4_error.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
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    "slideshow": {
     "slide_type": "skip"
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   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Sources of Error\n",
    "\n",
    "Error can come from many sources when using applying a numerical method:\n",
    " - Model/Data Error\n",
    " - Truncation Error\n",
    " - Floating Point Error\n",
    " \n",
    "**Goal:** Categorize and understand each type of error and explore some simple approaches to analyzing error."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Model and Data Error\n",
    "\n",
    "Errors in fundamental formulation\n",
    " - Lotka-Volterra - fractional rabbits, no extinctions, etc.\n",
    " - Data Error - Inaccuracy in measurement or uncertainties in parameters\n",
    " \n",
    "Unfortunatley we cannot control model and data error directly but we can use methods that may be more robust in the presense of these types of errors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Truncation Error\n",
    "\n",
    "Errors arising from approximating a function with a simpler function (e.g. $sin(x) \\approx x$ for $|x| \\approx 0$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Floating Point Error\n",
    "\n",
    "Errors arising from approximating real numbers with finite-precision numbers and arithmetic."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Basic Definitions\n",
    "\n",
    "Given a true value of a function $f$ and an approximate solution $\\hat{f}$ define:\n",
    "\n",
    "Absolute Error:  $e = |f - \\hat{f}|$\n",
    "\n",
    "Relative Error:  $r = \\frac{e}{|f|} = \\frac{|f - \\hat{f}|}{|f|}$\n",
    "\n",
    "Decimal precision p (number of significant digits) - $p$ is the precision if and only if$$\n",
    "\n",
    "$$x = \\text{round}(10^{-n} \\cdot x) \\cdot 10^n$$\n",
    "\n",
    "is true where\n",
    "\n",
    "$$n = \\text{floor}(\\log_{10} x) + 1 - p$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Absolute Error =  0.00828182845905\n",
      "Relative Error =  0.00304671442539\n",
      "2.71 = 2.71\n"
     ]
    }
   ],
   "source": [
    "f = numpy.exp(1)\n",
    "f_hat = 2.71\n",
    "\n",
    "# Error\n",
    "print \"Absolute Error = \", numpy.abs(f - f_hat)\n",
    "print \"Relative Error = \", numpy.abs(f - f_hat) / numpy.abs(f)\n",
    "\n",
    "# Precision\n",
    "p = 3\n",
    "n = numpy.floor(numpy.log10(f_hat)) + 1 - p\n",
    "print \"%s = %s\" % (f_hat, numpy.round(10**(-n) * f_hat) * 10**n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Truncation Error and Taylor's Theorem\n",
    "\n",
    "**Taylor's Theorem:**  Let $f(x) \\in C^{m+1}[a,b]$ and $x_0 \\in [a,b]$, then for all $x \\in (a,b)$ there exists a number $c = c(x)$ that lies between $x_0$ and $x$ such that\n",
