p04.ipynb

{
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  {
   "cell_type": "markdown",
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   "source": [
    "# p04: Periodic spectral differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We compute derivatives of following periodic functions on finite interval\n",
    "$$\n",
    "v(x) = \\max(0, 1-|x-\\pi|/2), \\qquad x \\in [0,2\\pi]\n",
    "$$\n",
    "and\n",
    "$$\n",
    "v(x) = \\exp(\\sin(x)), \\qquad x \\in [0,2\\pi]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import pi,inf,linspace,zeros,arange,sin,cos,tan,exp,maximum,abs\n",
    "from numpy.linalg import norm\n",
    "from scipy.linalg import toeplitz\n",
    "from matplotlib.pyplot import figure,subplot,plot,axis,title,text"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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   "source": [
    "# Set up grid and differentiation matrix:\n",
    "N = 24; h = 2*pi/N; x = h*arange(1,N+1);\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)\n",
    "row[0] = col[0]\n",
    "row[1:] = col[N-1:0:-1]\n",
    "D = toeplitz(col,row)\n",
    "\n",
    "figure(figsize=(10,6))\n",
    "\n",
    "# Differentiation of a hat function:\n",
    "v = maximum(0,1-abs(x-pi)/2)\n",
    "subplot(3,2,1)\n",
    "plot(x,v,'.-')\n",
    "axis([0, 2*pi, -.5, 1.5])\n",
    "title('function')\n",
    "subplot(3,2,2)\n",
    "plot(x,D.dot(v),'.-')\n",
    "axis([0, 2*pi, -1, 1])\n",
    "title('spectral derivative')\n",
    "\n",
    "# Differentiation of exp(sin(x)):\n",
    "v = exp(sin(x)); vprime = cos(x)*v;\n",
    "subplot(3,2,3)\n",
    "plot(x,v,'.-')\n",
    "axis([0, 2*pi, 0, 3])\n",
    "subplot(3,2,4)\n",
    "plot(x,D.dot(v),'.-')\n",
    "axis([0, 2*pi, -2, 2])\n",
    "error = norm(D.dot(v)-vprime,inf)\n",
    "text(1.5,1.4,\"max error=\"+str(error));"
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