"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "outputs": [
    {
     "data": {
      "text/plain": "array([None, None, None, None, None, None, None, None, None, None, None,\n       None, None, None, None, None, None, None, None, None], dtype=object)"
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Step 3: Initialize an array to store Maclaurin series terms\n",
    "ms = np.empty(20, dtype=object)\n",
    "ms"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "outputs": [
    {
     "data": {
      "text/plain": "array([-5.        , -4.97995992, -4.95991984, -4.93987976, -4.91983968,\n       -4.8997996 , -4.87975952, -4.85971944, -4.83967936, -4.81963928,\n       -4.7995992 , -4.77955912, -4.75951904, -4.73947896, -4.71943888,\n       -4.6993988 , -4.67935872, -4.65931864, -4.63927856, -4.61923848,\n       -4.5991984 , -4.57915832, -4.55911824, -4.53907816, -4.51903808,\n       -4.498998  , -4.47895792, -4.45891784, -4.43887776, -4.41883768,\n       -4.3987976 , -4.37875752, -4.35871743, -4.33867735, -4.31863727,\n       -4.29859719, -4.27855711, -4.25851703, -4.23847695, -4.21843687,\n       -4.19839679, -4.17835671, -4.15831663, -4.13827655, -4.11823647,\n       -4.09819639, -4.07815631, -4.05811623, -4.03807615, -4.01803607,\n       -3.99799599, -3.97795591, -3.95791583, -3.93787575, -3.91783567,\n       -3.89779559, -3.87775551, -3.85771543, -3.83767535, -3.81763527,\n       -3.79759519, -3.77755511, -3.75751503, -3.73747495, -3.71743487,\n       -3.69739479, -3.67735471, -3.65731463, -3.63727455, -3.61723447,\n       -3.59719439, -3.57715431, -3.55711423, -3.53707415, -3.51703407,\n       -3.49699399, -3.47695391, -3.45691383, -3.43687375, -3.41683367,\n       -3.39679359, -3.37675351, -3.35671343, -3.33667335, -3.31663327,\n       -3.29659319, -3.27655311, -3.25651303, -3.23647295, -3.21643287,\n       -3.19639279, -3.17635271, -3.15631263, -3.13627255, -3.11623246,\n       -3.09619238, -3.0761523 , -3.05611222, -3.03607214, -3.01603206,\n       -2.99599198, -2.9759519 , -2.95591182, -2.93587174, -2.91583166,\n       -2.89579158, -2.8757515 , -2.85571142, -2.83567134, -2.81563126,\n       -2.79559118, -2.7755511 , -2.75551102, -2.73547094, -2.71543086,\n       -2.69539078, -2.6753507 , -2.65531062, -2.63527054, -2.61523046,\n       -2.59519038, -2.5751503 , -2.55511022, -2.53507014, -2.51503006,\n       -2.49498998, -2.4749499 , -2.45490982, -2.43486974, -2.41482966,\n       -2.39478958, -2.3747495 , -2.35470942, -2.33466934, -2.31462926,\n       -2.29458918, -2.2745491 , -2.25450902, -2.23446894, -2.21442886,\n       -2.19438878, -2.1743487 , -2.15430862, -2.13426854, -2.11422846,\n       -2.09418838, -2.0741483 , -2.05410822, -2.03406814, -2.01402806,\n       -1.99398798, -1.9739479 , -1.95390782, -1.93386774, -1.91382766,\n       -1.89378758, -1.87374749, -1.85370741, -1.83366733, -1.81362725,\n       -1.79358717, -1.77354709, -1.75350701, -1.73346693, -1.71342685,\n       -1.69338677, -1.67334669, -1.65330661, -1.63326653, -1.61322645,\n       -1.59318637, -1.57314629, -1.55310621, -1.53306613, -1.51302605,\n       -1.49298597, -1.47294589, -1.45290581, -1.43286573, -1.41282565,\n       -1.39278557, -1.37274549, -1.35270541, -1.33266533, -1.31262525,\n       -1.29258517, -1.27254509, -1.25250501, -1.23246493, -1.21242485,\n       -1.19238477, -1.17234469, -1.15230461, -1.13226453, -1.11222445,\n       -1.09218437, -1.07214429, -1.05210421, -1.03206413, -1.01202405,\n       -0.99198397, -0.97194389, -0.95190381, -0.93186373, -0.91182365,\n       -0.89178357, -0.87174349, -0.85170341, -0.83166333, -0.81162325,\n       -0.79158317, -0.77154309, -0.75150301, -0.73146293, -0.71142285,\n       -0.69138277, -0.67134269, -0.65130261, -0.63126253, -0.61122244,\n       -0.59118236, -0.57114228, -0.5511022 , -0.53106212, -0.51102204,\n       -0.49098196, -0.47094188, -0.4509018 , -0.43086172, -0.41082164,\n       -0.39078156, -0.37074148, -0.3507014 , -0.33066132, -0.31062124,\n       -0.29058116, -0.27054108, -0.250501  , -0.23046092, -0.21042084,\n       -0.19038076, -0.17034068, -0.1503006 , -0.13026052, -0.11022044,\n       -0.09018036, -0.07014028, -0.0501002 , -0.03006012, -0.01002004,\n        0.01002004,  0.03006012,  0.0501002 ,  0.07014028,  0.09018036,\n        0.11022044,  0.13026052,  0.1503006 ,  0.17034068,  0.19038076,\n        0.21042084,  0.23046092,  0.250501  ,  0.27054108,  0.29058116,\n        0.31062124,  0.33066132,  0.3507014 ,  0.37074148,  0.39078156,\n        0.41082164,  0.43086172,  0.4509018 ,  0.47094188,  0.49098196,\n        0.51102204,  0.53106212,  0.5511022 ,  0.57114228,  0.59118236,\n        0.61122244,  0.63126253,  0.65130261,  0.67134269,  0.69138277,\n        0.71142285,  0.73146293,  0.75150301,  0.77154309,  0.79158317,\n        0.81162325,  0.83166333,  0.85170341,  0.87174349,  0.89178357,\n        0.91182365,  0.93186373,  0.95190381,  0.97194389,  0.99198397,\n        1.01202405,  1.03206413,  1.05210421,  1.07214429,  1.09218437,\n        1.11222445,  1.13226453,  1.15230461,  1.17234469,  1.19238477,\n        1.21242485,  1.23246493,  1.25250501,  1.27254509,  1.29258517,\n        1.31262525,  1.33266533,  1.35270541,  1.37274549,  1.39278557,\n        1.41282565,  1.43286573,  1.45290581,  1.47294589,  1.49298597,\n        1.51302605,  1.53306613,  1.55310621,  1.57314629,  1.59318637,\n        1.61322645,  1.63326653,  1.65330661,  1.67334669,  1.69338677,\n        1.71342685,  1.73346693,  1.75350701,  1.77354709,  1.79358717,\n        1.81362725,  1.83366733,  1.85370741,  1.87374749,  1.89378758,\n        1.91382766,  1.93386774,  1.95390782,  1.9739479 ,  1.99398798,\n        2.01402806,  2.03406814,  2.05410822,  2.0741483 ,  2.09418838,\n        2.11422846,  2.13426854,  2.15430862,  2.1743487 ,  2.19438878,\n        2.21442886,  2.23446894,  2.25450902,  2.2745491 ,  2.29458918,\n        2.31462926,  2.33466934,  2.35470942,  2.3747495 ,  2.39478958,\n        2.41482966,  2.43486974,  2.45490982,  2.4749499 ,  2.49498998,\n        2.51503006,  2.53507014,  2.55511022,  2.5751503 ,  2.59519038,\n        2.61523046,  2.63527054,  2.65531062,  2.6753507 ,  2.69539078,\n        2.71543086,  2.73547094,  2.75551102,  2.7755511 ,  2.79559118,\n        2.81563126,  2.83567134,  2.85571142,  2.8757515 ,  2.89579158,\n        2.91583166,  2.93587174,  2.95591182,  2.9759519 ,  2.99599198,\n        3.01603206,  3.03607214,  3.05611222,  3.0761523 ,  3.09619238,\n        3.11623246,  3.13627255,  3.15631263,  3.17635271,  3.19639279,\n        3.21643287,  3.23647295,  3.25651303,  3.27655311,  3.29659319,\n        3.31663327,  3.33667335,  3.35671343,  3.37675351,  3.39679359,\n        3.41683367,  3.43687375,  3.45691383,  3.47695391,  3.49699399,\n        3.51703407,  3.53707415,  3.55711423,  3.57715431,  3.59719439,\n        3.61723447,  3.63727455,  3.65731463,  3.67735471,  3.69739479,\n        3.71743487,  3.73747495,  3.75751503,  3.77755511,  3.79759519,\n        3.81763527,  3.83767535,  3.85771543,  3.87775551,  3.89779559,\n        3.91783567,  3.93787575,  3.95791583,  3.97795591,  3.99799599,\n        4.01803607,  4.03807615,  4.05811623,  4.07815631,  4.09819639,\n        4.11823647,  4.13827655,  4.15831663,  4.17835671,  4.19839679,\n        4.21843687,  4.23847695,  4.25851703,  4.27855711,  4.29859719,\n        4.31863727,  4.33867735,  4.35871743,  4.37875752,  4.3987976 ,\n        4.41883768,  4.43887776,  4.45891784,  4.47895792,  4.498998  ,\n        4.51903808,  4.53907816,  4.55911824,  4.57915832,  4.5991984 ,\n        4.61923848,  4.63927856,  4.65931864,  4.67935872,  4.6993988 ,\n        4.71943888,  4.73947896,  4.75951904,  4.77955912,  4.7995992 ,\n        4.81963928,  4.83967936,  4.85971944,  4.87975952,  4.8997996 ,\n        4.91983968,  4.93987976,  4.95991984,  4.97995992,  5.        ])"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Step 4: Create an array of 500 points from -5 to 5 for plotting\n",
