week3_lab.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "39711d53-0ad1-48a1-a58a-606e3245923f",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import minimize\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9c0786a-7187-4161-8ceb-ac77416fff71",
   "metadata": {},
   "source": [
    "### Question 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "13eca082-ab5b-4edf-9823-58b22978a947",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2, 2, 50)\n",
    "\n",
    "def f(x):\n",
    "    return x ** 4 + x ** 3 - x ** 2 - 5\n",
    "\n",
    "plt.plot(x, f(x), label=\"$y= x^5 + x^3 - x^2 -5$\")\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "2bdfe8a4-7fe4-4560-bf10-895b896f23d9",
   "metadata": {},
   "outputs": [],
   "source": [
    "def func(x_array):\n",
    "    return x_array ** 4 + x_array ** 3 - x_array ** 2 - 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "9760a24e-89d2-4543-8244-161355a1844a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The global minimum of the function is -1.1753916935675832\n"
     ]
    }
   ],
   "source": [
    "minima = minimize(func, x0=-2)\n",
    "print(f\"The global minimum of the function is {minima.x[0]}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "983ecc7a-2117-4401-b877-0c61c94e3bcb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The local minimum of the function is 0.42539232946828587\n"
     ]
    }
   ],
   "source": [
    "minima = minimize(func, x0=1.5)\n",
    "print(f\"The local minimum of the function is {minima.x[0]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "553befae-8206-4a43-8c08-ae516e6e2e8d",
   "metadata": {},
   "source": [
    "### Question 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "7155ab46-b039-46c1-b28b-2467d7c607b6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2, 2, 50)\n",
    "y = np.exp(-x)\n",
    "plt.plot(x, y)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e29a706b-0f37-42ee-819e-adc7c333f2c0",
   "metadata": {},
   "source": [
    "### Question 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "448e8352-e8eb-47b5-8be8-41e0d49fcaf9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = -x ** 3 + 4 * x ** 2 - 3 * x + 5\n",
    "plt.plot(x, y)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "4a7d0cbd-dd0e-4a35-b50f-ccab32e3dec4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def func2(x_array):\n",
    "    return -x_array ** 3 + 4 * x_array ** 2 - 3 * x_array + 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "079d96e5-f554-48e5-9afb-2982b6e4ceab",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The minimum of the function is 0.45141551064044644\n"
     ]
    }
   ],
   "source": [
    "minima2 = minimize(func2, x0=-2)\n",
    "print(f\"The minimum of the function is {minima2.x[0]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99075ab8-297b-4da4-80ae-2ff6937fd671",
   "metadata": {},
   "source": [
    "### Bisection Algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "77d45fd8-9006-429c-99e7-e095ca021459",
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "This is for getting the root (where y=0)\n",
    "\n",
    "def bisec_opt(func, a, b):\n",
    "    minimum = 0.345\n",
    "    while b - a > minimum:\n",
    "        alpha = (a + b) / 2\n",
    "        y_alpha = func(alpha)\n",
    "        if y_alpha > 0:\n",
    "            b = alpha\n",
    "        elif y_alpha < 0:\n",
    "            a = alpha\n",
    "        else:\n",
    "            return alpha\n",
    "    # Return the mean of the two\n",
    "    return (a + b) / 2\n",
    "\"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "id": "abe26a43-8981-480f-a6d5-886d4413a34a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def bisec_opt(func, a, b):\n",
    "    minimum = 0.0001\n",
    "    delta = 0.01\n",
    "    while b - a > minimum:\n",
    "        print(f\"current a: {a}------- current b: {b}\")\n",
    "        alpha = (a + b) / 2\n",
    "        alpha_gradient = (func(alpha + delta) - func(alpha - delta)) / 2 * delta\n",
    "        a_gradient = (func(a + delta) - func(a - delta)) / 2 * delta\n",
    "        print(f\"alpha_gradient: {alpha_gradient}------- a_gradient: {a_gradient}\")\n",
    "        if alpha_gradient * a_gradient < 0:\n",
    "            b = alpha\n",
    "        elif alpha_gradient * a_gradient > 0:\n",
    "            a = alpha\n",
    "        else:\n",
    "            return alpha\n",
    "    # Return the mean of the two\n",
