consolidated_taylor_perturbation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.sparse.linalg import eigs\n",
    "import scipy.sparse as sparse\n",
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "import sympy as sp\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "import Potapov_Code.Time_Delay_Network as networks\n",
    "from Potapov_Code.functions import spatial_modes\n",
    "from decimal import Decimal\n",
    "\n",
    "from sympy import init_printing\n",
    "init_printing() \n",
    "\n",
    "from sympy.utilities.autowrap import ufuncify\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import Potapov_Code.Time_Delay_Network"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from Potapov_Code.functions import gcd_lst"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X1 = Potapov_Code.Time_Delay_Network.Example3()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def _find_commensurate(delays):\n",
    "    '''\n",
    "    Find the 'gcd' but for Decimal numbers.\n",
    "\n",
    "    Args:\n",
    "        delays(list of Demicals): numbers whose gcd will be found.\n",
    "\n",
    "    Returns:\n",
    "        Decimal gcd.\n",
    "    '''\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\n",
    "\n",
    "Decimal_delays = map(lambda x: Decimal(str(x)),X1.delays)\n",
    "Decimal_gcd = _find_commensurate(Decimal_delays)\n",
    "\n",
    "xs = [sp.symbols('x_'+str(i)) for i in range(4) ]\n",
    "E_sym = sp.Matrix(np.zeros_like(X1.M1))\n",
    "for i,delay in enumerate(Decimal_delays):\n",
    "    E_sym[i,i] = xs[i]\n",
    "M1_sym = sp.Matrix(X1.M1)\n",
    "num, den = (E_sym - M1_sym).det().as_numer_denom()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Alternatively, we can Taylor expand about the $z$ variable only. This gives just Newton's method. To first order we get\n",
    "$$\n",
    "\\Delta z = - \\frac{f(z,T+\\Delta T)}{\\frac{\\partial f}{\\partial z (z,T+\\Delta T}}.\n",
    "$$\n",
    "This can be used as an iterative procedure with the substitution $z \\mapsto z + \\Delta_z$. In addition, dependence on $\\Delta T $ can be incorporated by evaluation $\\Delta T \\equiv \\Delta T(z + \\Delta z)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA2YAAAAcBAMAAAD4qrH3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( 1.0 x_{0}^{2} x_{1}^{2} x_{2} x_{3} + 0.32 x_{0}^{2} x_{1}^{2} - 0.72 x_{0} x_{1} x_{2} x_{3} - 0.8352 x_{0} x_{1} + 0.1296 x_{2} x_{3} + 0.2592, \\quad 1.0 x_{0} x_{1} - 0.36\\right )$$"
      ],
      "text/plain": [
       "⎛      2   2                2   2                                             \n",
       "⎝1.0⋅x₀ ⋅x₁ ⋅x₂⋅x₃ + 0.32⋅x₀ ⋅x₁  - - -0.72⋅x₀⋅x₁⋅x₂⋅x₃ - - -0.8352⋅x₀⋅x₁ + 0.\n",
       "\n",
       "                                     ⎞\n",
       "1296⋅x₂⋅x₃ + 0.2592, 1.0⋅x₀⋅x₁ - 0.36⎠"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num, den"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "z, z_Delta = sp.symbols('z dz',complex=True)\n",
    "Ts = [sp.symbols('T_'+str(i),real=True) for i in range(4)]\n",
    "x,y = sp.symbols('x y', real = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "num2 = num.subs({x: sp.exp(-z*T) for x,T in zip(xs,Ts)})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABD8AAAAVBAMAAACzn1g+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$0.2592 + 0.1296 e^{- T_{2} z} e^{- T_{3} z} - 0.8352 e^{- T_{0} z} e^{- T_{1} z} - 0.72 e^{- T_{0} z} e^{- T_{1} z} e^{- T_{2} z} e^{- T_{3} z} + 0.32 e^{- 2 T_{0} z} e^{- 2 T_{1} z} + 1.0 e^{- 2 T_{0} z} e^{- 2 T_{1} z} e^{- T_{2} z} e^{- T_{3} z}$$"
