numpy-array-operations.ipynb

{
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   "source": [
    "# Assignment 2: Using Numpy to Perform Basic Operations on Polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motivation\n",
    "The objective of this assignment is to develop a solid understanding of Numpy array operations. In this assignment you will:\n",
    "- Pick 5 interesting Numpy array functions by going through the documentation: https://numpy.org/doc/stable/reference/routines.html\n",
    "- Illustrate their usage with examples "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "A polynomial equation is of the form $a_nx^n  +  a_{n-1}x^{n-1}  +  ...  +  a_2x^2  +  a_1x  +  a_0 = 0$ where $a_n, a_{n-1}, ... a_1, a_0$ are coefficients of the equation. A polynomial equation thus can be represented using an array of numbers which represents its coefficients. For example `[2, 3, -4]` represent the polynomial $2x^2 + 3x - 4 = 0$. The Numpy library has a lot of functions which can be used to perform mathematical operations on polynomial equation such as add, multiply, divide two polynomials and so on.\n",
    "\n",
    "We will choose five functions and each of the function does the following: \n",
    "\n",
    "- The first function creates a polynomial; an array of numbers corresponding to an equation\n",
    "- The second function adds two polynomials\n",
    "- This function multiplies two polynomials\n",
    "- This function differentiates a given polynomial\n",
    "- The last function integrates the polynomial"
   ]
  },
  {
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   "source": [
    "!pip install jovian --upgrade -q"
   ]
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   "source": [
    "import jovian"
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    {
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       "window.require && require([\"base/js/namespace\"],function(Jupyter){Jupyter.notebook.save_checkpoint()})"
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     "text": [
      "[jovian] Updating notebook \"oyebamijimicheal10/numpy-array-operations\" on https://jovian.ai\u001b[0m\n",
      "[jovian] Committed successfully! https://jovian.ai/oyebamijimicheal10/numpy-array-operations\u001b[0m\n"
     ]
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     "data": {
      "text/plain": [
       "'https://jovian.ai/oyebamijimicheal10/numpy-array-operations'"
      ]
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     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "jovian.commit(project='numpy-array-operations')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "function1 = np.poly1d \n",
    "function2 = np.polyadd\n",
    "function3 = np.polymul\n",
    "function4 = np.polyder\n",
    "function5 = np.polyint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Function 1 -  np.poly1d\n",
    "\n",
    "This functions creates a polynomial equation using an array of coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   2\n",
      "2 x - 3 x - 3\n"
     ]
    }
   ],
   "source": [
    "# Example 1 - working\n",
    "poly1 = np.poly1d([2, -3, -3]) # A quadratic polynomial\n",
    "print(poly1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The coefficients `2, -3, -3` represent the equation $2x^2 - 3x - 3$ in that order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   3     2\n",
      "1 x + 3 x + 1 x - 2\n"
     ]
    }
   ],
   "source": [
    "# Example 2 - working\n",
    "poly2 = np.poly1d([1, 3, 1, -2]) # A cubic polynomial\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "Polynomial must be 1d only.",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_35/2586751491.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# Example 3 - breaking (to illustrate when it breaks)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mpoly3\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m6\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/opt/conda/lib/python3.9/site-packages/numpy/lib/polynomial.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, c_or_r, r, variable)\u001b[0m\n\u001b[1;32m   1234\u001b[0m         \u001b[0mc_or_r\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0matleast_1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_or_r\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1235\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mc_or_r\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1236\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Polynomial must be 1d only.\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1237\u001b[0m         \u001b[0mc_or_r\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtrim_zeros\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_or_r\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtrim\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'f'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1238\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_or_r\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: Polynomial must be 1d only."
