{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "import numpy\n",
    "import time\n",
    "import wendy\n",
    "%pylab inline\n",
    "import matplotlib.animation as animation\n",
    "from matplotlib import cm\n",
    "from IPython.display import HTML\n",
    "rcParams.update({'axes.labelsize': 17.,\n",
    "              'font.size': 12.,\n",
    "              'legend.fontsize': 17.,\n",
    "              'xtick.labelsize':15.,\n",
    "              'ytick.labelsize':15.,\n",
    "              'text.usetex': False,\n",
    "              'figure.figsize': [5,5],\n",
    "              'xtick.major.size' : 4,\n",
    "              'ytick.major.size' : 4,\n",
    "              'xtick.minor.size' : 2,\n",
    "              'ytick.minor.size' : 2,\n",
    "              'legend.numpoints':1})\n",
    "_SAVE_GIFS= True\n",
    "numpy.random.seed(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scaling of ``wendy`` with particle number *N*\n",
    "\n",
    "We will investigate how ``wendy`` scales with the number $N$ of particles using a self-gravitating disk. The following function initializes a self-gravitating disk:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def initialize_selfgravitating_disk(N):\n",
    "    totmass= 1.\n",
    "    sigma= 1.\n",
    "    zh= sigma**2./totmass # twopiG = 1. in our units\n",
    "    tdyn= zh/sigma\n",
    "    x= numpy.arctanh(2.*numpy.random.uniform(size=N)-1)*zh*2.\n",
    "    v= numpy.random.normal(size=N)*sigma\n",
    "    v-= numpy.mean(v)\n",
    "    m= numpy.ones_like(x)/N\n",
    "    return (x,v,m,tdyn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Time/collision for exact solver\n",
    "\n",
    "We believe that this should go as $\\log(N)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ns= numpy.round(10.**numpy.linspace(1.,5.,11)).astype('int')\n",
    "ntrials= 3\n",
    "T= numpy.empty((len(Ns),ntrials))\n",
    "ncoll= numpy.empty((len(Ns),ntrials))\n",
    "tdyn_fac_norm= 3000\n",
    "for ii,N in enumerate(Ns):\n",
    "    for jj in range(ntrials):\n",
    "        x,v,m,tdyn= initialize_selfgravitating_disk(N)\n",
    "        g= wendy.nbody(x,v,m,tdyn*(tdyn_fac_norm/N)**2.,maxcoll=10000000,full_output=True)\n",
    "        tx,tv,tncoll, time_elapsed= next(g)\n",
    "        if tncoll > 0:\n",
    "            T[ii,jj]= time_elapsed / tncoll\n",
    "            ncoll[ii,jj]= tncoll*(N/tdyn_fac_norm)**2.\n",
    "        else:\n",