      ],
      "text/plain": [
       "                 -T₂⋅z  -T₃⋅z              -T₀⋅z  -T₁⋅z            -T₀⋅z  -T₁⋅\n",
       "0.2592 + 0.1296⋅ℯ     ⋅ℯ      - - -0.8352⋅ℯ     ⋅ℯ      - - -0.72⋅ℯ     ⋅ℯ    \n",
       "\n",
       "z  -T₂⋅z  -T₃⋅z         -2⋅T₀⋅z  -2⋅T₁⋅z        -2⋅T₀⋅z  -2⋅T₁⋅z  -T₂⋅z  -T₃⋅z\n",
       " ⋅ℯ     ⋅ℯ      + 0.32⋅ℯ       ⋅ℯ        + 1.0⋅ℯ       ⋅ℯ       ⋅ℯ     ⋅ℯ     "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "num3 = num2.subs({sp.exp(-z*T): sp.cos(-x*T)*(1j*sp.sin(y*T)+sp.cos(y*T)) for T in Ts})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAE1oAAAAaBAMAAADyyvSXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nP3uyocAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.0 \\left(1.0 i \\sin{\\left (T_{0} y \\right )} + \\cos{\\left (T_{0} y \\right )}\\right)^{2} \\left(1.0 i \\sin{\\left (T_{1} y \\right )} + \\cos{\\left (T_{1} y \\right )}\\right)^{2} \\left(1.0 i \\sin{\\left (T_{2} y \\right )} + \\cos{\\left (T_{2} y \\right )}\\right) \\left(1.0 i \\sin{\\left (T_{3} y \\right )} + \\cos{\\left (T_{3} y \\right )}\\right) \\cos^{2}{\\left (T_{0} x \\right )} \\cos^{2}{\\left (T_{1} x \\right )} \\cos{\\left (T_{2} x \\right )} \\cos{\\left (T_{3} x \\right )} + 0.32 \\left(1.0 i \\sin{\\left (T_{0} y \\right )} + \\cos{\\left (T_{0} y \\right )}\\right)^{2} \\left(1.0 i \\sin{\\left (T_{1} y \\right )} + \\cos{\\left (T_{1} y \\right )}\\right)^{2} \\cos^{2}{\\left (T_{0} x \\right )} \\cos^{2}{\\left (T_{1} x \\right )} - 0.72 \\left(1.0 i \\sin{\\left (T_{0} y \\right )} + \\cos{\\left (T_{0} y \\right )}\\right) \\left(1.0 i \\sin{\\left (T_{1} y \\right )} + \\cos{\\left (T_{1} y \\right )}\\right) \\left(1.0 i \\sin{\\left (T_{2} y \\right )} + \\cos{\\left (T_{2} y \\right )}\\right) \\left(1.0 i \\sin{\\left (T_{3} y \\right )} + \\cos{\\left (T_{3} y \\right )}\\right) \\cos{\\left (T_{0} x \\right )} \\cos{\\left (T_{1} x \\right )} \\cos{\\left (T_{2} x \\right )} \\cos{\\left (T_{3} x \\right )} - 0.8352 \\left(1.0 i \\sin{\\left (T_{0} y \\right )} + \\cos{\\left (T_{0} y \\right )}\\right) \\left(1.0 i \\sin{\\left (T_{1} y \\right )} + \\cos{\\left (T_{1} y \\right )}\\right) \\cos{\\left (T_{0} x \\right )} \\cos{\\left (T_{1} x \\right )} + 0.1296 \\left(1.0 i \\sin{\\left (T_{2} y \\right )} + \\cos{\\left (T_{2} y \\right )}\\right) \\left(1.0 i \\sin{\\left (T_{3} y \\right )} + \\cos{\\left (T_{3} y \\right )}\\right) \\cos{\\left (T_{2} x \\right )} \\cos{\\left (T_{3} x \\right )} + 0.2592$$"
      ],
      "text/plain": [
       "                                 2                              2             \n",
       "1.0⋅(1.0⋅ⅈ⋅sin(T₀⋅y) + cos(T₀⋅y)) ⋅(1.0⋅ⅈ⋅sin(T₁⋅y) + cos(T₁⋅y)) ⋅(1.0⋅ⅈ⋅sin(T\n",
       "\n",
       "                                                   2          2               \n",
       "₂⋅y) + cos(T₂⋅y))⋅(1.0⋅ⅈ⋅sin(T₃⋅y) + cos(T₃⋅y))⋅cos (T₀⋅x)⋅cos (T₁⋅x)⋅cos(T₂⋅x\n",
       "\n",
       "                                                2                             \n",
       ")⋅cos(T₃⋅x) + 0.32⋅(1.0⋅ⅈ⋅sin(T₀⋅y) + cos(T₀⋅y)) ⋅(1.0⋅ⅈ⋅sin(T₁⋅y) + cos(T₁⋅y)\n",
       "\n",
       " 2    2          2                                                            \n",