     ]
    }
   ],
   "source": [
    "# Example 3 - breaking (to illustrate when it breaks)\n",
    "poly3 = np.poly1d([[1, 2, 3], [4, 5, 6]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The np.poly1d takes only one dimensional array as argument. \n",
    "Be sure to pass a one dimensional array**"
   ]
  },
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   "cell_type": "code",
   "execution_count": 21,
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    {
     "data": {
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       "window.require && require([\"base/js/namespace\"],function(Jupyter){Jupyter.notebook.save_checkpoint()})"
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     "text": [
      "[jovian] Updating notebook \"oyebamijimicheal10/numpy-array-operations\" on https://jovian.ai\u001b[0m\n",
      "[jovian] Committed successfully! https://jovian.ai/oyebamijimicheal10/numpy-array-operations\u001b[0m\n"
     ]
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     "data": {
      "text/plain": [
       "'https://jovian.ai/oyebamijimicheal10/numpy-array-operations'"
      ]
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    "jovian.commit()"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Function 2 - np.polyadd\n",
    "\n",
    "Given two polynomials, this function returns a single polynomial corresponding to their addition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    2\n",
      "-2 x + 6 x + 5\n"
     ]
    }
   ],
   "source": [
    "# Example 1 - working\n",
    "poly1 = np.poly1d([1, 2, 3])\n",
    "poly2 = np.poly1d([-3, 4, 2])\n",
    "poly3 = np.polyadd(poly1, poly2)\n",
    "print(poly3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Respective coefficients are added and the answer is returned"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   4     3     2\n",
      "3 x - 6 x + 7 x\n"
     ]
    }
   ],
   "source": [
    "# Example 2 - working\n",
    "poly1 = np.poly1d([3, -6, 7, -2, 1])\n",
    "poly2 = np.poly1d([2, -1])\n",
    "poly3 = np.polyadd(poly1, poly2)\n",
    "print(poly3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When two polynomials of unequal degree are added, the resulting polynomial's degree is of the largest polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "_binary_op_dispatcher() missing 1 required positional argument: 'a2'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_35/3629601038.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0mpoly1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m6\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mpoly2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mpoly3\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolyadd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mpoly1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpoly2\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<__array_function__ internals>\u001b[0m in \u001b[0;36mpolyadd\u001b[0;34m(*args, **kwargs)\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: _binary_op_dispatcher() missing 1 required positional argument: 'a2'"
     ]
    }
   ],
   "source": [
    "# Example 3 - breaking (to illustrate when it breaks)\n",
    "poly1 = np.poly1d([3, -5, 6])\n",
    "poly2 = np.poly1d([2])\n",
    "poly3 = np.polyadd([poly1, poly2])\n",
    "print(poly3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "np.polyadd function does not takes in list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The np.polyadd function can also be used to add two list which are not numpy array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "window.require && require([\"base/js/namespace\"],function(Jupyter){Jupyter.notebook.save_checkpoint()})"
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       "<IPython.core.display.Javascript object>"
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     "text": [
      "[jovian] Updating notebook \"oyebamijimicheal10/numpy-array-operations\" on https://jovian.ai\u001b[0m\n",
      "[jovian] Committed successfully! https://jovian.ai/oyebamijimicheal10/numpy-array-operations\u001b[0m\n"
     ]
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    {
     "data": {
      "text/plain": [
       "'https://jovian.ai/oyebamijimicheal10/numpy-array-operations'"
      ]
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   "source": [
    "## Function 3 - np.polymul\n",
    "\n",
    "This function returns the multiplication of two arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    4     3     2\n",
      "-3 x - 2 x + 1 x + 16 x + 6\n"
     ]
    }
   ],
   "source": [
    "# Example 1 - working\n",
    "poly1 = np.poly1d([1, 2, 3])\n",
    "poly2 = np.poly1d([-3, 4, 2])\n",
    "poly3 = np.polymul(poly1, poly2)\n",
    "print(poly3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is straight forward. The function evaluates the multiplication of the polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   5      4      3      2\n",
      "6 x - 15 x + 20 x - 11 x + 4 x - 1\n"
     ]
    }
   ],
   "source": [
    "# Example 2 - working\n",
    "poly1 = np.poly1d([3, -6, 7, -2, 1])\n",
    "poly2 = np.poly1d([2, -1])\n",
    "poly3 = np.polymul(poly1, poly2)\n",
    "print(poly3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function returns the expected result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "_binary_op_dispatcher() takes 2 positional arguments but 3 were given",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_35/1270562916.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0mpoly1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m7\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mpoly2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mpoly3\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolymul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpoly2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<__array_function__ internals>\u001b[0m in \u001b[0;36mpolymul\u001b[0;34m(*args, **kwargs)\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: _binary_op_dispatcher() takes 2 positional arguments but 3 were given"
     ]
    }
   ],
   "source": [
    "# Example 3 - breaking (to illustrate when it breaks)\n",
    "poly1 = np.poly1d([3, -6, 7, -2, 1])\n",
    "poly2 = np.poly1d([2, -1])\n",
    "poly3 = np.polymul(poly1, poly2, int)\n",
    "print(poly3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "np.polymul takes only two arguments"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the Algorithm and Data Structure course, an assignment was given to write a function which multiplies two polynomial using divide an conquer method. It is nice to know there is another method to do that aside the Fast Fourier Transform."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "window.require && require([\"base/js/namespace\"],function(Jupyter){Jupyter.notebook.save_checkpoint()})"
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    {
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     "output_type": "stream",
     "text": [
      "[jovian] Updating notebook \"oyebamijimicheal10/numpy-array-operations\" on https://jovian.ai\u001b[0m\n",