    "\n",
    "$$ f(x) = T_N(x) + R_N(x)$$\n",
    "\n",
    "where $T_N(x)$ is the Taylor polynomial approximation\n",
    "\n",
    "$$T_N(x) = \\sum^N_{n=0} \\frac{f^{(n)}(x_0)\\cdot(x-x_0)^n}{n!}$$\n",
    "\n",
    "and $R_N(x)$ is the residual (the part of the series we left off)\n",
    "\n",
    "$$R_N(x) = \\frac{f^{(n+1)}(c) \\cdot (x - x_0)^{n+1}}{(n+1)!}$$\n",
    "\n",
    "Another way to think about these results involves replacing $x - x_0$ with $\\Delta x$.  The primary idea here is that the residual $R_N(x)$ becomes smaller as $\\Delta x \\rightarrow 0$.\n",
    "\n",
    "$$T_N(x) = \\sum^N_{n=0} \\frac{f^{(n)}(x_0)\\cdot\\Delta x^n}{n!}$$\n",
    "\n",
    "and $R_N(x)$ is the residual (the part of the series we left off)\n",
    "\n",
    "$$R_N(x) = \\frac{f^{(n+1)}(c) \\cdot \\Delta x^{n+1}}{(n+1)!} \\leq M \\Delta x^{n+1} = O(\\Delta x^{n+1})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Example 1\n",
    "\n",
    "$f(x) = e^x$ with $x_0 = 0$\n",
    "\n",
    "Using this we can find expressions for the relative and absolute error as a function of $x$ assuming $N=2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$f'(x) = e^x, ~~~ f''(x) = e^x ~~~ f^{(n)}(x) = e^x$$\n",
    "\n",
    "$$T_N(x) = \\sum^N_{n=0} e^0 \\frac{x^n}{n!} ~~~~\\Rightarrow ~~~~ T_2(x) = 1 + x + \\frac{x^2}{2}$$\n",
    "\n",
    "$$R_N(x) = e^c \\frac{x^{n+1}}{(n+1)!} = e^c \\cdot \\frac{x^3}{6} ~~~~ \\Rightarrow ~~~~ R_2(x) \\leq \\frac{e^1}{6} \\approx 0.5$$\n",
    "\n",
    "$$e^1 = 2.718\\ldots$$\n",
    "\n",
    "$$T_2(1) = 2.5 \\Rightarrow e \\approx 0.2 ~~ r \\approx 0.1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We can also use the package sympy which has the ability to calculate Taylor polynomials built-in!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy\n",
    "x = sympy.symbols('x')\n",
    "f = sympy.symbols('f', cls=sympy.Function)\n",
    "\n",
    "f = sympy.exp(x)\n",
    "f.series(x0=0, n=6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Lets plot this numerically for a section of $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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cc1zQvbvd2RPxGG/POuuUD1yGaXQOreJaEZ92prCQNX/7G4sXLWL59u2EtmzJ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\nDh0qrYb49dfS6glntUXTpua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      "text/plain": [
       "<matplotlib.figure.Figure at 0x105a25fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = numpy.linspace(-1, 1, 100)\n",
    "T_N = 1.0 + x + x**2 / 2.0\n",
    "R_N = numpy.exp(1) * x**3 / 6.0\n",
    "\n",
    "plt.plot(x, T_N, 'r', x, numpy.exp(x), 'k', x, R_N, 'b')\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"$f(x)$, $T_N(x)$, $R_N(x)$\")\n",
    "plt.legend([\"$T_N(x)$\", \"$f(x)$\", \"$R_N(x)$\"], loc=2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Example 2\n",
