Random polynomials.ipynb

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   "source": [
    "## Exercise for the course [Python for MATLAB users](http://sese.nu/python-for-matlab-users/), by Olivier Verdier"
   ]
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     "text": [
      "Using matplotlib backend: MacOSX\n",
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab\n",
    "%matplotlib inline"
   ]
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   "source": [
    "Check out the formula for a [companion matrix on wikipedia](http://en.wikipedia.org/wiki/Companion_matrix).\n",
    "Define a function `companion` which accepts a vector in argument, and returns the corresponding companion matrix.\n",
    "You can use the command `diag` for that.\n",
    "The resulting matrix should be of complex type."
   ]
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   "source": [
    "diag?"
   ]
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   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "ones?"
   ]
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   "source": [
    "def companion(coefficients):\n",
    "    pass"
   ]
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   "source": [
    "C = companion(ones(3))\n",
    "assert(C.dtype == complex)\n",
    "assert(len(C) == 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fix a given size, say `size = 20`, and create a vector of length 20 with random, normally distributed, complex numbers. Use the `randn` function for that, and combine two random real vectors to get a random complex vector."
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": [
    "randn?"
   ]
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   "source": [
    "Now, fix a standard deviation, say `sigma = 1./10`, and use the random complex coefficients obtained above, multiplied by sigma, in the `companion` function. Use `eigvals` to compute the eigenvalues, and plot them on the complex plane. You can use the command `axis('equal')` to make sure the plot has the same dimensions in x and y."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, repeat that, say 200 times. Plot all the eigenvalues on the same figure. What do you observe? What happens when you change the standard deviation `sigma`?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "size = 20\n",
    "for i in range(200):\n",
    "    pass\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": []
  }
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