      "[jovian] Committed successfully! https://jovian.ai/oyebamijimicheal10/numpy-array-operations\u001b[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "'https://jovian.ai/oyebamijimicheal10/numpy-array-operations'"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "jovian.commit()"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Function 4 - np.polyder\n",
    "\n",
    "Calculus is one of the most important topics in mathematics. It's applications ranges from engineering, medicine, computer science, economics and every other professional discipline. Two main aspect of calculus are differentiation and integration. The np.polyder is used to compute the differential of a polynomial equation. $d/dx(ax^n) = anx^{n-1} n \\neq 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    3      2\n",
      "12 x - 18 x + 14 x - 2\n"
     ]
    }
   ],
   "source": [
    "# Example 1 - working\n",
    "poly1 = np.poly1d([3, -6, 7, -2, 1])\n",
    "poly2 = np.polyder(poly1)\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This equation is based on the formula $d/dx(ax^n) = anx^{n-1} n \\neq 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " \n",
      "0\n"
     ]
    }
   ],
   "source": [
    "# Example 2 - working\n",
    "poly1 = np.poly1d([3])\n",
    "poly2 = np.polyder(poly1)\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The derivation of a constant is zero. It worked as formulated!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "__init__() missing 1 required positional argument: 'c_or_r'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_35/1078525409.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# Example 3 - breaking (to illustrate when it breaks)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mpoly1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      3\u001b[0m \u001b[0mpoly2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolyder\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: __init__() missing 1 required positional argument: 'c_or_r'"
     ]
    }
   ],
   "source": [
    "# Example 3 - breaking (to illustrate when it breaks)\n",
    "poly1 = np.poly1d()\n",
    "poly2 = np.polyder(poly1)\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We cannot take the derivative of nothing. We need to pass a valid polynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This computes the derivative of polynomial equations only. It cannot be used for logarithmic, exponential, trigonometric etc. functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
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       "window.require && require([\"base/js/namespace\"],function(Jupyter){Jupyter.notebook.save_checkpoint()})"
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     "output_type": "stream",
     "text": [
      "[jovian] Updating notebook \"oyebamijimicheal10/numpy-array-operations\" on https://jovian.ai\u001b[0m\n",
      "[jovian] Committed successfully! https://jovian.ai/oyebamijimicheal10/numpy-array-operations\u001b[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "'https://jovian.ai/oyebamijimicheal10/numpy-array-operations'"
      ]
     },
     "execution_count": 58,
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     "output_type": "execute_result"
    }
   ],
   "source": [
    "jovian.commit()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Function 5 - np.polyint\n",
    "\n",
    "The inverse of differentiation is integration. Integration is arguably the most important aspect of calculus. Basic integration operation can be performed on polynomials using the np.polyint function. The integration formula for polynomial is \n",
    "$$\n",
    "\\int x^n = \\frac{x^{n+1}}{n+1} + C \n",
    "$$\n",
    "$n \\neq -1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      4          3       2\n",
      "0.25 x + 0.6667 x - 1.5 x + 2 x\n"
     ]
    }
   ],
   "source": [
    "# Example 1 - working\n",
    "poly1 = np.poly1d([1, 2, -3, 2])\n",
    "poly2 = np.polyint(poly1)\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result is based on the formula \n",
    "$$\n",
    "\\int x^n = \\frac{x^{n+1}}{n+1} + C \n",
    "$$\n",
    "$n \\neq -1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   2\n",
      "2 x + 2 x\n"
     ]
    }
   ],
   "source": [
    "# Example 2 - working\n",
    "poly1 = np.poly1d([4, 2])\n",
    "poly2 = np.polyint(poly1)\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The functions returns the integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "__init__() missing 1 required positional argument: 'c_or_r'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_35/1929361757.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# Example 3 - breaking (to illustrate when it breaks)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mpoly1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoly1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      3\u001b[0m \u001b[0mpoly2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolyint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoly2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: __init__() missing 1 required positional argument: 'c_or_r'"
     ]
    }
   ],
   "source": [
    "# Example 3 - breaking (to illustrate when it breaks)\n",
    "poly1 = np.poly1d()\n",
    "poly2 = np.polyint(poly1)\n",
    "print(poly2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We cannot take the derivative of nothing. We need to pass a valid polynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can verify that differentiation operation is the inverse of integration by either calling the integration function on the result of a differential operation and vice versa"
   ]
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   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
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     "text": [
      "[jovian] Updating notebook \"oyebamijimicheal10/numpy-array-operations\" on https://jovian.ai\u001b[0m\n",
      "[jovian] Committed successfully! https://jovian.ai/oyebamijimicheal10/numpy-array-operations\u001b[0m\n"
     ]
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     "data": {
      "text/plain": [
       "'https://jovian.ai/oyebamijimicheal10/numpy-array-operations'"
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     "output_type": "execute_result"
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    "jovian.commit()"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "I have been able to cover the following;\n",
    "- create a polynomial using an array\n",
    "- add two polynomials\n",
    "- multiply\n",
    "- differentiate\n",
    "and \n",
    "- integrate polynomial equations\n",
    "\n",
    "However, the functions forms part of the old polynomial API. I need to get familiar with the new API defined in numpy.polynomial. \n",
    "Where to go from here? I think the answer to that will be '100 numpy exercises'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reference Links\n",
    "Provide links to your references and other interesting articles about Numpy arrays:\n",
    "* Numpy official tutorial : https://numpy.org/doc/stable/user/quickstart.html"
   ]
  },
  {
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   "execution_count": null,
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