{ "metadata": { "name": "", "signature": "sha256:00fdd0435e8b52e99288dbec76fb0fcabd9d2b2a8fa9f1d4b5fe2da202858524" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Barycentric Lagrange interpolation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us interpolate the Runge function on $[-1,+1]$\n", "$$\n", "f(x) = \\frac{1}{1 + 16 x^2}\n", "$$\n", "using the Barycentric formula\n", "$$\n", "p(x) = \\frac{ \\sum_{i=0}^N \\frac{w_i}{x - x_i} f_i }{ \\sum_{i=0}^N \\frac{w_i}{x - x_i} }\n", "$$\n", "where the weight $w_i$ is given by\n", "$$\n", "w_i = \\frac{1}{\\prod_{j=0,j\\ne i}^N (x_i - x_j)}\n", "$$" ] }, { "cell_type": "code", "collapsed": true, "input": [ "import numpy as np\n", "from matplotlib import pyplot as plt" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "code", "collapsed": true, "input": [ "def fun(x):\n", " f = 1.0/(1.0+16.0*x**2)\n", " return f" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 14 }, { "cell_type": "code", "collapsed": true, "input": [ "def LagrangeInterpolation(X,Y,x):\n", " nx = np.size(x)\n", " nX = np.size(X)\n", " w = np.ones(nX)\n", " for i in range(nX):\n", " for j in range(nX):\n", " if i != j:\n", " w[i] = w[i]/(X[i]-X[j])\n", " num= np.zeros(nx)\n", " den= np.zeros(nx)\n", " eps=1.0e-14\n", " for i in range(nX):\n", " num = num + Y[i]*w[i]/((x-X[i])+eps)\n", " den = den + w[i]/((x-X[i])+eps)\n", " f = num/den\n", " return f" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [ "xmin, xmax = -1.0, +1.0\n", "N = 20" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We first interpolate on uniformly spaced points." ] }, { "cell_type": "code", "collapsed": false, "input": [ "X = np.linspace(xmin,xmax,20)\n", "Y = fun(X)\n", "x = np.linspace(xmin,xmax,100)\n", "fi = LagrangeInterpolation(X,Y,x)\n", "fe = fun(x)\n", "plt.plot(x,fe,'b--',x,fi,'r-',X,Y,'o')\n", "plt.legend((\"True function\",\"Interpolation\",\"Data\"),'upper center')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 27, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 27 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next, we interpolate on Chebyshev points." ] }, { "cell_type": "code", "collapsed": true, "input": [ "X = cos(np.linspace(0.0,pi,N))\n", "Y = fun(X)\n", "x = np.linspace(xmin,xmax,100)\n", "fi = LagrangeInterpolation(X,Y,x)\n", "fe = fun(x)\n", "plt.plot(x,fe,'b--',x,fi,'r-',X,Y,'o')\n", "plt.legend((\"True function\",\"Interpolation\",\"Data\"),'upper right')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 26, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "svg": [ "\n", "\n", "\n", "\n" ], "text": [ "" ] } ], "prompt_number": 26 } ], "metadata": {} } ] }