    "\n",
    "$f(x) = \\frac{1}{x} ~~~~~~ x_0  = 1$, approximate with $\\hat{f}(x) = T_2(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$f'(x) = -\\frac{1}{x^2} ~~~~~~~ f''(x) = \\frac{2}{x^3} ~~~~~~~ f^{(n)}(x) = \\frac{(-1)^n n!}{x^{n+1}}$$\n",
    "\n",
    "$$T_N(x) = \\sum^N_{n=0} (-1)^n (x-1)^n ~~~~ \\Rightarrow ~~~~ T_2(x) = 1 - (x - 1) + (x - 1)^2$$\n",
    "\n",
    "$$R_N(x) = \\frac{(-1)^{n+1}(x - 1)^{n+1}}{c^{n+2}} ~~~~ \\Rightarrow ~~~~ R_2(x) = \\frac{-(x - 1)^{3}}{c^{4}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+3pyS++abpsVx003w8MPw93+vdc7dUViISEiqq4P33oO//tWMJL/7bvjmN81A\nwP79ra6d/1FYiEjIO33aHKb685/NmuYPPwzf/rZZ9EkMhYWEpMjISKqrq62uRsCIiIigqqrK6mr0\nisOH4U9/MiUyEh55xIRHqJ+Kq7CQkGSz2dDfQPeF4ufV2GimHfn9703/xp13QkaGGfwXiutzKCwk\nJIXizu9qhPrnVV1t+jYyM82suY8+alocobSsbE/DIgRzVURCVUQEfO97UFAAq1ebFQKnTjVjNz7+\n2KwaKB1Ty0ICWqh/U/aUPq/2amvhjTfg9dfNCoFPPGE6xcPCrK6Zb6hlIeJHampqWLRoEZmZmaxd\nu5Y+ffrw29/+lszMTBYtWkRBQYHb57Db7eTmdm9GnaysrKutcsgKCzOtjd27YdUqsyZHdLQJjQMH\nrK6deItDQpu//g2sXbvWUVtb63A4HI6ioiJHUlKS676cnJxuPcerr77a7dez2+2OtWvXun2cv35e\n/ubECYfjJz9xOEaOdDgWLXI43nvP4WhosLpW3gH0qGmploWIDyQmJjJ06FAAcnJyWLhwoeu+2NhY\ntz+fk5NDUlKS28c5xcTEsH37ds8rKh0aMwZeegmOHDGn2/74xzBtGqxcaearCkUKCxEfiImJcV1u\nGxYt7+vM2rVrmT9/fqvbCgoKyM3NZdmyZeTm5pKent7q/qioKIqL/X0l5MByzTWm/2LHDhMUH3xg\nDlH9+MdQXm517XqXwkLEx/Lz89vt+AHS09PJzMwETP9Eeno6Z850vBBlbW0tsbGxpKamkpOTQ2pq\nKitWrGj1mNjYWOx2u/ffgGCzwbx58M47sHmzGSk+eTI8/rgZ/BcKFBYS3Gw275QestvthIeHd3jf\nY489xpo1awDTKnj++eddh67ajrIOCwsjLCwMu91OcnIy0L6FEh4eTk1NTY/rKt1zww3wm9+YBZtG\njIBbbzUz3+bnW10z31JYSHBzOLxTeignJ4cFCxZ0eF9kZCRJSUkUFBSQl5dHfHx8q/taqq2tpaam\nhqysLFdfRtszqmpqajoNJvG+ESPgZz8zYzVuvdUsCXvnnWa52GCksBDxgdraWjIzM1m1ahW1tbXt\nduy5ubkkJCSwbNkyXnnllXY/Hx4eTm1trev6qlWrWLNmjatzPDc3t11H+fbt25kzZ44P3o10ZcgQ\neOopKCoyg/seeQS+8hUzhXowDWnpZ3UFRIJRWFgYGRkZZGRkuH0c0O7w0YMPPkhOTg5paWkAPP30\n0916XedanC6/AAAJjElEQVRhLOl9AwfCd79rOsRXr4YnnzRra7z4Itx111UdzfQLalmI9LL8/HxW\nrlzp6sx+7rnn2h0+SkhI8Gh22NzcXB566CGv1lN6pl8/+MY3YO9e+OEP4emnzeJMH3wQ2C0Nq7Mu\nDagBYoHMDu5fATwLZHRyf9MYEwlVwT59RW5uLqmpqV0+pra2lry8PLePg+D/vPxRY6NZQ/yll8wh\nq5//HBYssK6lEYizziYCMUAWJgzygLZzIFQBlcBjwMcdPIfCIsRp5+cZfV7WaWiAt94yh6VGjYKX\nX4bbb+/9egTi3FDpgHPVGjvQ0SkjGcD1dBwUIiIBo29f+PrXYd8++M534FvfgnvugZ07ra5Z91gZ\nFuGYloNTR+tXRQKpQPd690RE/Fy/fvAP/2DGadx1l1kz/OtfN2dT+TOrz4Zy1xRy9lMsxIRGuyk4\nX3zxRdfllJQUUlJSvFQ1ERHfGTgQvv9908r49a9NJ7hzHqrhw733Ohs3bmTjxo1X/Ty+7rPo6LzB\nKkw/xXIgGxMASzD9F79q87POxz6N6Qhv28mtPosQp2PwntHn5b8qKkzn91//asZtPPUUDB7s/dcJ\nxA7uBCAZEwBPY4JjJ+bwVA2mJZEH1GKC5c2m+1tSWIQ47fw8o8/L/xUVwbJlsHWr6QT/5je9u1Z4\nIIYFmNaDndanzuZhQgTMqbVgWh3/u4OfV1iEOO38PKPPK3Bs2WLGaVy6ZA5TeevMqUANi6ulsAhx\n2vl5Rp9XYHE4zGjwZ581fRorVkA3ZrjvUiCeOisSMlatWkVmZmar5U+1bKq4Y7PBQw+Z5V1nzYLk\nZHjhBair6/26KCxEeoFzAaSWM9BmZWV1a9Q1mJX3FBiha/BgExK7doHdDlOnmhZHbzYSFRYiPpaf\nn8+cOXOIjo52TRyoZVOlJ8aNM2dL/eUv8MorcMcdZpBfb1BYSFCz2WxeKT1VXFzM8uXLsdlsraYp\n17KpcjVuv90s9frAA5CSAj/6EZw969vXVFhIUHM4HF4pPeVczW7p0qUkJCR0+jgtmyqe6tsXvvc9\n07KorDSHptau9d2hKYWFiI/Z7fZ260xo2VTxlhEj4A9/gDffNJMULl5s+jW8TWEh4kM1NTXtVrQD\nLZsq3nfbbVBQYPoxbrzR9Glcvuy951dYiPhQTk4OixYtane7lk0VX+jfH555BvLyzFrgSUnwxRfe\neW6rJxIUCVp2u528vDyef/75dvdp2VTxpehoszLf6tVw//2wZAn88pdmmdeeUstCxAcKCgrIysri\nxhtv7HAHr2VTxdecA/r27TOD+GbOhI8+uorn817VLKHpPkJcoE9f4e1lU90J9M9Lem79enjsMThy\nRHNDSQjSzs8z+rxC27lzcN11CgsJQdr5eUafl2giQRER8RmFhYiIuKWwEBERtxQWIiLilsJCxIeK\ni4tZuHAh6enpZGVlkZmZyeOPP+7R7LFaJEn8gcJCxIdiYmKw2Wy8+uqrpKWlkZGRQWJiIs8++2y3\nn0OLJIk/UFiI+FhVVRXR0dGu6zk5OR1OAdIRLZIk/kJzQ4n4UH5+PrGxsRQUFJCXl0dNTQ1vvfVW\nt39+7dq1vP76661uKygooKqqiuzsbBYuXMjKlStbPadzkaS2U5yLXA21LER8KCcnh4ceeoiEhAQy\nMjJYvXp1q9lmAdLT08nMzARM/0R6ejpnzpzp8Pm0SJJYRWEhQc1m807pqZycHBITE13Xa2pqqK6u\nbvWYxx57jDVr1gCmVfD888+7Jh/UIkniLxQWEtQcDu+UnrLb7a36K5zXW7YuIiMjSUpKch2qio+P\nb3VfS1okSayisBDxgeLiYp599llsNhs7d+503f7AAw+Qm5tLXl4eYGadTUhIYNmyZbzyyivtnkeL\nJIm/UAe3iA/ExMSwYsWKdv0Jq1ev7vDxYWFhAO0OH2mRJPEXalmIWCQ/P5+VK1e6OrOfe+65doeP\ntEiS+AtNUS4BLVSm3PbWIkmh8nlJ53o6RbnCQgKadn6e0eclWs9CRER8RmEhIiJuKSxERMQthYWI\niLilcRYS0CIiIpwddtINERERVldBAlSg/5fpbCgREQ8E8tlQK7q4Lw1IBTJ6qS5+Y+PGjVZXwaf0\n/gKb3l/osToslmICoSPOqTqd60km+L46/iPY/1j1/gKb3l/osTosVgGdTbyfDjjncrYDC3qlRiIi\n0o7VYdGVcKDlpDhRVlVERCTU+UMH93pgUQe3vw6sBAow/RYLgWVtHlMITPJp7UREgksREOfpD/n6\n1NmOOqargKxu/GwN4Fz5JQKo7OAxHr9hEREJTOvbXHfO0ZxAc9g8DcQjIiKWsLrPYgmQDDza4rac\npq1zrchUTCtjJ8Et2E8h7ur9ZTSV5b1UF1/o6v05dW/lIv/T1XtLxPx9BuvfZjD874UUd7+wQP+F\nLsX0v3TE+c8I5v0F4inEXb2/VCCm6fJbTdcDTVfvz2kB7VvRgcDde3urafs0wfe3mUDze0olMN+f\nuy9i3d53Wt2y6A534y0SMKfW5jZtA/EXGuynEHf1/mJpfk/2puuBpqv35xSoUw109d6WANubLv+K\n5qMBgcTd787Z6ogl8N5fKuZITSam/m2/iHk0li0QwqI7O8tA/oW6E+ynEGc2FTB/vNu7eGygSqD5\nHzKYJGP+HhMI3ENsXSkAijH/f91f29Z/uPsi5tEX0UAIC3c7y0D/hXaHP5zi7GuJwA6Cs28q0v1D\nAtZpmr+gdTYbQ6AKxxyiysB8oYnp+uF+x90XMY++iAZCWEDXO8tA/4W6051TiINBKvCc1ZXwgWBt\nVYD5WyxuulwDzLGwLr6QgRnrlQU8gDnsFoi6+iLW7S+igRAW7naWwfILbct5CvFqmpuPMUC2NdXx\nuvAWl5dijnlDYHZwd8T5/mIx37iXYv6OA7FPrS3ne1tL899mOLDNmup4Xcu/zTNN21zMvigQdfZF\nLOi+iHY23iK8xW0tBeIZUUswzcGWpxDntbicQWCf7dXV+1vQdF9h03Z+71bNK9z9/sD87g4TeOOF\nuvO3mQa80puV8iJ37+9pAvvU4KUtLju/iAX1WLaOdpbB9AsVEfG2zr6IBdMXURERERERERERERER\nEREREREREREREREREREREREREREJaSuARsz0CdmYSSBFgkIorJMg0pvewszDU01wTrkuIiJeEI5p\nXQTi7LkiItJLnsHMgFxodUVERMQ/rcQsXRmDCYt1QJilNRIRERERERERERERERERERERERERERER\nERERr/r/KxmhkikUnEcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a1d0050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = numpy.linspace(0.8, 2, 100)\n",
    "T_N = 1.0 - (x-1) + (x-1)**2\n",
    "R_N = -(x-1.0)**3 / (1.1**4)\n",
    "\n",
    "plt.plot(x, T_N, 'r', x, 1.0 / x, 'k', x, R_N, 'b')\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"$f(x)$, $T_N(x)$, $R_N(x)$\")\n",
    "\n",