    "            T[ii,jj]= numpy.nan\n",
    "            ncoll[ii,jj]= numpy.nan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x106ed02e8>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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gAjjnPp6D8oiIFLWgoenMbPZ4L6RbnENEpJwEvT2/AmgysyTQO+x5Ay4H0i3QISJSFoKG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\niATw/wFHGC2B/93BugAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111a9d278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Ns,numpy.nanmean(T,axis=1)*10.**6,'o')\n",
    "p= numpy.polyfit(numpy.log(Ns),numpy.log(numpy.nanmean(T,axis=1)*10.**6/numpy.log(Ns)),deg=1)\n",
    "plot(Ns,numpy.exp(numpy.polyval(p,numpy.log(Ns)))*numpy.log(Ns))\n",
    "pyplot.text(10,4.5,r'$\\Delta t  \\approx %.2f\\,\\mu\\mathrm{s} \\times N^{%.2f}\\log N$' % (numpy.exp(p[1]),p[0]),size=16.)\n",
    "gca().set_xscale('log')\n",
    "xlim(5,150000)\n",
    "ylim(0.,5.)\n",
    "xlabel(r'$N$')\n",
    "ylabel(r'$\\Delta t / \\mathrm{collision}\\,(\\mu\\mathrm{s})$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The behavior seems to be more like $(\\log N)^2$ or $N^{1/4}$, probably because the implementation of the binary search tree to determine the next collision is not optimal.\n",
    "\n",
    "## Number of collisions per dynamical time\n",
    "\n",
    "In a dynamical time, every particle will cross every other particle, so the total number of collisions per dynamical time should scale as $N^2$, which is indeed what we observe:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x106f3eba8>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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7oufdbYfwm5Kt+HZmOpZMGYF0VzIEQLorGYsnD2d9luJeq6vgOgVXwY2sL2vq\ncfOT7yD13CQUzrsa3ZK4uCLFh1BWwW2zRRtoSkQir4amZtz7UjnqGprw9IzRTLJErWivdPCc7xwG\nIjIqwvG0SURyjJ88EeGQX4stedOzgu2jU0ZgSO9zrQ6HyLbaS7SrVPVtn+2c1u4oIj8xJ6RWHz8L\nwGxVLYFnoMS0SD4fte3NzV9g2Ts78N1xF3IFW6J2tPddL0VEVsEzybcAuEFEAq2kIACmAngslCcX\nkTwAxUby9N2/wHjODHgGRJSrarnxHDD254XyXGSeHYdOYr4xWcxDN3MFW6L2tJlojRUVSuBJbDD+\nXdvK3YNem0REcuAZWZYLoNjv2BoAi43EChEpBjDe5/gsAGtUtQoUdfUNTZi7vAwJCYLf3Z7JyWKI\ngtDu1QvfUWAiAt8Jv32JSFsze/k/ZgmAEhEZH+BwjqpO9dmuEpEcb6tXVQtEZKnvPoqe//faZmz5\n8jj++B9j0K9XN6vDIYoJIfWj9VtVoYfv5N+qWtHRYIyWrn9L1Q1gvIhkGXVaACgDMLujz0ehWf3R\nHqwu3Yv7rh+Crw/rbXU4RDEj5AELIjJTRD6Hp5VbLiKHReRuk+IJ1JPgMDxliRx8VZ5w4eyETBH0\n6b5jePi1zbhqcCp+mDPU6nCIYkqoS9lMAQBVHaKqqaqaAk/yE5OSbauTh6vqEnguzuXCs7T5YhOe\nj4JwrL4B96wog6tbIp68LRMJnbiCLVEowpkmcZnfdg2AZSJixgTbgeq8qT7PVWDcbO2CHJlMVbFg\nzUbsOVqHl2ddifPO7WJ1SEQxJ9RE29Z43cMdCcTgRuDyQdBlAqNXwizv9oABA0wIK379/t0dePOT\nL/HTmy7BmIG2Wq2IKGaEWqNNDTQs19gXqH9tSIxeBP5/zRnw6wLWzmMUqGq29yctLa2jYcWt0p1H\n8OhftuDGS8/H3V8bZHU4RDEr1KVslhlL2Si+amWO9hzSCSbFVCIiWd5+tAAywunGJSITAUwcMsRW\nS5nFjEMnTuHelyqQ3isZ+VNHQoR1WaJwhdzrQFWnASiAp556BEBeqEnW6KqVB09PgjxjJJjXTADT\nRSTXuE9YtV9VXa+qs3r27BnOr8e1pmbFD1/egCO1p/H0jCz0TE60OiSimObYaRJ9WrQzt23bZnU4\nMeU3xVvx5FvbkDdlOKaPYY2bKBDTpkmMZWzRhuf/Kg/gt29vQ+7ofpiW3d/qcIgcwbGJlisshK7a\nXYcfrdqMp/IXAAAPuklEQVSAYed3xy9uuZx1WSKTODbRskUbmtONzZi3ohwNTYqnZ2QhOYmTxRCZ\nhVPiEwDg1298hg173Hh6RhYy0jiJN5GZHNuiZekgeK9v3If//ddO3HX1INw0vI/V4RA5TliJVkSe\nCXTbTlg6CM7nB05g4dqNGH1hLyy66WKrwyFypPYWZ7y+lQUaU1u5TTGk9nQj7llRhi6JCXjq9kwk\nJjj2Cw6Rpdqr0QqAh4yFEBWeeWBL0HLOA1t2xOXIsLapKn766mZsO3ACL9w1Fn16JlsdEpFjtbeU\nzVsAfCf7zoRn3a7RRsmgCoEngbGcqq4HsD47O9uMWcUcZ+WHe/BqRTV+lDMUX7uI80EQRVKocx1U\nAKgQkTGqOldEBsEzXNZbp90OoNxv5VyyicKKauQXVWKfuw4KYNgF3XHf9WzxE0Vam4lWRB4FcAie\nlWg3+BxSwLOemIhUqepc4/6DAGSJyLMAjqrqogjFTSEqrKjGonWbUNfQdGbfrsMn8aeP92FSZrqF\nkRE5X3ulgwd9kucceBJsDQCXiPRQ1WPwqdH6LOT4SgRjDgprtC3lF1W2SLIAUN/QjPyiSiZaoggL\nZRXcVwBARHoCWANgkYj0gicJT4an1XssksGGgjXalva560LaT0TmCWcpmxoRcXvLAiKyGp5W7lm9\nE1R1p5nBUvhSz03CoROnz9rf18XeBkSRFu4Q3Bbdu1rpnZAFYGf4oZFZ9h+rR31DEwQtX7jkxATM\nnzDMqrCI4ka4PdSPtnIbgKd3gqquC/OxyUQNTZ7JYpoVWPiNYUh3JUMApLuSsXjycNZniaIg3Bbt\nwlZu2wYvhnnk/WULSncdxf/cOgq3jErHnOvi+/+DyAphtWiNJcbPum0nnOsAeGPTF3ju3R347rgL\nccsotlyJrMLB7Q61/eAJLFi7EaP6u/DTmy+1OhyiuMZE60C1pxsxd3kZEhMET8/IQlJnvsxEVuLE\n3w7jP1kMu28RWY9NHYdZ/sFuThZDZDNMtA6yYY8bv1j/Ka4bloZ7v87eBUR24dhEG29L2Rw9eRrz\nVpQjrXsXPDFtFDp14gq2RHbh2EQbT927mpoV96/agIPHT+GZO7LQ65wkq0MiIh+OTbTx5Ldvb8M/\nth7Ez751KUb0s+U87ERxLaZ6HYjILOPmaFWdbWkwNvH3rQfxP29tw+SsdNw+doDV4RBRADHTohWR\nHHhmBCsAsF1EFlgdk9X2Hq3F/S9XYNj53fGrScMhwroskR1ZmmhFJM9IoP77F4hIrvFvlrE7A0Cu\ncbsKwOBoxWlHpxqbMG9FOZqaFM/cMRrJSQlWh0RErbCkdGAk1yx4Emex37E1ABararmxXQxgvNGS\n9Rrv/3vx5pevf4aP99bg2TuyMOi8c6wOh4jaYEmLVlVLVHUJPC1TfzneJGuo8m31ikiG8RhrIxym\nbRVWVOPF93dh1r9l4BuX97E6HCJqh61qtEZC9U++bnhasF6z4/lCWOWXx7Fo3SaMHZiCBZy0mygm\n2CrRAgjUN+kwPPVZiEgugMXG7bNqu053vL4Bc5eX4ZwunfHU7ZnonGC3l4+IArHbX2pKaweMxLoM\nQJmIHIWnxhs3VBULX9mIXUdq8dTtmejdo6vVIRFRkOzWj/ZIgH2pgKeuC6BXdMOxj9+/uwNvbPoS\ni755Ma7MSLU6HCIKgd0SrRuByweBLpoFZAxq8A5swIABsd+J/6OdR/DoX7bgxkvPx6x/y7A6HCIK\nka1KB0ar1b98kIEQunKpaoGqZnt/0tJie6rAg8dPYd6KcvTrlYzHpo3koASiGGSrRGso8RmkAAAZ\nRgIOiRNm72psasYPVlbgWH0DnrljNHp0TbQ6JCIKgyWJVkSyRCQPQA6APL/htDMBTDdGhuUZ2yFz\nwuxdjxdvxXtVh/HLScNxSZ8eVodDRGGypEZrDEgoR4ClylXV7bM/7EEJsb7cePGn+/HM/23HbWMH\nIHd0P6vDIaIOsGPpwBSx3KLddfgkHli9AZen98DPJnIFW6JY59hEG6s12vqGJsxdXo5OInhmxmh0\nTeRkMUSxzm7du0yjqusBrM/Ozg6rxhtthRXVyC+qRLW7DgAw82uD0D+lm8VREZEZHNuijSWFFdVY\ntG7TmSQLAMvf343CimoLoyIiszg20cZS6SC/qBJ1DU0t9tU1NCG/qNKiiIjITI5NtLF0MWyfT0s2\nmP1EFFscm2hjRd3pJnROCDzaq68rOcrREFEkODbRxkLpQFUxf+3HaGhSJPlNeZicmID5nG+WyBEc\nm2hjoXTw1Nuf4/WNX2DhNy7GktwRSHclQwCku5KxePJwTMpMtzpEIjKBY7t32d2bm7/A48Vb8e3M\ndMy5NgMiwsRK5FCObdHa2Sf7avCjVR9jVH8XFk/mMuFETufYRGvXGu3B46cw8/lSuLolouA7HPlF\nFA8cm2jtWKM91diE2S+W4kjtaSz7TjZ6d+dyNETxgDXaKFFVLFq3CeW73fjd7Vm4PN0+HwBEFFmO\nbdHazbJ3qrCuvBo/zLkIN4/oY3U4RBRFTLRR8PaW/Vj8ly24eXgf/OD6i6wOh4iizLGJ1i4Xw7bu\nP44frNyAy/r2wGNTR6JTJ/YwIIo3jk20drgYduTkadz9fCmSkxKw7DvZSE5iDwOieMSLYRFyurEZ\nc5eX4ctj9Vg160r06cl5C4jilWNbtFZSVfzsT5/ggx1HsGTKCGQO6GV1SERkISbaCHjhvV1Y+eFu\n3HPdYA6rJSImWrO9s+0gHnn9U4y/9Hz85EbOvkVETLSmqjp4AvNWlOOi3ufiiemj2MOAiAA4ONFG\nu3tXTW0D7n6+FJ0TOmHZd7JxbhdeZyQiD8cm2mh272psasa9K8ux52gtnr1jNFevJaIW2Owywa/e\n+AzvbDuEvCnDMXZQitXhEJHNxFyLVkSyrI7B18oPd+OP/9yJ718zCNPHDLA6HCKyoZhKtCKSA2CN\n1XF4vV91GA8Xbsa1Q9Ow6JsXWx0OEdmUpYlWRPKM5Om/f4GI5Br/nmnBqmoJgKqoBtmK3YdrMXd5\nGS5M7Ybf3p6Jzgkx9ZlFRFFkSY3WSK5ZAHIBFPsdWwNgsaqWG9vFAMZHPcg2HK9vwN0vfIRmBZ77\n7hj06JpodUhEZGOWNMNUtURVlyBw6zTHm2QNVYFavVZpalb88OUN2H7wJJ6ekYVB551jdUhEZHO2\n+r5rJFT/5OuGjVq0S4q24K0tB/BfEy/F1UPOszocIooBduve5Qqw7zCAMdEOxFdhRTXyiypR7a4D\nAFw9OAV3jhtoZUhEFENs1aIF0GYnVBHJBZBhXCQLlJRNV1hRjUXrNp1JsgBQttuNworqaDw9ETmA\n3RLtkQD7Ur03VHWtqg5W1SWq6o5GQPlFlahraGqxr76hGflFldF4eiJyALuVDtwIXD4IukuXiMwC\nMMu7PWBAxwYR7PNpyQazn4jIn61atEY/Wf/yQQb8uoC18xgFqprt/UlLS+tQTH1dgVdGaG0/EZE/\nWyVaQ4nfMNsMIwGHxKzZu+ZPGIbkxJZrfSUnJmD+BM41S0TBsSTRikiWiOQByAGQJyILfA7PBDDd\nGBmWZ2yHzKzZuyZlpmPx5OFIdyVDAKS7krF48nCunEBEQRNVtTqGiBCRiQAmDhkyZOa2bdusDoeI\nHEZEylQ1O5j72rF0YAo7LDdORAQ4ONFGe4UFIqLWODbRskVLRHbh2ERLRGQXjk20LB0QkV04NtGy\ndEBEduHYREtEZBd2m+vANN5+tABqReQzv8M9AdQEue17+zwAh0wK0f85w71va8cC7W9vXyydd1vH\ngzn31s7Vf9usc7fjeftv2/01t9t5Xxjk/QBVdfQPgIL29rW17Xe7NJJxhXPf1o4Fc97tnKutz7uj\n597auUbq3O143rH2msfaefv+xEPpYH0Q+9raDvT7Zgjlcdu6b2vHgjlv/32xdN5tHQ/1NW/v/WAG\nO563/7bdX/NYO+8zHDsENxJEpFSDHHLnJPF63kD8njvP21zx0KI1U4H/Dr+ZxpyqxXmLyCzjZ6lV\nAUWR/7nnGD950VrlwyJnvdcBwJjoycn8X++lIuIy3u9hv95s0XaAsZjkUlUdbHUs0eJdQFNVq7yz\nrqlnRWPHMz5UF6nqVONDpkxVAyYkJzLOf5mqjrY6lmgRkaPGzYUdea3ZovVhtFLOWtrcWKMs1/j3\nTAtWPfPkBr36g12FeN4ZAHKN21UAYvpDJpRzV9VyVZ1q3CUDQMjzJNtFqO91QwoCLzcVM8I475mq\n2qujH6iO7d4VCuM/PgueBFLsd2wNgMWqWm5sF8NGy593RDjn7feGG+//e7GiI6+5sVzSGlWNuQ/Z\ncM/bSD6l0Y3WPB14vTO8v9uRb25s0cLTMjX+EwP94eR4XwBDVaBPxFjUkfMWkQzjMdZGOMyI6Mi5\nGx82o2PxfdCB807RKC2IGgnhnrd6FoItAeD2W6AgJEy0bfDWI/12u+GQFm1rgjzv2ao6O3pRRUdb\n526sDOL9WlkGwDHn3955I4Zbs21p57xzjW8vQAfLZEy0bQt0lfEwPPU5J2vzvEUkF8Bi43bMtera\n0da55+Cr194FB9TnfbR13hkAcozXPcNhr3lb510FYLWxLwOeD9ewMNG2zX9F3hZ83ngLHNbVp9Xz\nNv7IlgEoM67IOq17W6vnbnz1TDFe98EwPmwcoq3zXutTImrzbyIGtXXe5QCmeV/vjlwQ48WwtgW6\nwprqvWG8+WKyRtmOVs/bqFf1im44UdXea+79Y3Pa697meQOOfb9H5fVmi7ZtbgT+auGkr4yBxOt5\nA/F77jzvlkw9bybaNhitN/+vFhmI0S5NwYrX8wbi99x53i2Yft5MtO0r8evAnGG8OE4Xr+cNxO+5\n87w9TD9v1mhxpjP2dHiuKqeIyCqfzskzASwy+o2OMbYdIV7PG4jfc+d5W3PenOuAiCjCWDogIoow\nJloioghjoiUiijAmWiKiCGOiJSKKMCZaIqIIY6IlIoowJlqKayKSYSzAp8ZM+/7H13iPxclCnBQB\nHLBAcc8YEbQQwCwAvfxXEhCRPFVdaElw5Ahs0RJ55tRdCM9MTrN8DxjzDH9kRVDkHEy0RF+th1WA\ns5enyQZQfvavEAWPiZboK0vhWTHDfyYnp8/JShHGREtxzSgNHAEAI6GWA1hkaVDkOEy0FO9yAPjO\nPboYQK5FsZBDcT5aincpvr0MVHWtiMBYZroKLZMwUVjYoiU6m/eiGOuzZAomWopbvvVZP0vh6fI1\nOLoRkVMx0VI8m4YAXbdUtRyessH2qEdEjsRES3FHRFzGcNulAJYaI8P85YH1WTIJh+ASEUUYW7RE\nRBHGREtEFGFMtEREEcZES0QUYUy0REQRxkRLRBRhTLRERBHGREtEFGFMtEREEfb/AQa785EhNi1V\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111dcd1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Ns,numpy.nanmean(ncoll,axis=1),'o-')\n",
    "gca().set_xscale('log')\n",
    "gca().set_yscale('log')\n",
    "xlim(5,150000)\n",
    "#ylim(0.,2.)\n",
    "xlabel(r'$N$')\n",
    "ylabel(r'$\\#\\ \\mathrm{of\\ collisions} / t_{\\mathrm{dyn}}$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The full, exact algorithm to run a system for a dynamical time therefore runs in order $N^2\\,\\log N$ time, which becomes prohibitively slow for large $N$.\n",
    "\n",
    "## Approximate solver\n",
    "\n",
    "``wendy`` also contains an approximate solver, which computes the gravitational force exactly at each time step, but uses leapfrog integration to increment the dynamics. The gravitational force can be computed exactly for $N$ particles using a sort and should therefore scale as $N \\log N$. The number of time steps necessary to conserve energy to something like 1 part in $10^6$ should be relatively insensitive to $N$, so the overall algorithm should scale as $N\\log N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ns= numpy.round(10.**numpy.linspace(1.,6.,11)).astype('int')\n",
    "ntrials= 3\n",
    "T= numpy.empty((len(Ns),ntrials))\n",
    "dE= numpy.empty((len(Ns),ntrials))\n",
    "E= numpy.empty((len(Ns),ntrials))\n",
    "tdyn_fac_norm= 3000\n",
