p02.ipynb

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  {
   "cell_type": "markdown",
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   "source": [
    "# p02: Convergence of periodic spectral method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We repeat program P1 using periodic spectral method to compute derivative of\n",
    "$$\n",
    "u(x) = \\exp(\\sin(x)), \\qquad x \\in [-\\pi,\\pi]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from scipy.linalg import toeplitz\n",
    "from numpy import pi,arange,exp,sin,cos,zeros,tan,inf\n",
    "from numpy.linalg import norm\n",
    "from matplotlib.pyplot import figure,loglog,grid,xlabel,ylabel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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   "source": [
    "figure()\n",
    "for N in range(2,100,2):\n",
    "    h = 2.0*pi/N\n",
    "    x = -pi + arange(1,N+1)*h\n",
    "    u = exp(sin(x))\n",
    "    uprime = cos(x)*u #Exact derivative\n",
    "    col = zeros(N)\n",
    "    col[1:] = 0.5*(-1.0)**arange(1,N)/tan(arange(1,N)*h/2.0)\n",
    "    row = zeros(N); row[0] = col[0]; row[1:] = col[N-1:0:-1]\n",
    "    D = toeplitz(col,row)\n",
    "    error = norm(D.dot(u)-uprime,inf)\n",
    "    loglog(N,error,'or')\n",
    "    \n",
    "grid(True)\n",
    "xlabel('N')\n",
    "ylabel('error');"
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