p05.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p05: Repetition of p04 via FFT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_format='svg'\n",
    "#  For complex v, delete \"real\" commands.\n",
    "from numpy import pi,inf,linspace,maximum,abs,zeros,arange,real,sin,cos,exp\n",
    "from numpy.fft import fft,ifft\n",
    "from numpy.linalg import norm\n",
    "from matplotlib.pyplot import figure,subplot,plot,axis,title,text"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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    "# Set up grid and differentiation matrix:\n",
    "N = 24; h = 2*pi/N; x = h*arange(1,N+1);\n",
    "\n",
    "# wave numbers\n",
    "ik = 1j*zeros(N)\n",
    "ik[0:N//2+1] = 1j*arange(0,N//2+1)\n",
    "ik[N//2+1:]  = 1j*arange(-N//2+1,0,1)\n",
    "\n",
    "# Differentiation of a hat function:\n",
    "v = maximum(0.0,1.0-abs(x-pi)/2.0)\n",
    "v_hat = fft(v)\n",
    "w_hat = ik * v_hat; w_hat[N//2] = 0.0\n",
    "w = real(ifft(w_hat))\n",
    "\n",
    "figure(figsize=(10,6))\n",
    "\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,w,'.-')\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",
    "v_hat = fft(v)\n",
    "w_hat = ik * v_hat; w_hat[N//2] = 0.0\n",
    "w = real(ifft(w_hat))\n",
    "\n",
    "subplot(3,2,3)\n",
    "plot(x,v,'.-')\n",
    "axis([0, 2*pi, 0, 3])\n",
    "subplot(3,2,4)\n",
    "plot(x,w,'.-')\n",
    "axis([0, 2*pi, -2, 2])\n",
    "error = norm(w-vprime,inf)\n",
    "text(1.5,1.4,\"max error=\"+str(error));"
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