       ") ⋅cos (T₀⋅x)⋅cos (T₁⋅x) - - -0.72⋅(1.0⋅ⅈ⋅sin(T₀⋅y) + cos(T₀⋅y))⋅(1.0⋅ⅈ⋅sin(T₁\n",
       "\n",
       "                                                                              \n",
       "⋅y) + cos(T₁⋅y))⋅(1.0⋅ⅈ⋅sin(T₂⋅y) + cos(T₂⋅y))⋅(1.0⋅ⅈ⋅sin(T₃⋅y) + cos(T₃⋅y))⋅c\n",
       "\n",
       "                                                                              \n",
       "os(T₀⋅x)⋅cos(T₁⋅x)⋅cos(T₂⋅x)⋅cos(T₃⋅x) - - -0.8352⋅(1.0⋅ⅈ⋅sin(T₀⋅y) + cos(T₀⋅y\n",
       "\n",
       "                                                                              \n",
       "))⋅(1.0⋅ⅈ⋅sin(T₁⋅y) + cos(T₁⋅y))⋅cos(T₀⋅x)⋅cos(T₁⋅x) + 0.1296⋅(1.0⋅ⅈ⋅sin(T₂⋅y)\n",
       "\n",
       "                                                                        \n",
       " + cos(T₂⋅y))⋅(1.0⋅ⅈ⋅sin(T₃⋅y) + cos(T₃⋅y))⋅cos(T₂⋅x)⋅cos(T₃⋅x) + 0.2592"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "num_real,num_imag = num3.expand().as_real_imag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARYAAAAUBAMAAAC63A/DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEEKsiZlS7\nme+E9sVtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAC1UlEQVRIDZ1UTWsTURQ9iTP5mDR1Ggi6kbYi\nEV1FcOGuAXduMl0YFCzkH3S666qtdNNds3KhYAYEd2K3omDBTRdCgxUspWB140JBpWrFr3jvm+k0\nc99MaHJJ3rx3zz2X8947MyhdxrFjkNpjN1WFDyqgX2MAUgP5fxurf1amHJ2UjLQ/rXQ3lr/3p5CM\nxoBaTAejEyh6euNk5DrwHtjpTxlCSw5od2A19cbJyBLyf4GH/SlDaLkGPF5DytUbJyKGB+s3cKo/\nZQgtNWDVRUbvi0TEAHIHQFPn9FIOtcydvlO2RWn2jI37cvtEUHcvag+X5Ir4IIslREgJtGQdazJd\nFcVmYQ/jIqd8znePcllCtFaIIRtxIVkM5u4bnaMo09u050CL6ZoHhZYovFes4qXIKS1895ZTWJIY\nZQkxtid0gC2Gm1jWEaYUveJ6qCWLE7QQ4Y7a+CpySgvffboJGmUwgsKETNOaLIYLqMv9+kYqOtaX\nUAtw0tYbzLrGN5llv/DdL9bwUWI+Eq+FXVFBu6lxfCP1nAtQr2lV2EJqT2ZZC9897fCixHwkVoty\nBfDc1TjcDFiohedScMdjvhgbGFniwt5gLXz3dReTvXl/zkisFnYFRRIldYnAwLuza5u4i+y+IoTD\nLmY9mWMtm7Q7Opd3GkEhSstiNeyiJiPKXGaHLlcg3Izcux5qmR+b2/GAsyrPmIr56bctmWvAfNG9\nYrNffkiCjygt6fOHTdTz1tT+bZo8on8UCSgw6NyCcwmI2mu9RSfeCkD/wefC4b9HAvQh9R498eeR\nMeNkakAMslDL/5JazAg1V83S1qM59U5zVarD3xcB+nT6RAKOP4+MV58+o4OPQdIt64PUMhehpr0R\nG4jmQi2YmaG+AlT0/KufNrLE1OJzt4tYxJi+0ZJavAg9M/aa1tHckRZVKkCVU4N1NBWzZET5pXRO\n1PdZDlLbp00MVKqgVPkPmZXItVvYhvIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ x, \\quad y, \\quad \\left [ T_{0}, \\quad T_{1}, \\quad T_{2}, \\quad T_{3}\\right ]\\right ]$$"
      ],
      "text/plain": [
       "[x, y, [T₀, T₁, T₂, T₃]]"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[x,y]+[Ts]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f_r =  ufuncify( [x,y]+Ts, num_real)\n",
    "f_i =  ufuncify( [x,y]+Ts, num_imag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.14730830564958425-0.40346430265728966j)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_r(1,2,3,4,5,6)+f_i(1,2,3,4,5,6)*1j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWwAAAAPBAMAAAAypujKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nNsn/j8C/8Yop/XPH4poAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$0.147308305649584 - 0.40346430265729 i$$"
      ],
      "text/plain": [
       "0.147308305649584 - - -0.40346430265729⋅ⅈ"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## testing\n",
    "num3.subs({x:1,y:2,Ts[0]:3,Ts[1]:4,Ts[2]:5,Ts[3]:6}).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#Consolidating\n",
    "def get_real_imag_func(expression):\n",
    "    '''\n",
    "    Takes an expression in terms of Ts and sp.exp(-z*T) for T in Ts. \n",
