2. Root-finding.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## Minima and roots of a function\n",
    "\n",
    "\n",
    "Demos finding minima and roots of a function.\n",
    "\n",
    "SciPy optimize provides functions for minimizing (or maximizing) objective functions, possibly subject to constraints. It includes solvers for nonlinear problems (with support for both local and global optimization algorithms), linear programing, constrained and nonlinear least-squares, root finding and curve fitting.\n",
    "\n",
    "1. Scalar Functions Optimization\n",
    "2. Local (Multivariate) Optimization\n",
    "3. Global Optimization\n",
    "4. Least-squares and Curve Fitting\n",
    "5. Root finding"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from scipy import optimize\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Define the function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.arange(-10, 10, 0.1)\n",
    "def f(x):\n",
    "    return x**2 + 10*np.sin(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Find minima"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Global minima found [-1.30641113]\n",
      "Local minimum found 3.8374671194983834\n"
     ]
    }
   ],
   "source": [
    "# Global optimization\n",
    "grid = (-10, 10, 0.1)\n",
    "xmin_global = optimize.brute(f, (grid, ))\n",
    "print(\"Global minima found %s\" % xmin_global)\n",
    "\n",
    "# Constrain optimization\n",
    "xmin_local = optimize.fminbound(f, 0, 10)\n",
    "print(\"Local minimum found %s\" % xmin_local)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Root finding"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First root found [0.]\n",
      "Second root found [-2.47948183]\n"
     ]
    }
   ],
   "source": [
    "root = optimize.root(f, 1)  # our initial guess is 1\n",
    "print(\"First root found %s\" % root.x)\n",
    "root2 = optimize.root(f, -2.5)\n",
    "print(\"Second root found %s\" % root2.x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot function, minima, and roots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    "\n",
    "fig = plt.figure(figsize=(6, 4))\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "# Plot the function\n",
    "ax.plot(x, f(x), 'b-', label=\"f(x)\")\n",
    "\n",
    "# Plot the minima\n",
    "xmins = np.array([xmin_global[0], xmin_local])\n",
    "ax.plot(xmins, f(xmins), 'go', label=\"Minima\")\n",
    "\n",
    "# Plot the roots\n",
    "roots = np.array([root.x, root2.x])\n",
    "ax.plot(roots, f(roots), 'kv', label=\"Roots\")\n",
    "\n",
    "# Decorate the figure\n",
    "ax.legend(loc='best')\n",
    "ax.set_xlabel('x')\n",
    "ax.set_ylabel('f(x)')\n",
    "ax.axhline(0, color='gray')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": 1
}