commensurate_roots.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finding commensurate roots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In many applications the poles of the transfer function can be found by solving a polynomial. If the denominator is a polynoamial in $x = exp(-zT_{\\text{gcd}})$ for some delay $T_{\\text{gcd}}$ then this can be done. In the more general case network when the delays $T_1,...,T_n$ are not commensurate, a different method is needed, such as the root-finding algorithm implemented in Roots.py. Here we show how from the commensurate delays we can find roots in a desired frequency range."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 683,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as la"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 684,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import Potapov_Code.Time_Delay_Network as networks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 685,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: pylab import has clobbered these variables: ['roots']\n",
      "`%matplotlib` prevents importing * from pylab and numpy\n"
     ]
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%pylab inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 686,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "from sympy import init_printing\n",
    "init_printing() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 687,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from fractions import gcd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 688,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## To identify commensurate delays, we must use a decimal and NOT a binary representation of the delays,\n",
    "## as done by standard python floats/longs\n",
    "\n",
    "from decimal import Decimal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 689,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "    X = networks.Example3()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 690,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAN8AAAAUBAMAAAD/+PVoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ 0.1, \\quad 0.23, \\quad 0.1, \\quad 0.17\\right ]$$"
      ],
      "text/plain": [
       "[0.1, 0.23, 0.1, 0.17]"
      ]
     },
     "execution_count": 690,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.delays"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## get gcd for a list of integers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 691,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def gcd_lst(lst):\n",
    "    l = len(lst)\n",
    "    if l == 0:\n",
    "        return None\n",
    "    elif l == 1:\n",
    "        return lst[0]\n",
    "    elif l == 2:\n",
    "        return gcd(lst[0],lst[1])\n",
    "    else:\n",
    "        return gcd(lst[0],gcd_lst(lst[1:]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 692,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 692,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd_lst([1,2,3,4,5,6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 693,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIpm7MhCriUTv3c12VLge\nopIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABOSURBVAgdY2BUMnZgYAhjYH/BwJDKwDCTgWEWA0Oe\nA8O+ABAJBOsCgATHcxCTKwFEKoEIHgUQeYmBUYCBRYGBR4BBqrwoi4Fh37t3rxgAK5QOlzv7snYA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 693,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd_lst([2,4,8,6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 694,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAIom7VJlmdt1E7xDNIS4hGwAA\nAAlwSFlzAAAOxAAADsQBlSsOGwAAAFJJREFUCB1jYBBSMmFgYAxg8E1gYL/CwNvAwLmSYe8BBiAA\nijAwcDcBCUaNaCDJwKA1AURyrWZgFGBgBiq+DSK5DBg4LjKwJTD4FjAwTA21ZAAAM4UOK1ZklQ8A\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$3$$"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 694,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd_lst([3,27,300,6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## find the shortest commensurate delay"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 695,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def find_commensurate(delays):\n",
    "    mult = min([d.as_tuple().exponent for d in delays])\n",
    "    power = 10**-mult\n",
    "    delays = map(lambda x: x*power,delays)\n",
    "    int_gcd = gcd_lst(delays)\n",
    "    return int_gcd/power"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 696,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "delays = [Decimal('1.14'),Decimal('532.23423'),Decimal('0.06'),Decimal('0.1')]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 697,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Decimal('0.00001')"
      ]
     },
     "execution_count": 697,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gcd_delays = find_commensurate(delays)\n",
    "gcd_delays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 701,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Decimal('114'), Decimal('53223.423'), Decimal('6'), Decimal('1E+1')]"
      ]
     },
     "execution_count": 701,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(lambda z: z / gcd_delays, delays)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 702,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Decimal('0.01')"
      ]
     },
