p12.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p12: Accuracy of Chebyshev spectral differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import zeros,pi,inf,linspace,arange,abs,dot,exp\n",
    "from scipy.linalg import toeplitz,norm\n",
    "from matplotlib.pyplot import figure,subplot,semilogy,title,xlabel,ylabel,axis,grid\n",
    "from chebPy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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    "Nmax = 50\n",
    "E = zeros((4,Nmax))\n",
    "for N in range(1,Nmax+1):\n",
    "    D,x = cheb(N)\n",
    "    \n",
    "    v = abs(x)**3          # 3rd deriv in BV\n",
    "    vprime = 3.0*x*abs(x)\n",
    "    E[0][N-1] = norm(dot(D,v)-vprime,inf)\n",
    "    \n",
    "    v = exp(-(x+1.0e-15)**(-2))   # C-infinity\n",
    "    vprime = 2.0*v/(x+1.0e-15)**3\n",
    "    E[1][N-1] = norm(dot(D,v)-vprime,inf)\n",
    "    \n",
    "    v = 1.0/(1.0+x**2)     # analytic in a [-1,1]\n",
    "    vprime = -2.0*x*v**2\n",
    "    E[2][N-1] = norm(dot(D,v)-vprime,inf)\n",
    "    \n",
    "    v = x**10\n",
    "    vprime = 10.0*x**9   # polynomial\n",
    "    E[3][N-1] = norm(dot(D,v)-vprime,inf)\n",
    "\n",
    "\n",
    "titles = [\"$|x|^3$\", \"$\\exp(-x^{-2})$\", \\\n",
    "          \"$1/(1+x^2)$\", \"$x^{10}$\"]\n",
    "figure(figsize=(10,10))\n",
    "for iplot in range(4):\n",
    "    subplot(2,2,iplot+1)\n",
    "    semilogy(arange(1,Nmax+1,),E[iplot][:],'.-')\n",
    "    title(titles[iplot])\n",
    "    xlabel('N')\n",
    "    ylabel('error')\n",
    "    axis([0,Nmax,1.0e-16,1.0e3])\n",
    "    grid('on')"
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