Discrete Derivatives.ipynb

{
 "metadata": {
  "name": "Discrete Derivatives"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy.sparse as sps\n",
      "def filterprint(domain, prec=1, fmt=\"f\", predicate=lambda x: x == 0, blank='- '):\n",
      "    \"\"\"\n",
      "    Pretty print a NumPy array, hiding values which match a predicate\n",
      "    (default: x == 0). predicate must be callable.\n",
      "\n",
      "    Aliased to fp.\n",
      "\n",
      "    Print an array with three decimal places in scientific notation when the\n",
      "    value is larger than 2, and '-' otherwise.\n",
      "\n",
      "        fp(domain, 3, 'e', lambda x: x > 2)\n",
      "\n",
      "    \"\"\"\n",
      "    if hasattr(domain, \"todense\"):\n",
      "        domain = domain.todense()\n",
      "    # if domain.ndim == 1: # Print 1-D vectors as columns\n",
      "        # domain = domain[:,np.newaxis]\n",
      "    tmp = \"% .{0}{1}\".format(prec, fmt)\n",
      "    domain = np.atleast_2d(domain)\n",
      "    xdim, ydim = np.shape(domain)\n",
      "    pad = max(len(tmp % x) for x in domain.flat)\n",
      "    fmt = \"% {pad}.{prec}{fmt}\".format(pad=pad, prec=prec, fmt=fmt)\n",
      "    bstr = \"{:>{pad}}\".format(blank, pad=pad)\n",
      "    for i in range(xdim):\n",
      "        for j in range(ydim):\n",
      "            if not predicate(domain[i,j]):\n",
      "                print fmt % domain[i,j],\n",
      "            else:\n",
      "                print bstr,\n",
      "        print\n",
      "    return\n",
      "fp = filterprint"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These functions generate tri-diagonal matrices. When applied, they will compute (discrete) derivatives like https://en.wikipedia.org/wiki/Finite_difference_method"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def D(dim):\n",
      "    '''Discrete first derivative operator with no boundary.'''\n",
      "    operator = np.zeros((3, dim))\n",
      "    operator[0,2:]  =  0.5\n",
      "    operator[2,:-2] = -0.5\n",
      "    return sps.dia_matrix((operator, (1,0,-1)), shape=(dim,dim))\n",
      "\n",
      "def D2(dim):\n",
      "    '''Discrete second derivative operator with no boundary.'''\n",
      "    operator = np.zeros((3, dim))\n",
      "    operator[0,2:]  =  1\n",
      "    operator[1,1:-1]  = -2\n",
      "    operator[2,:-2] =  1\n",
      "    return sps.dia_matrix((operator, (1,0,-1)), shape=(dim,dim))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fst_deriv = D(10)\n",
      "snd_deriv = D2(10)\n",
      "print \"First derivative\"\n",
      "fp(fst_deriv)\n",
      "print \"Second derivative\"\n",
      "fp(snd_deriv)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "First derivative\n",
        "  -    -    -    -    -    -    -    -    -    - \n",
        "-0.5   -   0.5   -    -    -    -    -    -    - \n",
        "  -  -0.5   -   0.5   -    -    -    -    -    - \n",
        "  -    -  -0.5   -   0.5   -    -    -    -    - \n",
        "  -    -    -  -0.5   -   0.5   -    -    -    - \n",
        "  -    -    -    -  -0.5   -   0.5   -    -    - \n",
        "  -    -    -    -    -  -0.5   -   0.5   -    - \n",
        "  -    -    -    -    -    -  -0.5   -   0.5   - \n",
        "  -    -    -    -    -    -    -  -0.5   -   0.5\n",
        "  -    -    -    -    -    -    -    -    -    - \n",
        "Second derivative\n",
        "  -    -    -    -    -    -    -    -    -    - \n",
        " 1.0 -2.0  1.0   -    -    -    -    -    -    - \n",
        "  -   1.0 -2.0  1.0   -    -    -    -    -    - \n",
        "  -    -   1.0 -2.0  1.0   -    -    -    -    - \n",
        "  -    -    -   1.0 -2.0  1.0   -    -    -    - \n",
        "  -    -    -    -   1.0 -2.0  1.0   -    -    - \n",
        "  -    -    -    -    -   1.0 -2.0  1.0   -    - \n",
        "  -    -    -    -    -    -   1.0 -2.0  1.0   - \n",
        "  -    -    -    -    -    -    -   1.0 -2.0  1.0\n",
        "  -    -    -    -    -    -    -    -    -    - \n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For linear functions $y = x$, the derivative $dy/dx$ is 1 everywhere."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.arange(10)\n",
      "y = x\n",
      "dx = 1\n",
      "print \"Our domain\"\n",
      "fp(a)\n",
      "print \"1st Deriv\"\n",
      "fp(fst_deriv.dot(y) * (1/dx)) \n",
      "print \"2nd Deriv\"\n",
      "fp(snd_deriv.dot(y) * (1/dx**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Our domain\n",
        "  -   1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0  9.0\n",
        "1st Deriv\n",
        "  -   1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0   - \n",
        "2nd Deriv\n",
        "  -    -    -    -    -    -    -    -    -    - \n"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Quadratic functions have derivative 2 as expected.  Note first deriv is now linear"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.arange(10)\n",
      "y = x**2.0\n",
      "dx = np.hstack((0,np.diff(x)))\n",
      "print \"Our domain\"\n",
      "fp(y)\n",
      "print \"1st Deriv\"\n",
      "fp(fst_deriv.dot(y) * (1/dx))\n",
      "print \"2nd Deriv\"\n",
      "fp(snd_deriv.dot(y) * (1/dx**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Our domain\n",
        "   -    1.0   4.0   9.0  16.0  25.0  36.0  49.0  64.0  81.0\n",
        "1st Deriv\n",
        "   -    2.0   4.0   6.0   8.0  10.0  12.0  14.0  16.0    - \n",
        "2nd Deriv\n",
        "  -   2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0   - \n"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These have been forward examples. We don't invert anything."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}