     "execution_count": 702,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## in general converting floats to Decimal shoudl be avoided because floats are stored in binary form.\n",
    "## converting to a string first will round the number.\n",
    "## Good practice would be to specify the delays in Decimals, then convert to floats later.\n",
    "\n",
    "gcd_delays = find_commensurate(map(lambda x: Decimal(str(x)),X.delays)) \n",
    "gcd_delays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 703,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Decimal('0.1'), Decimal('0.23'), Decimal('0.1'), Decimal('0.17')]"
      ]
     },
     "execution_count": 703,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Decimal_delays =map(lambda x: Decimal(str(x)),X.delays)\n",
    "Decimal_delays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 704,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "E = sp.Matrix(np.zeros_like(X.M1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 705,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = sp.symbols('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 706,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for i,delay in enumerate(Decimal_delays):\n",
    "    E[i,i] = x**int(delay / gcd_delays)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 708,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAABlCAMAAAAlOLwSAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}x^{10} & 0.0 & 0.0 & 0.0\\\\0.0 & x^{23} & 0.0 & 0.0\\\\0.0 & 0.0 & x^{10} & 0.0\\\\0.0 & 0.0 & 0.0 & x^{17}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 10               ⎤\n",
       "⎢x    0.0  0.0  0.0⎥\n",
       "⎢                  ⎥\n",
       "⎢      23          ⎥\n",
       "⎢0.0  x    0.0  0.0⎥\n",
       "⎢                  ⎥\n",
       "⎢           10     ⎥\n",
       "⎢0.0  0.0  x    0.0⎥\n",
       "⎢                  ⎥\n",
       "⎢                17⎥\n",
       "⎣0.0  0.0  0.0  x  ⎦"
      ]
     },
     "execution_count": 708,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 709,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "M1 = sp.Matrix(X.M1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 710,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfsAAABkCAMAAAC2EmnCAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nsG7Dbz1A7+jsZv4DrQa9//XkLyMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0.0 & -0.9 & 0.0 & 0.0\\\\-0.4 & 0.0 & 0.916515138991168 & 0.0\\\\0.0 & 0.0 & 0.0 & -0.8\\\\0.916515138991168 & 0.0 & 0.4 & 0.0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       0.0         -0.9         0.0         0.0 ⎤\n",
       "⎢                                                ⎥\n",
       "⎢      -0.4         0.0   0.916515138991168  0.0 ⎥\n",
       "⎢                                                ⎥\n",
       "⎢       0.0         0.0          0.0         -0.8⎥\n",
       "⎢                                                ⎥\n",
       "⎣0.916515138991168  0.0          0.4         0.0 ⎦"
      ]
     },
     "execution_count": 710,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 711,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgEAAABlCAMAAADjwsA2AAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\ngeJOOUYq3Hj58AVUTyp+xAOQ4R637H/zi6ZXW0/YygAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}x^{10} & 0.9 & 0 & 0\\\\0.4 & x^{23} & -0.916515138991168 & 0\\\\0 & 0 & x^{10} & 0.8\\\\-0.916515138991168 & 0 & -0.4 & x^{17}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        10                                      ⎤\n",
       "⎢       x            0.9          0            0 ⎥\n",
       "⎢                                                ⎥\n",
       "⎢                     23                         ⎥\n",
       "⎢       0.4          x    -0.916515138991168   0 ⎥\n",
       "⎢                                                ⎥\n",
       "⎢                                 10             ⎥\n",
       "⎢        0            0          x            0.8⎥\n",
       "⎢                                                ⎥\n",
       "⎢                                              17⎥\n",
       "⎣-0.916515138991168   0          -0.4         x  ⎦"
      ]
     },
     "execution_count": 711,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E - M1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 712,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sp.apart((E - M1).det())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 713,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = sp.Poly(expr, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 714,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfkAAAAcBAMAAACQZttSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\operatorname{Poly}{\\left( 1.0 x^{60} + 0.32 x^{33} - 0.36 x^{27} - 0.72, x, domain=\\mathbb{R} \\right)}$$"
      ],
      "text/plain": [
       "Poly(1.0*x**60 + 0.32*x**33 - 0.36*x**27 - 0.72, x, domain='RR')"
      ]
     },
     "execution_count": 714,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 715,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAADZ4AAAAUBAMAAABFFNTxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\left [ 1.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.32, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad -0.36, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad 0.0, \\quad -0.72\\right ]$$"
      ],
      "text/plain": [
       "[1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.\n",
