{
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   "source": [
    "# Bauer-Fike Eigenvalue Sensitivity Bound\n",
    "\n",
    "Copyright (C) 2019 Andreas Kloeckner\n",
    "\n",
    "<details>\n",
    "<summary>MIT License</summary>\n",
    "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
    "of this software and associated documentation files (the \"Software\"), to deal\n",
    "in the Software without restriction, including without limitation the rights\n",
    "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
    "copies of the Software, and to permit persons to whom the Software is\n",
    "furnished to do so, subject to the following conditions:\n",
    "\n",
    "The above copyright notice and this permission notice shall be included in\n",
    "all copies or substantial portions of the Software.\n",
    "\n",
    "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n",
    "IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n",
    "FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n",
    "AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n",
    "LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n",
    "OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n",
    "THE SOFTWARE.\n",
    "</details>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as la"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the Bauer-Fike eigenvalue sensitivity bound, an important observation is that, given a diagonalized matrix\n",
    "$$X^{- 1} A  X = D$$\n",
    "that is perturbed by an additive perturbation $E$\n",
    "$$X^{- 1} (A + E) X = D + F,$$\n",
    "and if we suppose that $\\mu$ is an eigenvalue of $A+E$ (and $D+F$), we have\n",
    "$$\\|(\\mu I - D)^{- 1}\\|^{- 1} = | \\mu - \\lambda _k |,$$\n",
    "where $\\lambda_k$ is the eigenvalue of $A$ (diagonal entry of $D$) closest to $\\mu$.\n",
    "\n",
    "This notebook illustrates this latter fact. To that end, let the following be $D$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 0, 0, 0, 0],\n",
       "       [0, 1, 0, 0, 0, 0],\n",
       "       [0, 0, 2, 0, 0, 0],\n",
       "       [0, 0, 0, 3, 0, 0],\n",
       "       [0, 0, 0, 0, 4, 0],\n",
       "       [0, 0, 0, 0, 0, 5]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = np.diag(np.arange(6))\n",
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = 2.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.1,  0. ,  0. ,  0. ,  0. ,  0. ],\n",
       "       [ 0. ,  1.1,  0. ,  0. ,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0.1,  0. ,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0. , -0.9,  0. ,  0. ],\n",
       "       [ 0. ,  0. ,  0. ,  0. , -1.9,  0. ],\n",
       "       [ 0. ,  0. ,  0. ,  0. ,  0. , -2.9]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu * np.eye(6) - D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.476,  0.   ,  0.   ,  0.   ,  0.   ,  0.   ],\n",
       "       [ 0.   ,  0.909,  0.   ,  0.   ,  0.   ,  0.   ],\n",
       "       [ 0.   ,  0.   , 10.   ,  0.   ,  0.   ,  0.   ],\n",
       "       [-0.   , -0.   , -0.   , -1.111, -0.   , -0.   ],\n",
       "       [-0.   , -0.   , -0.   , -0.   , -0.526, -0.   ],\n",
       "       [-0.   , -0.   , -0.   , -0.   , -0.   , -0.345]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.inv(mu * np.eye(6) - D).round(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.999999999999991"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.norm(la.inv(mu * np.eye(6) - D), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For all $p$-norms, the norm of a diagonal matrix is the biggest (abs. value) diagonal entry, so we can also use, say, the $\\infty$ norm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.999999999999991"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.norm(la.inv(mu * np.eye(6) - D), np.inf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10000000000000009"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/ la.norm(la.inv(mu * np.eye(6) - D), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this matches the distance between $\\mu$ and the closest entry of $D$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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