    "for ii,N in enumerate(Ns):\n",
    "    tnleap= int(numpy.ceil((1000*tdyn_fac_norm)/N))\n",
    "    for jj in range(ntrials):\n",
    "        x,v,m,tdyn= initialize_selfgravitating_disk(N)\n",
    "        E[ii,jj]= wendy.energy(x,v,m)\n",
    "        g= wendy.nbody(x,v,m,tdyn*(tdyn_fac_norm/N),approx=True,\n",
    "                       nleap=tnleap,full_output=True)\n",
    "        tx,tv, time_elapsed= next(g)\n",
    "        T[ii,jj]= time_elapsed*(N/tdyn_fac_norm)\n",
    "        dE[ii,jj]= (E[ii,jj]-wendy.energy(tx,tv,m))/E[ii,jj]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The algorithm indeed scales close to $N\\log N$ with a fixed time step:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x111d98390>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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5eYSb6621RV/0Z6yqq6ttZ2en12GIVx6+Eh74Lsw5DWquhQn7ex2RlIlsEvFY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\nO1fMHVU/AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111d5b128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Ns,numpy.nanmean(T,axis=1)*10.**3,'o')\n",
    "p= numpy.polyfit(numpy.log(Ns),numpy.log(numpy.nanmean(T,axis=1)*10.**3/numpy.log(Ns)),deg=1)\n",
    "plot(Ns,numpy.exp(numpy.polyval(p,numpy.log(Ns)))*numpy.log(Ns))\n",
    "pyplot.text(10,10.**4.8,r'$\\Delta t  \\approx %.2f \\,\\mu\\mathrm{s} \\times N^{%.2f}\\log N$' % \n",
    "            (numpy.exp(p[1])*10.**3.,p[0]),size=16.)\n",
    "gca().set_xscale('log')\n",
    "gca().set_yscale('log')\n",
    "xlabel(r'$N$')\n",
    "ylabel(r'$\\Delta t / t_{\\mathrm{dyn}}\\,(\\mathrm{ms})$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, with the same time step, energy is much better conserved for large $N$ than for small $N$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x111ab56a0>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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rmNmGpJWU155Iarr7U3d/JGl7iEPXLhwbmCrcwKe8cgtfd2+GYD1Nebnh7kcXnp+mjY4v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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111d7f1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Ns,numpy.nanmean(numpy.fabs(dE),axis=1)*10.**3,'o')\n",
    "gca().set_xscale('log')\n",
    "gca().set_yscale('log')\n",
    "ylabel(r'$\\Delta E$')\n",
