{ "metadata": { "name": "", "signature": "sha256:09a5903e9f0b807879ad1054ff3110fe28f1bf98235094b91100393c95cc3b3b" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Runge phenomenon" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider interpolating the following two functions on $[-1,1]$\n", "$$\n", "f_1(x) = \\exp(-5x^2), \\qquad f_2(x) = \\frac{1}{1 + 16 x^2}\n", "$$\n", "We will try uniformly spaced points and Chebyshev points." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import exp,linspace,polyfit,polyval,cos,pi\n", "from matplotlib.pyplot import figure,plot,legend,axis,text" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 49 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us first plot the two functions." ] }, { "cell_type": "code", "collapsed": false, "input": [ "xmin, xmax = -1.0, +1.0\n", "\n", "f1 = lambda x: exp(-5.0*x**2)\n", "f2 = lambda x: 1.0/(1.0+16.0*x**2)\n", "\n", "xx = linspace(xmin,xmax,100,True)\n", "figure(figsize(8,4))\n", "plot(xx,f1(xx),xx,f2(xx))\n", "legend((\"$1/(1+16x^2)$\", \"$\\exp(-5x^2)$\"))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 50, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 50 }, { "cell_type": "code", "collapsed": false, "input": [ "def interp(f,points):\n", " xx = linspace(xmin,xmax,100,True);\n", " ye = f(xx);\n", "\n", " figure(figsize(10,8))\n", " for i in range(1,7):\n", " N = 2*i\n", " subplot(3,2,i)\n", " if points == 'uniform':\n", " x = linspace(xmin,xmax,N+1,True)\n", " else:\n", " theta = linspace(0,pi,N+1, True)\n", " x = cos(theta)\n", " y = f(x);\n", " P = polyfit(x,y,N);\n", " yy = polyval(P,xx);\n", " plot(x,y,'o',xx,ye,'--',xx,yy)\n", " axis([xmin, xmax, -1.0, +1.1])\n", " text(-0.1,0.0,'N = '+str(N))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 51 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Interpolate $f_1(x)$ on uniformly spaced points." ] }, { "cell_type": "code", "collapsed": false, "input": [ "interp(f1,'uniform')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 52 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Interpolate $f_2(x)$ on uniformly spaced points." ] }, { "cell_type": "code", "collapsed": false, "input": [ "interp(f2,'uniform')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 53 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above results are not good. Let us try $f_2(x)$ on Chebyshev points." ] }, { "cell_type": "code", "collapsed": false, "input": [ "interp(f2,'chebyshev')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 54 }, { "cell_type": "markdown", "metadata": {}, "source": [ "What about interpolating $f_1(x)$ on Chebyshev points ?" ] }, { "cell_type": "code", "collapsed": false, "input": [ "interp(f1,'chebyshev')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 55 } ], "metadata": {} } ] }