    "    return (a + b) / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "955d4bfc-c186-46cd-b950-7d83b3d4438d",
   "metadata": {},
   "outputs": [],
   "source": [
    "def bisec_opt(func, a, b):\n",
    "    minimum = 0.345\n",
    "    while b - a > minimum:\n",
    "        alpha = (a + b) / 2\n",
    "        y_alpha = func(alpha)\n",
    "        if y_alpha > 0:\n",
    "            b = alpha\n",
    "        elif y_alpha < 0:\n",
    "            a = alpha\n",
    "        else:\n",
    "            return alpha\n",
    "    # Return the mean of the two\n",
    "    return (a + b) / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "id": "107a68ed-1933-4853-a7ab-94189940b77b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "current a: -2------- current b: 2\n",
      "alpha_gradient: -0.00030001000000000387------- a_gradient: -0.003100009999999962\n",
      "current a: 0.0------- current b: 2\n",
      "alpha_gradient: 0.0001999899999999988------- a_gradient: -0.00030001000000000387\n",
      "current a: 0.0------- current b: 1.0\n",
      "alpha_gradient: 2.498999999999807e-05------- a_gradient: -0.00030001000000000387\n",
      "current a: 0.0------- current b: 0.5\n",
      "alpha_gradient: -0.00011875999999999998------- a_gradient: -0.00030001000000000387\n",
      "current a: 0.25------- current b: 0.5\n",
      "alpha_gradient: -4.2197499999998554e-05------- a_gradient: -0.00011875999999999998\n",
      "current a: 0.375------- current b: 0.5\n",
      "alpha_gradient: -7.4318749999990885e-06------- a_gradient: -4.2197499999998554e-05\n",
      "current a: 0.4375------- current b: 0.5\n",
      "alpha_gradient: 9.072031250001444e-06------- a_gradient: -7.4318749999990885e-06\n",
      "current a: 0.4375------- current b: 0.46875\n",
      "alpha_gradient: 8.933203125005562e-07------- a_gradient: -7.4318749999990885e-06\n",
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      "current a: 0.4453125------- current b: 0.453125\n",
      "alpha_gradient: -1.1742456054708583e-06------- a_gradient: -3.2509667968749767e-06\n",
      "current a: 0.44921875------- current b: 0.453125\n",
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      "current a: 0.451171875------- current b: 0.453125\n",
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      "current a: 0.451171875------- current b: 0.45166015625\n",
      "alpha_gradient: -1.0113248825049937e-08------- a_gradient: -1.393182373066182e-07\n",
      "current a: 0.451416015625------- current b: 0.45166015625\n",
      "alpha_gradient: 5.4475834367906376e-08------- a_gradient: -1.0113248825049937e-08\n",
      "current a: 0.451416015625------- current b: 0.4515380859375\n",
      "alpha_gradient: 2.2182410361892836e-08------- a_gradient: -1.0113248825049937e-08\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.451446533203125"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bound_a = -2\n",
    "bound_b = 2\n",
    "bisec_opt(lambda x: -x**3 + 4*x**2 - 3*x + 5, bound_a, bound_b)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4aa20961-aca1-4005-a5bf-df17c570d2c5",
   "metadata": {},
   "source": [
    "### Newton-Raphson Optimization Algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "251c2576-f2ba-4f5e-afda-a9a8c1ec583a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_value(function, x):\n",
    "    return function(x)\n",
    "\n",
    "def newton_raphson(function, i):\n",
    "    minimum = 0.015\n",
    "    f_prime = sp.diff(function, x)\n",
    "    i_next = 0\n",
    "    while True:\n",
    "        y_val = get_value(lambda x_val: function.subs(x, x_val), i)\n",
    "        f_prime_val = get_value(lambda x_val: f_prime.subs(x, x_val), i)\n",
    "        i_next = i - (y_val / f_prime_val)\n",
    "        if abs(i - i_next) < minimum:\n",
    "            break\n",
    "        else:\n",
    "            i = i_next\n",
    "    \n",
    "    return round(float(i_next), 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "cf22b8ce-37dc-445b-a82f-956bb718a502",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.426"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sp.symbols('x')\n",
    "# cost_function = -x ** 3 + 4 * x ** 2 - 3 * x + 5 Struggles with complex function like this\n",
    "cost_function = x ** 4 + x ** 3 - x ** 2 - 5\n",
    "guess_point = 1\n",
    "newton_raphson(cost_function, guess_point)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8f0c4c98-0684-4d92-bbcc-bbb11cbb42bd",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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