       "0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.32, 0.0, 0.0, 0.0,\n",
       " 0.0, 0.0, -0.36, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, \n",
       "0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.72]"
      ]
     },
     "execution_count": 715,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poly_coeffs = a.all_coeffs()\n",
    "poly_coeffs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 716,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x122c258d0>"
      ]
     },
     "execution_count": 716,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaYAAAGgCAYAAADhKMMeAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1224f3dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "roots = np.roots(poly_coeffs)\n",
    "\n",
    "plt.figure(figsize =(6,6))\n",
    "fig = plt.scatter(roots.real,roots.imag)\n",
    "plt.title('Roots of the above polynomial',{'fontsize':24})\n",
    "plt.xlabel('Real part',{'fontsize':24})\n",
    "plt.ylabel('Imaginary part',{'fontsize':24})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Going from $x = exp(-zT_{\\text{gcd}})$ to solutions z. (Note the sign convention!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The roots repeat with a period of $2 \\pi / T_{\\text{gcd}} \\approx 628$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 717,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "zs = np.asarray(map(lambda r: np.log(r) / - float(gcd_delays), roots))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 718,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x122c6b590>"
      ]
     },
     "execution_count": 718,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAakAAAGgCAYAAAAQI5iTAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nBykzMystBykzMystBykzMyut/w9rfmq1CAGYvgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x122550610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize =(6,6))\n",
    "fig = plt.scatter(zs.real,zs.imag)\n",
    "plt.title('Roots of the above polynomial',{'fontsize':24})\n",
    "plt.xlabel('Real part',{'fontsize':24})\n",
    "plt.ylabel('Imaginary part',{'fontsize':24})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 555,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def find_instances_in_range(z,freq_range):\n",
    "    T = 2.*np.pi / float(lcm)\n",
    "    if z.imag >= freq_range[0] and z.imag <= freq_range[1]:\n",
    "        lst_in_range = [z]\n",
    "        num_below = int((z.imag - freq_range[0])/T )\n",
    "        num_above = int((freq_range[1] - z.imag)/T )\n",
    "        lst_in_range += [z + 1j * disp for disp in (np.asarray(range(num_above))+1) * T]\n",
    "        lst_in_range += [z - 1j * disp for disp in (np.asarray(range(num_below))+1) * T]\n",
    "        return lst_in_range\n",
    "    elif z.imag > freq_range[1]:\n",
    "        min_dist = (int((z.imag - freq_range[1])/T)+1) * T\n",
    "        max_dist = int((z.imag - freq_range[0]) / T) * T\n",
    "        if min_dist > max_dist:\n",
    "            return []\n",
    "        else:\n",
    "            return find_instances_in_range(z - 1j*min_dist,freq_range)\n",
    "    else:\n",
    "        min_dist = (int((freq_range[0] - z.imag)/T)+1) * T\n",
    "        max_dist = int((freq_range[1] - z.imag)/T)  * T\n",
    "        if min_dist > max_dist:\n",
    "            return []\n",
    "        else:\n",
    "            return find_instances_in_range(z + 1j*min_dist,freq_range)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 556,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAPBAMAAAAok50oAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n2q0Gj0YPjQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$104.71975512$$"
      ],
      "text/plain": [
       "104.71975512"
      ]
     },
     "execution_count": 556,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = zs[40]\n",
    "z.imag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 562,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## test cases..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 558,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.65220723077943077-1151.9173063162575j),\n",
       " (0.65220723077943077-1780.2358370342163j),\n",
       " (0.65220723077943077-2408.5543677521746j)]"
      ]
     },
     "execution_count": 558,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "find_instances_in_range(z,(-3000,-1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 559,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.65220723077943077+104.71975511965982j),\n",
       " (0.65220723077943077+733.03828583761845j),\n",
       " (0.65220723077943077-523.59877559829886j),\n",
       " (0.65220723077943077-1151.9173063162575j),\n",
       " (0.65220723077943077-1780.235837034216j),\n",
       " (0.65220723077943077-2408.5543677521746j)]"
      ]
     },
     "execution_count": 559,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "find_instances_in_range(z,(-3000,1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 560,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.65220723077943077+3246.3124087094534j),\n",
       " (0.65220723077943077+3874.6309394274122j),\n",
       " (0.65220723077943077+4502.9494701453705j)]"
      ]
     },
