{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parasitic solutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find $x$ such that\n",
    "$$\n",
    "\\exp(-x) = 99 x\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0100004983870833\n"
     ]
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "eq1 = exp(-x) - 99*x\n",
    "r1 = solve(eq1,x)\n",
    "for r in r1:\n",
    "    print(N(r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us make a Taylor expansion of $\\exp(-x)$ around $x=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$1 - x + \\frac{x^{2}}{2} - \\frac{x^{3}}{6} + \\frac{x^{4}}{24} - \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "         2    3    4     5        \n",
       "        x    x    x     x     ⎛ 6⎞\n",
       "1 - x + ── - ── + ── - ─── + O⎝x ⎠\n",
       "        2    6    24   120        "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(-x),x,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We truncate this at the quadratic term and consider the equation\n",
    "$$\n",
    "1 - x + \\frac{x^2}{2} = 99 x \\qquad \\Longrightarrow \\qquad x^2 - 200 x + 2 = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0100005000500063\n",
      "199.989999499950\n"
     ]
    }
   ],
   "source": [
    "eq2 = x**2 - 200*x + 2\n",
    "r2 = solve(eq2,x)\n",
    "for r in r2:\n",
    "    print(N(r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The simplied problem has more solutions than the original problem. One of the solutions is the correct one while the other is a spurious or parasitic 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": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}