p21.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p21: Eigenvalues of Mathieu operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import pi,arange,sin,cos,zeros,diag,sort,real\n",
    "from scipy.linalg import toeplitz\n",
    "from numpy.linalg import eig\n",
    "from itertools import cycle\n",
    "from matplotlib.pyplot import figure,plot,xlabel,ylabel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 360x720 with 1 Axes>"
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     },
     "metadata": {
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     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 42; h = 2.0*pi/N; x = h*arange(1,N+1)\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 = toeplitz(col)\n",
    "\n",
    "ne = 11 # number of eigenvalues to plot\n",
    "qq = arange(0.0, 15.0, 0.2)\n",
    "data= zeros((len(qq),ne))\n",
    "i = 0\n",
    "for q in qq:\n",
    "    evals,evecs = eig(-D2 + 2.0*q*diag(cos(2.0*x)))\n",
    "    e = real(sort(evals))\n",
    "    data[i,:] = e[0:ne]\n",
    "    i = i + 1\n",
    "    \n",
    "figure(figsize=(5,10))\n",
    "lines=cycle([\"-\",\"--\"])\n",
    "for i in range(ne):\n",
    "    plot(qq,data[:,i],next(lines))\n",
    "xlabel(\"q\")\n",
    "ylabel(\"$\\lambda$\");"
   ]
  }
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