variance of the round-off error

roundoff.ipynb

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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def q(x, N=10**5, a=1.0):\n",
    "    \"\"\"step size\n",
    "\n",
    "    Args:\n",
    "        x (int): bit number\n",
    "        N (int, optional): full well. Defaults to 10**5.\n",
    "        a (float): factor for digitization ~ 1.0\n",
    "    \"\"\"\n",
    "    return a*N*2**-x\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sigma2_roundoff(x, N=10**5, a=1.0):\n",
    "    return q(x,N,a)**2/12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
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    {
     "data": {
      "text/plain": [
       "(49.67053731282552, 3.104408582051595, 0.1940255363782247)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma2_roundoff(12), sigma2_roundoff(14), sigma2_roundoff(16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(24.4140625, 6.103515625, 1.52587890625)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q(12), q(14), q(16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(77.61021455128987, 4.850638409455617, 0.30316490059097606)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "af=1/0.8\n",
    "sigma2_roundoff(12,a=af), sigma2_roundoff(14,a=af), sigma2_roundoff(16,a=af)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(30.517578125, 7.62939453125, 1.9073486328125)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q(12,a=af), q(14,a=af), q(16,a=af)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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