ExampleNumpyNotebook.ipynb

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    "#### <p style=\"text-align: center; \"> PDS part Two </p>\n",
    "# <p style=\"text-align: center;\"> *Example Numpy/Matplotlib Notebook* </p>\n",
    "---\n",
    " \n",
    " "
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    "## Python project: Zeros of polynomials with 0,1 coefficients\n",
    "\n",
    "$$ a_n z^n +  a_{n-1}z^{n-1} + \\cdots + a_1 z^1 + a_0=0\\\\$$ \n",
    "$$a_i\\in \\{0,1\\}$$"
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   "source": [
    "## Python project: additional references\n",
    "* \"A Digital Slice of Pi\", W. Wayt Gibbs. *Scientific American Magazine*, May 2003.\n",
    "* https://johncarlosbaez.wordpress.com/2011/12/11/the-beauty-of-roots/ (blog)\n"
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    "## What are zeros $z=x+iy$ of polynomials?\n",
    "Numbers of the form $x+iy$, where $x$ and $y$ are any real numbers, are called *complex numbers*.\n",
    "\n",
    "Given a polynomial $$P(z)=a_nz^n + \\cdots + a_1z+a_0$$\n",
    "the complex numbers $z=x+iy$ for which $P(z)=0$ are called the *roots* (also known as *zeros*) of the polynomial.\n",
    "\n"
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    "We wil use the numpy function \"roots\", which, given the coefficients $(a_n, \\cdots, a_1, a_0)$ of a polynomial $$P(z)=a_nz^n + \\cdots + a_1z+a_0$$ returns the complex numbers $z$ for which $P(z)=0$.\n",
    "\n",
    "We will demonstate in the following cells how to use Numpy and Matplotlib to visualize such zeros. First import the modules we need."
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   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt"
   ]
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     "text": [
      "Help on function roots in module numpy.lib.polynomial:\n",
      "\n",
      "roots(p)\n",
      "    Return the roots of a polynomial with coefficients given in p.\n",
      "    \n",
      "    The values in the rank-1 array `p` are coefficients of a polynomial.\n",
      "    If the length of `p` is n+1 then the polynomial is described by::\n",
      "    \n",
      "      p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n]\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    p : array_like\n",
      "        Rank-1 array of polynomial coefficients.\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    out : ndarray\n",
      "        An array containing the complex roots of the polynomial.\n",
      "    \n",
      "    Raises\n",
      "    ------\n",
      "    ValueError\n",
      "        When `p` cannot be converted to a rank-1 array.\n",
      "    \n",
      "    See also\n",
      "    --------\n",
      "    poly : Find the coefficients of a polynomial with a given sequence\n",
      "           of roots.\n",
      "    polyval : Compute polynomial values.\n",
      "    polyfit : Least squares polynomial fit.\n",
      "    poly1d : A one-dimensional polynomial class.\n",
      "    \n",
      "    Notes\n",
      "    -----\n",
      "    The algorithm relies on computing the eigenvalues of the\n",
      "    companion matrix [1]_.\n",
      "    \n",
      "    References\n",
      "    ----------\n",
      "    .. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*.  Cambridge, UK:\n",
      "        Cambridge University Press, 1999, pp. 146-7.\n",
      "    \n",
      "    Examples\n",
      "    --------\n",
      "    >>> coeff = [3.2, 2, 1]\n",
      "    >>> np.roots(coeff)\n",
      "    array([-0.3125+0.46351241j, -0.3125-0.46351241j])\n",
      "\n"
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   "source": [
    "help(np.roots)"
   ]
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    "So what is $i$?"
   ]
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    "So what is  i? It is a zero of the polynomial\n",
    "$$P(z)=z^2+1$$\n",
    " \n",
    "That is,"
   ]
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    "$$ i^2 + 1 = 0$$\n"
   ]
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   "source": [
    "So, $i$ is a square root of $-1$, since\n",
    "$$i^2 = -1$$\n"
   ]
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    "One plots complex numbers $z=x+iy$ as points $(x,y)$ in the $xy$-plane. \n",
    "\n",
    "The complex number $z=0+i1$ is denoted simply $i$.\n",
    "\n",
    "Let's plot the zeros of $$z^2+1=0$$"
   ]
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    "coeff = [1, 0, 1]"
   ]
  },
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   "source": [
    "Solutions = np.roots([1,0,1])"
   ]
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ad2eeae5f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(Solutions.real, Solutions.imag)\n",
    "plt.ylabel('Imaginary')\n",
    "plt.xlabel('Real')\n",
    "plt.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now, you try plotting the zeros to the polynomial equation\n",
    "$$P(z)=z^{13} + 1$$\n",
    "\n",
    "in the cells below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "nbpresent": {
     "id": "48dc6ed7-80f6-4257-9adb-bd9493ddb077"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Interested in interacting with these images?\n",
    "\n",
    "\n",
    "Move on to the Bokeh workbook."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": "3.5.2"
  },
  "nbpresent": {
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