Fixed point problem

Fixed point problem.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step1: Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import warnings\n",
    "warnings.simplefilter(\"ignore\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step2: Define f and plot it to make sure it's converging"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sin(x)/x\n",
    "\n",
    "def plot(lower=-1, upper=200):\n",
    "    ys, xs = [], []\n",
    "    plt.figure(figsize=(6, 4))\n",
    "    for x in range(lower, upper):\n",
    "        try:\n",
    "            ys.append(f(x))\n",
    "            xs.append(x)\n",
    "        except:\n",
    "            continue\n",
    "    plt.xlabel(\"x\")        \n",
    "    plt.ylabel(\"f(x)\")        \n",
    "    plt.plot(xs, ys)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function above is clearly converging"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 3: Solve the fixed point problem by repeatedly applying the function infinite times\n",
    "\n",
    "Note: Instead of infinity, we just apply it `10000` times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fixed_point(INF = 10000):\n",
    "    x0 = np.random.randn()\n",
    "    for _ in range(INF):\n",
    "        x0 = f(x0)\n",
    "    return x0    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8767262153950625\n"
     ]
    }
   ],
   "source": [
    "solution = fixed_point()\n",
    "print(solution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def verify(solution):\n",
    "    print(f\"Values of function: f{f(solution)} | Value of x: {solution}\")\n",
    "    if f(solution) == solution:\n",
    "        return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Values of function: f0.8767262153950625 | Value of x: 0.8767262153950625\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "verify(solution)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}