    "plt.legend([\"$T_N(x)$\", \"$f(x)$\", \"$R_N(x)$\"], loc=8)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Symbols and Definitions\n",
    "\n",
    "Big-O notation: $f(x) = \\text{O}(g(x))$ as $x \\rightarrow a$ if and only if $|f(x)| \\leq M |g(x)|$ as $|x - a| < \\delta$ for $M$ and $a$ positive.\n",
    "\n",
    "In practice we use Big-O notation to say something about how the terms we may have left out of a series might behave.  We saw an example earlier of this with the Taylor's series approximations:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Example:\n",
    "$f(x) = \\sin x$ with $x_0 = 0$ then\n",
    "\n",
    "$$T_N(x) = \\sum^N_{n=0} (-1)^{n} \\frac{x^{2n+1}}{(2n+1)!}$$\n",
    "\n",
    "We can actually write $f(x)$ then as\n",
    "\n",
    "$$f(x) = x - \\frac{x^3}{6} + \\frac{x^5}{120} + O(x^7)$$\n",
    "\n",
    "This becomes more useful when we look at this as we did before with $\\Delta x$:\n",
    "\n",
    "$$f(x) = \\Delta x - \\frac{\\Delta x^3}{6} + \\frac{\\Delta x^5}{120} + O(\\Delta x^7)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**We can also develop rules for error propagation based on Big-O notation:**\n",
    "\n",
    "In general, there are two theorems that do not need proof and hold when the value of x is large: \n",
    "\n",
    "Let\n",
    "$$f(x) = p(x) + O(x^n)$$\n",
    "$$g(x) = q(x) + O(x^m)$$\n",
    "$$k = \\max(n, m)$$ then\n",
    "$$f+g = p + q + O(x^k)$$\n",
    "$$f \\cdot g = p \\cdot q + O(x^{n\\cdot m})$$\n",
    "\n",
    "On the other hand, if we are interested in small values of x, say Δx, the above expressions can be modified as follows: \n",
    "\n",
    "$$f(\\Delta x) = p(\\Delta x) + O(\\Delta x^n)$$\n",
    "$$g(\\Delta x) = q(\\Delta x) + O(\\Delta x^m)$$\n",
    "$$r = \\min(n, m)$$ then\n",
    "\n",
    "$$f+g = p + q + O(\\Delta x^r)$$\n",
    "\n",
    "$$f \\cdot g = p \\cdot q + p \\cdot O(\\Delta x^m) + q \\cdot O(\\Delta x^n) + O(\\Delta x^{n+m}) = p \\cdot q + O(\\Delta x^r)$$\n",
    "\n",
    "**Note 1:** In this case we suppose that at least the polynomial with $k = \\max(n, m)$ has the following form: \n",
    "$$p(\\Delta x) = 1 + p_1x + p_2x^2 + ...$$\n",
    "or $$q(\\Delta x) = 1 + q_1x + q_2x^2 + ...$$\n",
    "so that there is an O(1) term that guarantees the existence of $O(\\Delta x^r)$ in the final product. \n",
    "\n",
    "To get a sense of why we care most about the power on $\\Delta x$ when considering convergence the following figure shows how different powers on the convergence rate can effect how quickly we converge to our solution.  Note that here we are plotting the same data two different ways.  Plotting the error as a function of $\\Delta x$ is a common way to show that a numerical method is doing what we expect and exhbits the correct convergence behavior.  