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    "# p08: Eigenvalues of harmonic oscillator \n",
    "\n",
    "$$\n",
    "-u'' + x^2 u = \\lambda u, \\qquad x \\in \\mathbb{R}\n",
    "$$\n",
    "\n",
    "The exact eigenvalues are $1,2,5,7,\\ldots$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import pi,arange,linspace,sin,zeros,diag,sort\n",
    "from scipy.linalg import toeplitz\n",
    "from numpy.linalg import eig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-------- N = 6 --------\n",
      "    0.46147291699547\n",
      "    7.49413462105052\n",
      "    7.72091605300656\n",
      "   28.83248377834012\n",
      "-------- N = 12 --------\n",
      "    0.97813728129861\n",
      "    3.17160532064719\n",
      "    4.45593529116679\n",
      "    8.92452905811993\n",
      "-------- N = 18 --------\n",
      "    0.99997000149931\n",
      "    3.00064406679583\n",
      "    4.99259532440772\n",
      "    7.03957189798150\n",
      "-------- N = 24 --------\n",
      "    0.99999999762904\n",
      "    3.00000009841086\n",
      "    4.99999796527328\n",
      "    7.00002499815654\n",
      "-------- N = 30 --------\n",
      "    0.99999999999998\n",
      "    3.00000000000074\n",
      "    4.99999999997560\n",
      "    7.00000000050862\n",
      "-------- N = 36 --------\n",
      "    0.99999999999999\n",
      "    3.00000000000000\n",
      "    5.00000000000000\n",
      "    6.99999999999999\n"
     ]
    }
   ],
   "source": [
    "L = 8.0\n",
    "for N in range(6,37,6):\n",
    "    h = 2.0*pi/N; x = h*linspace(1,N,N); x = L*(x-pi)/pi\n",
    "    col = zeros(N)\n",
    "    col[0] = -pi**2/(3.0*h**2) - 1.0/6.0\n",
    "    col[1:] = -0.5*(-1.0)**arange(1,N)/sin(0.5*h*arange(1,N))**2\n",
    "    D2 = (pi/L)**2 * toeplitz(col)\n",
    "    evals,evecs = eig(-D2 + diag(x**2))\n",
    "    eigenvalues = sort(evals)\n",
    "    print(\"-------- N = %d --------\" % N)\n",
    "    for e in eigenvalues[0:4]:\n",
    "        print(\"%20.14f\" % e)"
   ]
  }
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