     "execution_count": 560,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "find_instances_in_range(z,(3000,5000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 561,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAUBAMAAABCNWFYAAAAG1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAB4Gco9AAAACHRSTlMAdt3NMolEZgN4ymIAAAAJcEhZcwAADsQAAA7EAZUrDhsA\nAAAXSURBVAgdYxAyKVZjCGMAIpoQQipCagCy4Q1mVUJFwQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\right ]$$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 561,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "find_instances_in_range(z,(3000,1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 567,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def find_roots_in_range(roots,freq_range):\n",
    "    return np.concatenate([find_instances_in_range(r,freq_range) for r in roots])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 571,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "roots_in_range = find_roots_in_range(zs,(-1000,1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 581,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x122ed59d0>"
      ]
     },
     "execution_count": 581,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbAAAALvCAYAAAAEfpCVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nlwgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1227d62d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize =(6,12))\n",
    "fig = plt.scatter(roots_in_range.real,roots_in_range.imag,label='extended')\n",
    "plt.title('Roots of the above polynomial',{'fontsize':24})\n",
    "plt.xlabel('Real part',{'fontsize':24})\n",
    "plt.ylabel('Imaginary part',{'fontsize':24})\n",
    "plt.scatter(zs.real,zs.imag,label='original',c='yellow')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scaling of finding roots of polynomial using Python\n",
    "\n",
    "The best we can hope for is $O(n^2)$ because this is a cost of an iteration in the QR algorithm, and ideally the number of iterations would not change much throughout the convergence process."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 285,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sample_pol_length = [10,50,100,1000,2000,3000]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 274,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 275,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ts = [time.clock()]\n",
    "for pol_length in sample_pol_length:\n",
    "    pol = [0]*(pol_length+1)\n",
    "    pol[0] = 1\n",
    "    pol[pol_length] = 5\n",
    "    np.roots(pol)\n",
    "    ts.append(time.clock())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 276,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "delta_ts = [ts[i+1]-ts[i] for i in range(len(ts)-1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 281,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 281,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEHCAYAAACncpHfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115cc57d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(sort(sample_pol_length),sort(delta_ts))\n",
    "plt.loglog()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## very roughly estimating the slope of the above log-log plot gives the exponent in $O(n^p)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 300,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.1371719506769975"
      ]
     },
     "execution_count": 300,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.log(delta_ts[-1])-np.log(delta_ts[0])) / (np.log(sample_pol_length[-1]) - np.log(sample_pol_length[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Extracting Potapov Projectors for Commensurate systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The slowest part of the Potapov analysis is generating the Potapov projection vectors once the roots of known. The runtime of this procedure is $O(n^2)$ because all of the previously known projectors must be invoked when finding each new projector. This results in slow performance when the number of modes is large (more than 1000 modes or so begins taking a substantial amount of time to process). Is there a way to reduce this overhead when the system is commensurate? \n",
    "\n",
    "There are two ways to address this issue. The first is to simply note that in the high-Q approximation linear modes don't interact much so that we can just consider a diagonal linear Hamiltonian to begin with. However, this yields no information for the B/C matrices, which still depend on the Potapov vectors. Although we can still get the spatial mode location, it would be nice to actually have an efficient method to compute the Potapov projectors.\n",
    "\n",
    "When the roots become periodic in the frequency direction of the complex plane, we might anticipate the Potapov vectors corresponding to them would also have the same periodicity (i.e. all poles $\\{  p_k + i n T | n \\in \\mathbb{R}\\}$ have the same Potapov vectors where $T$ is the periodicity for the system and $p_k$ is a pole).\n",
    "We can test this assertion by using a modified limiting procedure described below.\n",
    "\n",
    "In our original procedure we evaluated each $v_k$ in the product\n",
    "\\begin{align}\n",
    "T(z) = \\prod_k \\left(I - v_k^\\dagger v_k + v_k^\\dagger v_k \\left( \\frac{z + \\bar p_k}{z - p_k} \\right) \\right).\n",
    "\\end{align}\n",