Since errors can get small quickly it is very common to plot these sorts of plots on a log-log scale to easily visualize the results.  Note that if a method was truly of the order $n$ that they will be a linear function in log-log space with slope $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lr5k5s0n9QnV197DzzuPFyEi/VnfrSDmHIAh+RXk5TJyozGTOqrsb4umvn6bl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\nhBBCCCGEEMKV/h8UoYVWbfHD2wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a1e13d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dx = numpy.linspace(1.0, 1e-4, 100)\n",
    "\n",
    "fig = plt.figure()\n",
    "fig.set_figwidth(fig.get_figwidth() * 2.0)\n",
    "axes = []\n",
    "axes.append(fig.add_subplot(1, 2, 1))\n",
    "axes.append(fig.add_subplot(1, 2, 2))\n",
    "\n",
    "for n in xrange(1, 5):\n",
    "    axes[0].plot(dx, dx**n, label=\"$\\Delta x^%s$\" % n)\n",
    "    axes[1].loglog(dx, dx**n, label=\"$\\Delta x^%s$\" % n)\n",
    "\n",
    "axes[0].legend(loc=2)\n",
    "axes[1].set_xticks([10.0**(-n) for n in xrange(5)])\n",
    "axes[1].set_yticks([10.0**(-n) for n in xrange(16)])\n",
    "axes[1].legend(loc=4)\n",
    "for n in xrange(2):\n",
    "    axes[n].set_title(\"Growth of Error vs. $\\Delta x^n$\")\n",
    "    axes[n].set_xlabel(\"$\\Delta x$\")\n",
    "    axes[n].set_ylabel(\"Estimated Error\")\n",
    "    axes[n].set_title(\"Growth of different\")\n",
    "    axes[n].set_xlabel(\"$\\Delta x$\")\n",
    "    axes[n].set_ylabel(\"Estimated Error\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Horner's Method for Evaluating Polynomials\n",
    "\n",
    "Given \n",
    "\n",
    "$$P_N(x) = a_0 + a_1 x + a_2 x^2 + \\ldots + a_N x^N$$ \n",
    "\n",
    "$$P_N(x) = p_1 x^N + p_2 x^{N-1} + p_3 x^{N-2} + \\ldots + p_{N+1}$$\n",
    "\n",
    "want to find best way to evaluate $P_N(x)$.  First consider two ways to write $P_3$:\n",
    "\n",
    "$$ P_3(x) = p_1 x^3 + p_2 x^2 + p_3 x + p_4$$\n",
    "\n",
    "and using nested multiplication:\n",
    "\n",
    "$$ P_3(x) = ((p_1 x + p_2) x + p_3) x + p_4$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Consider how many operations it takes for each...\n",
    "\n",
    "$$ P_3(x) = p_1 x^3 + p_2 x^2 + p_3 x + p_4$$\n",
    "\n",
    "$$P_3(x) = \\overbrace{p_1 \\cdot x \\cdot x \\cdot x}^3 + \\overbrace{p_2 \\cdot x \\cdot x}^2 + \\overbrace{p_3 \\cdot x}^1 + p_4$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Adding up all the operations we can in general think of this as a pyramid\n",
    "\n",
    "![Original Count](./images/horners_method_big_count.png)\n",
    "\n",
    "We can estimate this way that the algorithm written this way will take approximately $O(N^2 / 2)$ operations to complete."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Looking at our other means of evaluation:\n",
    "\n",
    "$$ P_3(x) = ((p_1 x + p_2) x + p_3) x + p_4$$\n",
