{ "metadata": { "name": "", "signature": "sha256:bada4cabf82efe263c03de3293740cc4197a0206e2566fd92d061613ffe98fd4" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Function $\\omega_N(x)$ arising in interpolation error" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a given set of points $x_0, x_1, \\ldots, x_N \\in [-1,+1]$ we plot the function\n", "$$\n", "\\omega_N(x) = (x-x_0)(x-x_1) \\ldots (x-x_N), \\qquad x \\in [-1,+1]\n", "$$\n", "for uniform and chebyshev points." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import linspace,pi,cos\n", "from matplotlib.pyplot import plot,legend,title" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "def omega(x,xp):\n", " f = 1.0\n", " for z in xp:\n", " f = f * (x-z)\n", " return f" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "M = 1000\n", "xx = linspace(-1.0,1.0,M)\n", "\n", "N = 16\n", "xu = linspace(-1.0,1.0,N+1) # uniform points\n", "xc = cos(linspace(0.0,pi,N+1)) # chebyshev points\n", "fu = 0*xx\n", "fc = 0*xx\n", "for i in range(M):\n", " fu[i] = omega(xx[i],xu)\n", " fc[i] = omega(xx[i],xc)\n", "plot(xx,fu,'b-',xx,fc,'r-')\n", "legend((\"Uniform\",\"Chebyshev\"))\n", "title(\"N = \"+str(N))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 12, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 12 } ], "metadata": {} } ] }