polynomial/Polynomial in Numpy.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  Method 1:  Polynomials as arrays (Like MATLAB)\n",
    "\n",
    "Polynomials are represented as arrays in order of descending coefficients\n",
    "\n",
    "For example, the array [5, -3, 2, 1] represents\n",
    "$$5x^3 - 3x^2 + 2x + 1$$\n",
    "\n",
    "There are suites of function for manipulating these polynomials.\n",
    "- polyval --  Numerically evaluate at point(s)\n",
    "- roots   --  Find toor\n",
    "- polyder --  Derivative\n",
    "- poltiny --  Integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.44951388+0.68320679j  0.44951388-0.68320679j -0.29902775+0.j        ]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\moone\\Anaconda3\\lib\\site-packages\\numpy\\core\\_asarray.py:83: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y)')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "#  Create an array representing a polynomial\n",
    "P1 = [5, -3, 2, 1]\n",
    "\n",
    "#  Calculate and print the roots\n",
    "R1 = np.roots(P1)\n",
    "print(R1)\n",
    "\n",
    "#  Use polyval to evaulate an a numer of x-values\n",
    "x = np.linspace(-1, 1, 100)\n",
    "y = np.polyval(P1, x)\n",
    "\n",
    "#  Plot the function\n",
    "plt.plot(x, y, 'k')\n",
    "plt.plot(R1[-1], 0, 'o')\n",
    "plt.grid(True)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Poly1d\n",
    "\n",
    "Coefficient order is the same as in above\n",
    "\n",
    "Poly1d returns an object\n",
    "\n",
    "Some class atributes are calculated on creation\n",
    "- roots\n",
    "\n",
    "Member functions are\n",
    "- deriv\n",
    "- integ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([15, -6,  2])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "x = np.linspace(-1, 1, 100)\n",
    "\n",
    "#  Create a poly1d object\n",
    "P2 = np.poly1d(  [5, -3, 2, 1] )\n",
    "\n",
    "#   Plot it\n",
    "plt.plot(x, P2(x), 'k')\n",
    "plt.grid(True)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y)')\n",
    "\n",
    "#  Print the roots\n",
    "P2.roots\n",
    "\n",
    "#  Print out the coefficients\n",
    "P2.deriv().coef"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  Polynomial class\n",
    "\n",
    "<a href=\"https://numpy.org/doc/stable/reference/routines.polynomials.html\">Documentation</A>\n",
    "\n",
    "Coefficients are in ascending order\n",
    "\n",
    "There are numerous function for arithmetic (add, muiltply, etc.) calculus, and building polynomials in different ways."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.29902775+0.j          0.44951388-0.68320679j  0.44951388+0.68320679j]\n"
     ]
    },
    {
     "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": [
    "#  [5, -3, 2, 1]  Remember for this, the order of the coefficients is reversed.\n",
    "#  Create a Polynomial object\n",
    "P3 = np.polynomial.Polynomial( [1, 2, -3, 5] )\n",
    "\n",
    "x = np.linspace(-1, 1, 100)\n",
    "\n",
    "#  Plot it\n",
    "plt.plot(x, P3(x), 'k')\n",
    "plt.grid(True)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y)')\n",
    "\n",
    "#  print the roots\n",
    "print( P3.roots() )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  An example multiplying polynomials\n",
    "If we wish to multiply two polynomials together, for example, \n",
    "$$2x^2 + x$$\n",
    "and\n",
    "$$-x^3 + 2x^2 + 1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Polynomial([ 0.,  1.,  2.,  2.,  3., -2.], domain=[-1.,  1.], window=[-1.,  1.])]\n"
     ]
    }
   ],
   "source": [
    "#  Define the two polynomals\n",
    "A = np.polynomial.Polynomial( [0, 1, 2] )\n",
    "B = np.polynomial.Polynomial( [1, 0, 2, -1] )\n",
    "\n",
    "#  Find the product\n",
    "C = np.polynomial.polynomial.polymul(A, B)\n",
    "print(C)"
   ]
  },
  {
   "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}