    "This was done by multiplying by $z-p_k$ and taking $z \\to p_k$. Now suppose all $v_k$ are the same for some subset of poles (ones related by a period). In this case we can instead multiply by $\\prod_n (z_n - p_k - i n T)$ (multiplying by some factor to normalize both sides) and then take the limit $z_n \\to p_k^{(n)} \\equiv p_k + i n T$ (the limits will commute when $T(z)$ is meromorphic. We can also introduce a notion of taking the limits at the same time for these poles by using the same $\\delta-\\epsilon$, etc., since they are just translations of each other ):\n",
    "\\begin{align}\n",
    "\\lim_{z_n \\to p_k^{(n)}}\\left[ T(z) \\prod_n (z - p_k - i n T) \\right]= \\lim_{z_n \\to p_k^{(n)}}\\prod_n v_k^\\dagger v_k (z + \\bar p_k - i n T) ( z - p_k - i n T).\n",
    "\\end{align}\n",
    "If all the $v_k$ vectors are the same on the right hand side (let's also say they are normal), then the right hand side turns into\n",
    "\\begin{align}\n",
    "v_k^\\dagger v_k \\lim_{z_n \\to p_k^{(n)}}\\prod_n  (z_n + \\bar p_k^{(n)}) ( z - p_k^{(n)})\n",
    "= \n",
    "v_k^\\dagger v_k \\lim_{z \\to p_k}\\left(\\prod_n  (z_n + \\bar p_k) ( z - p_k)\\right)^n.\n",
    "\\end{align}\n",
    "This suggests we can just use the same $v_k$ vectors as for the particular root and it's periodic copies as long as we arrange the Potapov terms for the root and it's periodic copies adjacent to each other in the expansion.\n",
    "\n",
    "Another note is that the Potapov expansion can be fast to evaluate because matrix multiplicaiton only needs to be done between different commutative subspaces (i.e. all the periodic copies belong in one commensurate subset)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 639,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x12cfa4150>"
      ]
     },
     "execution_count": 639,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYwAAAEACAYAAACgS0HpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nM8vFAcPMzHL5fxwfeEBuA3M2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x12359f610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## The poles of the transfer function, no multiplicities included\n",
    "poles = -zs\n",
    "plt.scatter(poles.real,poles.imag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I ended up not using the stuff below. The Potapov vectors are implemented in Potapov_Code.Potapov.py."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 719,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "T = X.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 720,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from Potapov_Code.functions import limit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 761,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Potapov_prod(z,poles,vecs,N):\n",
    "    '''\n",
    "    Original Potapov product.\n",
    "    '''\n",
    "    R = np.asmatrix(np.eye(N))\n",
    "    for pole_i,vec in zip(poles,vecs):\n",
    "        Pi = vec*vec.H\n",
    "        R = R*(np.eye(N) - Pi + Pi * ( z + pole_i.conjugate() )/( z - pole_i) )\n",
    "    return R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 762,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def get_Potapov_vecs(T,poles):\n",
    "    '''\n",
    "    Original method to get vectors.\n",
    "    '''\n",
    "    N = T(0).shape[0]\n",
    "    found_vecs = []\n",
    "    for pole in poles:\n",
    "        L = (la.inv(Potapov_prod(pole,poles,found_vecs,N)) *\n",
    "            limit(lambda z: (z-pole)*T(z),pole) ) ## Current bottleneck O(n^2).\n",
    "        [eigvals,eigvecs] = la.eig(L)\n",
    "        index = np.argmax(map(abs,eigvals))\n",
    "        big_vec = np.asmatrix(eigvecs[:,index])\n",
    "        found_vecs.append(big_vec)\n",
    "    return found_vecs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def get_Potapov_vecs_commensurate(T,commensurate_poles):\n",
    "    '''\n",
    "    Original method to get vectors.\n",
    "    '''\n",
    "    N = T(0).shape[0]\n",
    "    found_vecs = []\n",
    "    for pole in poles:\n",
    "        L = (la.inv(Potapov_prod(pole,poles,found_vecs,N)) *\n",
    "            limit(lambda z: (z-pole)*T(z),pole) ) ## Current bottleneck O(n^2).\n",
    "        [eigvals,eigvecs] = la.eig(L)\n",
    "        index = np.argmax(map(abs,eigvals))\n",
    "        big_vec = np.asmatrix(eigvecs[:,index])\n",
    "        found_vecs.append(big_vec)\n",
    "    return found_vecs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 763,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_tot = 21"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 793,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def periodic_poles_prod(z,pole,gcd_delays,n_tot = 21):\n",
    "    '''\n",
    "    Return the normalized product :math:`prod_n (z - pole - i * n * T_{\\text{gcd}})`.\n",
    "    '''\n",
    "    n_range = np.asarray(range(n_tot)) - n_tot/2\n",
    "    return np.exp(np.sum([1j*np.angle(z-pole - 1j*n*2*np.pi / gcd_delays) for n in n_range]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 794,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.60804934203776106-0.793899236457276j)"
      ]
     },
     "execution_count": 794,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## testing...\n",
    "periodic_poles_prod(10000j-100,poles[0],float(gcd_delays),n_tot=33)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  },
  "widgets": {
   "state": {},
   "version": "1.1.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}