    "x_values = np.linspace(-5, 5, 500)\n",
    "x_values"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "outputs": [
    {
     "data": {
      "text/plain": "array([[0., 0., 0., ..., 0., 0., 0.],\n       [0., 0., 0., ..., 0., 0., 0.],\n       [0., 0., 0., ..., 0., 0., 0.],\n       ...,\n       [0., 0., 0., ..., 0., 0., 0.],\n       [0., 0., 0., ..., 0., 0., 0.],\n       [0., 0., 0., ..., 0., 0., 0.]])"
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Step 5: Create a 2D array to store function values and approximations for plotting\n",
    "function_and_approximations = np.zeros([21, 500])\n",
    "function_and_approximations"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "outputs": [],
   "source": [
    "# Step 6: Evaluate and store the function values at each point in x_values\n",
    "function_evaluation = sym.lambdify(x, f, 'numpy')\n",
    "function_evaluation\n",
    "function_and_approximations[20, :] = function_evaluation(x_values)"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 : 0\n",
      "2 : 0\n",
      "3 : 0\n",
      "4 : 0.5*x**3\n",
      "5 : 0\n",
      "6 : -0.0208333333333333*x**5\n",
      "7 : 0\n",
      "8 : 0.000260416666666667*x**7\n",
      "9 : 0\n",
      "10 : -1.55009920634921e-6*x**9\n",
      "11 : 0\n",
      "12 : 5.38228891093474e-9*x**11\n",
      "13 : 0\n",
      "14 : -1.2232474797579e-11*x**13\n",
      "15 : 0\n",
      "16 : 1.96033249961201e-14*x**15\n",
      "17 : 0\n",
      "18 : -2.33372916620478e-17*x**17\n",
      "19 : 0\n",
      "20 : 2.14497166011469e-20*x**19\n"
     ]
    }
   ],
   "source": [
    "# Step 7: Perform Taylor series expansion around x = 0\n",
    "for degree in range(20):\n",
    "    # Calculate the nth derivative of f(x) at x = 0\n",
    "    ms[degree] = f.diff(x, degree).subs(x, 0) * (\n",
    "            x ** degree) / (np.math.factorial(degree))\n",
    "\n",
    "    # Convert the symbolic expression to a NumPy function\n",
    "    taylor_series_function = sym.lambdify(x, ms[degree], 'numpy')\n",
    "\n",
    "    # Evaluate and store the approximation at each point in x_values\n",
    "    function_and_approximations[degree, :] = taylor_series_function(x_values)\n",
    "\n",
    "    # Print the Taylor series term for better understanding\n",
    "    print(degree + 1, \":\", ms[degree])"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "outputs": [
    {
     "data": {
      "text/plain": "<Figure size 640x480 with 1 Axes>",
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting\n",
    "plt.plot(x_values, function_and_approximations[0, :], label=\"Degree 1\")\n",
    "plt.plot(x_values, function_and_approximations[4, :], label=\"Degree 5\")\n",
    "plt.plot(x_values, function_and_approximations[9, :], label=\"Degree 10\")\n",
    "plt.plot(x_values, function_and_approximations[20, :], label=\"Function\")\n",
    "\n",
    "# Add labels and legend for clarity\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.legend()\n",
    "\n",
    "# Display the plot\n",
    "plt.show()"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false
   }
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}