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   "source": [
    "# Minimum supercell problem\n",
    "\n",
    "For all the q-vectors that fits in a given supercell, phonon can be computed from that same cell.\n",
    "However, to maximize the computational efficiency, we need to compute the q-vectors at the smallest supercell possible where that q-vector fits in.\n",
    "This leads us the the minimum supercell problem.\n",
    "More generically, at arbitrary order, we need to find the supercell that fits in all the q-vectors of a given Q of that order.\n",
    "\n",
    "A brute-force search might gives us the answer relatively easily at most of the times, a mathematical approach can maximize the efficiency of this search as well.\n",
    "\n",
    "The details of the algorithms are described in the paper Phys. Rev. B 100, 014303 (2019).\n",
    "\n",
    "Here we simply shows how to use it.\n",
    "\n",
    "In the new version, this feature will simply be a function that returns the supa provided Q, \n",
    "all the mathematical operations will be performed using `smith_normal_form` module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from mincell import minCell"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "[[2 0 0]\n",
      " [0 1 0]\n",
      " [0 0 1]]\n"
     ]
    }
   ],
   "source": [
    "m = minCell(\"1/2 0 0\")\n",
    "print m.mult\n",
    "print m.supa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "[[1 1 0]\n",
      " [0 2 0]\n",
      " [0 0 1]]\n"
     ]
    }
   ],
   "source": [
    "m = minCell(\"1/2 1/2 0\")\n",
    "print m.mult\n",
    "print m.supa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "[[1 0 1]\n",
      " [0 1 1]\n",
      " [0 0 2]]\n"
     ]
    }
   ],
   "source": [
    "m = minCell(\"1/2 1/2 1/2\")\n",
    "print m.mult\n",
    "print m.supa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n",
      "[[1 1 1]\n",
      " [0 2 0]\n",
      " [0 0 2]]\n"
     ]
    }
   ],
   "source": [
    "m = minCell(\"1/2 1/2 0; 0 1/2 1/2\")\n",
    "print m.mult\n",
    "print m.supa"
   ]
  }
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