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   "source": [
    "# p08: Eigenvalues of harmonic oscillator \n",
    "\n",
    "$$\n",
    "-u'' + x^2 u = \\lambda u, \\qquad x \\in \\mathbb{R}\n",
    "$$"
   ]
  },
  {
   "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": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 6\n",
      "   4.614729169954720e-01\n",
      "   7.494134621050521e+00\n",
      "   7.720916053006558e+00\n",
      "   2.883248377834012e+01\n",
      "N = 12\n",
      "   9.781372812986103e-01\n",
      "   3.171605320647185e+00\n",
      "   4.455935291166790e+00\n",
      "   8.924529058119932e+00\n",
      "N = 18\n",
      "   9.999700014993121e-01\n",
      "   3.000644066795828e+00\n",
      "   4.992595324407724e+00\n",
      "   7.039571897981499e+00\n",
      "N = 24\n",
      "   9.999999976290406e-01\n",
      "   3.000000098410861e+00\n",
      "   4.999997965273280e+00\n",
      "   7.000024998156542e+00\n",
      "N = 30\n",
      "   9.999999999999772e-01\n",
      "   3.000000000000737e+00\n",
      "   4.999999999975596e+00\n",
      "   7.000000000508624e+00\n",
      "N = 36\n",
      "   9.999999999999927e-01\n",
      "   2.999999999999998e+00\n",
      "   5.000000000000000e+00\n",
      "   6.999999999999988e+00\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(\"%24.15e\" % e)"
   ]
  }
 ],
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