    "    Here :math:`z = x + i y` is a complex number.\n",
    "    '''\n",
    "    D = {sp.exp(-z*T): sp.cos(-x*T)*(1j*sp.sin(y*T)+sp.cos(y*T)) for T in Ts}\n",
    "    expression2 = expression.subs(D)\n",
    "    num_real,num_imag = expression2.expand().as_real_imag()\n",
    "    f_r =  ufuncify( [x,y]+Ts, num_real)\n",
    "    f_i =  ufuncify( [x,y]+Ts, num_imag)\n",
    "    return lambda x,y,Ts: f_r(x,y,*Ts)+f_i(x,y,*Ts)*1j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F = get_real_imag_func(num2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.14730830564958425-0.40346430265728966j)"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "##  testing\n",
    "F(1,2,(3,4,5,6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.33226636779973417+1.7256217430980794j)"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "get_real_imag_func(sp.diff(num2,z))(1,2,(3,4,5,6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def blah(args):\n",
    "    for i in range(500):\n",
    "        F(*args)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ls = [[0.]*5]*5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ls[1][2] = 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABWUAAAAUBAMAAAAXe0LaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ \\left [ 0.0, \\quad 0.0, \\quad 4, \\quad 0.0, \\quad 0.0\\right ], \\quad \\left [ 0.0, \\quad 0.0, \\quad 4, \\quad 0.0, \\quad 0.0\\right ], \\quad \\left [ 0.0, \\quad 0.0, \\quad 4, \\quad 0.0, \\quad 0.0\\right ], \\quad \\left [ 0.0, \\quad 0.0, \\quad 4, \\quad 0.0, \\quad 0.0\\right ], \\quad \\left [ 0.0, \\quad 0.0, \\quad 4, \\quad 0.0, \\quad 0.0\\right ]\\right ]$$"
      ],
      "text/plain": [
       "[[0.0, 0.0, 4, 0.0, 0.0], [0.0, 0.0, 4, 0.0, 0.0], [0.0, 0.0, 4, 0.0, 0.0], [0\n",
       ".0, 0.0, 4, 0.0, 0.0], [0.0, 0.0, 4, 0.0, 0.0]]"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ls"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ls2 = np.zeros((5,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.]])"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ls2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ls2[2,3] = 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  4.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.]])"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ls2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  0.,  0.,  4.,  0.])"
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ls2[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "delays = [1,4,5,6,7]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ8AAAAUBAMAAABhZ6XhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 1.0, \\quad 4.0, \\quad 5.0, \\quad 10.0, \\quad 7.0\\right ]$$"
      ],
      "text/plain": [
       "[1.0, 4.0, 5.0, 10.0, 7.0]"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(sum,zip(ls2[2],delays))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAAPBAMAAABzWwAZAAAAIVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAADdcGRXAAAACnRSTlMAVO8Qq5l2zWbd0hufRgAAAAlwSFlzAAAOxAAA\nDsQBlSsOGwAAADpJREFUGBljYMAEjMoQMTANIkxCFoNFwDSEwwYRYQDTCAKoDMEZeDVWBUjuYXVa\n7sHA6cAApsEEhtcBrYgcUf0WzaQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.11$$"
      ],
      "text/plain": [
       "1.11"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round(1.111111,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}