Using external libraries.ipynb

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   "source": [
    "There are a number of tremendously useful external libraries that you can install and use. **If you are using Anaconda** then most of these are already installed.\n",
    "\n",
    "If not, open a terminal (Mac OSX) or a command prompt (Windows) and type:\n",
    "\n",
    "```\n",
    "pip install <name of library>\n",
    "```\n",
    "\n",
    "replacing `<name of library>` with the actual name of the library. Here are some examples of libraries:\n",
    "\n",
    "- Sympy: symbolic computation library\n",
    "- Numpy: high performance numeric library\n",
    "\n",
    "For example to install the above we'd write:\n",
    "\n",
    "```\n",
    "pip install sympy\n",
    "pip install numpy\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sym  # Sympy is a library for symbolic mathematics\n",
    "sym.init_printing()  # Use LaTex to "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x, y = sym.symbols('x, y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
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      "text/latex": [
       "$$\\left ( x, \\quad y\\right )$$"
      ],
      "text/plain": [
       "(x, y)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "x, y"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "We can solve linear equations using `solveset`:"
   ]
  },
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   "execution_count": 9,
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    {
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      "text/latex": [
       "$$\\left\\{- \\frac{\\sqrt{2} i}{2}, \\frac{\\sqrt{2} i}{2}\\right\\}$$"
      ],
      "text/plain": [
       "⎧-√2⋅ⅈ   √2⋅ⅈ⎫\n",
       "⎨──────, ────⎬\n",
       "⎩  2      2  ⎭"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solveset(2 *  x ** 2 + 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "array = np.array([[1, 2], [3, 4]])  # Creating a numpy array\n",
    "array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2, -1],\n",
       "       [ 0,  1]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "array - 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5, 0, 3],\n",
       "       [3, 7, 9],\n",
       "       [3, 5, 2]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)  # numpy has it's own random number generator\n",
    "A = np.random.randint(0, 10, (3, 3))  # We can easily create a 3 by 3 array of random numbers\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0, 1, 0]), array([2, 2, 2]))"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.random.randint(0, 2, 3)\n",
    "v = np.random.randint(2, 4, 3)\n",
    "u, v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "38"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(np.dot(u, A), v)  # Matrix mulitplication u * A * v"
   ]
  }
 ],
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