    "xlabel(r'$N$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can design the time step such that energy is conserved to about the same degree for different $N$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ns= numpy.round(10.**numpy.linspace(1.,6.,11)).astype('int')\n",
    "ntrials= 3\n",
    "T= numpy.empty((len(Ns),ntrials))\n",
    "E= numpy.empty((len(Ns),ntrials))\n",
    "dE= numpy.empty((len(Ns),ntrials))\n",
    "tdyn_fac_norm= 10**4\n",
    "for ii,N in enumerate(Ns):\n",
    "    tnleap= int(numpy.ceil(((3000.*tdyn_fac_norm)/N)**1.))\n",
    "    tdt= tdyn*(tdyn_fac_norm/N)**.15/10.\n",
    "    for jj in range(ntrials):\n",
    "        x,v,m,tdyn= initialize_selfgravitating_disk(N)\n",
    "        E[ii,jj]= wendy.energy(x,v,m)\n",
    "        g= wendy.nbody(x,v,m,tdt,approx=True,\n",
    "                       nleap=tnleap,full_output=True)\n",
    "        tx,tv, time_elapsed= next(g)\n",
    "        T[ii,jj]= time_elapsed/tdt\n",
    "        dE[ii,jj]= (E[ii,jj]-wendy.energy(tx,tv,m))/E[ii,jj]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1120ee2e8>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114d97da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Ns,numpy.nanmean(numpy.fabs(dE),axis=1)*10.**3,'o')\n",
    "gca().set_xscale('log')\n",
    "#gca().set_yscale('log')\n",
    "ylabel(r'$\\Delta E$')\n",
    "xlabel(r'$N$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, the algorithm for integrating the self-gravitating $\\mathrm{sech}^2$ disk scales much better, approximately as $N^{1/4}\\log N$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x112394a58>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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K7WCmUVOiFZHs7e5xygTrlsCh74cr7oVD3+t3VIGlRCsi2XnlGVg+D159Bk79nLPFzNh9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\nosAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1120f6780>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Ns,numpy.nanmean(T,axis=1),'o')\n",
    "p= numpy.polyfit(numpy.log(Ns),numpy.log(numpy.nanmean(T,axis=1)/numpy.log(Ns)),deg=1)\n",
    "plot(Ns,numpy.exp(numpy.polyval(p,numpy.log(Ns)))*numpy.log(Ns))\n",
    "pyplot.text(10,10.**1.8,r'$\\Delta t  \\approx %.2f \\mathrm{s} \\times N^{%.2f}\\log N\\,$' % \n",
    "            (numpy.exp(p[1]),p[0]),size=16.)\n",
    "gca().set_xscale('log')\n",
    "gca().set_yscale('log')\n",
    "xlabel(r'$N$')\n",
    "ylabel(r'$\\Delta t / t_{\\mathrm{dyn}}\\,(\\mathrm{s})$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
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