rjleveque/hw0python.ipynb

## hw0python.ipynb
{
 "metadata": {
  "name": "hw0python"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Homework 0:  Interpolation examples (Python version)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Developed by Randy LeVeque for a course on Approximation Theory and Spectral Methods at the University of Washington.\n",
      "\n",
      "See <http://faculty.washington.edu/rjl/classes/am590a2013/codes.html> for more IPython Notebook examples."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is a demonstration of using an IPython notebook to combine code with\n",
      "descriptions.\n",
      "\n",
      "It also demonstrates how to use pychebfun from within the notebook.\n",
      "\n",
      "## Sample problem\n",
      "\n",
      "Determine the quadratic polynomial $p(x)$ that interpolates the data\n",
      "$$\n",
      "(-1,3),~~(0,-1),~~(1,2)\n",
      "$$\n",
      "first using the Vandermonde matrix and then using the Lagrange form of\n",
      "interpolating polynomial.\n",
      "\n",
      "Note that if the notebook is started with --pylab option then numpy and matplotlib are already imported."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Interpolation data for all the examples:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xj = array([-1., 0., 1.])\n",
      "yj = array([3., -1., 2.])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Using monomials"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For a quadratic interpolation we use basis functions $1,~x,$ and $x^2$.\n",
      "\n",
      "Define the Vandermonde matrix: The columns are the basis functions evaluated at the interpolation points."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = vstack((xj**0, xj, xj**2)).T\n",
      "print A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 1. -1.  1.]\n",
        " [ 1.  0.  0.]\n",
        " [ 1.  1.  1.]]\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solve the system for the monomial coefficients:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from numpy.linalg import solve\n",
      "c = solve(A,yj)\n",
      "print c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-1.  -0.5  3.5]\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot the resulting polynomial on a fine grid:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = linspace(-1, 1, 1001)\n",
      "p = c[0] + c[1]*x + c[2]*x**2\n",
      "plot(x,p)\n",
      "hold(True)\n",
      "plot(xj,yj,'b.',markersize=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10f0b0850>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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2LLB5M+Drq3RErs/S3HndEf+OHTua/ZlWq0VpaSl8fHxw9uxZdOrUqcnjfP/4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\n"
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Lagrange form\n",
      "\n",
      "Evaluate the Lagrange basis functions on the fine grid for plotting:\n",
      "\n",
      "\\begin{align}\n",
      "L_1(x) &= \\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)},\\\\\\\\\n",
      "L_2(x) &= \\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)},\\\\\\\\\n",
      "L_3(x) &= \\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\n",
      "\\end{align}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Evaluate the Lagrange basis functions on the fine grid for plotting:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L1 = (x-xj[1])*(x-xj[2]) / ((xj[0]-xj[1])*(xj[0]-xj[2]))\n",
      "L2 = (x-xj[0])*(x-xj[2]) / ((xj[1]-xj[0])*(xj[1]-xj[2]))\n",
      "L3 = (x-xj[0])*(x-xj[1]) / ((xj[2]-xj[0])*(xj[2]-xj[1]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this form, the data values are the coefficients"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = yj[0]*L1 + yj[1]*L2 + yj[2]*L3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot this version (should look the same as before!)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "clf()\n",
      "plot(x,p)\n",
      "hold(True)\n",
      "plot(xj,yj,'b.',markersize=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10f14ed90>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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2LLB5M+Drq3RErs/S3HndEf+OHTua/ZlWq0VpaSl8fHxw9uxZdOrUqcnjfP/4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\n"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Using pychebfun\n",
      "\n",
      "To use pychebfun, you must download or clone from https://github.com/olivierverdier/pychebfun and add the location to the python path.  See the README file at the bottom of the github page."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pychebfun as P"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use pychebfun to plot the function\n",
      "$f(x) = \\left(\\frac{1}{1 + 25x^2}\\right)^{10}$\n",
      "and the function\n",
      "$g(x) = 5 \\int_0^x f(s)\\, ds.$\n",
      "\n",
      "What degree polynomial approximation is used for each?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = P.Chebfun.identity()\n",
      "f = (1/(1+25*x**2))**10;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f.plot()\n",
      "\n",
      "g = 5 * f.integrate()\n",
      "g = g - g(-1.)  # because f.integrate gives integral from 0 rather than from -1.\n",
      "g.plot()\n",
      "\n",
      "legend(['f','g'])\n",
      "print \"The degree of f is %i \" % (len(f.chebyshev_coefficients()) - 1)\n",
      "print \"The degree of g is %i \" % (len(g.chebyshev_coefficients()) - 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The degree of f is 234 \n",
        "The degree of g is 201 \n"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "See <http://nbviewer.ipython.org/6724986> for more pychebfun examples."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    }
   ],
   "metadata": {}
  }
 ]
}