    "\n",
    "Here we find that the method is $O(N)$ (the 2 is usually ignored in these cases).  The important thing is that the first evaluation is $O(N^2)$ and the second $O(N)$!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Algorithm\n",
    "\n",
    "Fill in the function and implement Horner's method:\n",
    "```python\n",
    "def eval_poly(p, x):\n",
    "    \"\"\"Evaluates polynomial given coefficients p at x\n",
    "    \n",
    "    Function to evaluate a polynomial in order N operations.  The polynomial is defined as\n",
    "    \n",
    "    P(x) = p[0] x**n + p[1] x**(n-1) + ... + p[n-1] x + p[n]\n",
    "    \n",
    "    The value x should be a float.\n",
    "    \"\"\"\n",
    "    pass\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "```python\n",
    "def eval_poly(p, x):\n",
    "    \"\"\"Evaluates polynomial given coefficients p at x\n",
    "    \n",
    "    Function to evaluate a polynomial in order N operations.  The polynomial is defined as\n",
    "    \n",
    "    P(x) = p[0] x**n + p[1] x**(n-1) + ... + p[n-1] x + p[n]\n",
    "    \n",
    "    The value x should be a float.\n",
    "    \"\"\"\n",
    "    \n",
    "    y = p[0]\n",
    "    for coefficient in p[1:]:\n",
    "        y = y * x + coefficient\n",
    "    \n",
    "    return y\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "or an alternative version that allows `x` to be a vector of values:\n",
    "```python\n",
    "def eval_poly(p, x):\n",
    "    \"\"\"Evaluates polynomial given coefficients p at x\n",
    "    \n",
    "    Function to evaluate a polynomial in order N operations.  The polynomial is defined as\n",
    "    \n",
    "    P(x) = p[0] x**n + p[1] x**(n-1) + ... + p[n-1] x + p[n]\n",
    "    \n",
    "    The value x can by a NumPy ndarray.\n",
    "    \"\"\"\n",
    "    \n",
    "    y = numpy.ones(x.shape) * p[0]\n",
    "    for coefficient in p[1:]:\n",
    "        y = y * x + coefficient\n",
    "    \n",
    "    return y\n",
    "```\n",
    "This version calculates each `y` value simultaneously making for much faster code!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a68d590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def eval_poly(p, x):\n",
    "    \"\"\"Evaluates polynomial given coefficients p at x\n",
    "    \n",
    "    Function to evaluate a polynomial in order N operations.  The polynomial is defined as\n",
    "    \n",
    "    P(x) = p[0] x**n + p[1] x**(n-1) + ... + p[n-1] x + p[n]\n",
    "    \n",
    "    The value x can by a NumPy ndarray.\n",
    "    \"\"\"\n",
    "    \n",
    "    y = numpy.ones(x.shape) * p[0]\n",
    "    for coefficient in p[1:]:\n",
    "        y = y * x + coefficient\n",
    "    \n",
    "    return y\n",
    "\n",
    "p = [1, -3, 10, 4, 5, 5]\n",
    "x = numpy.linspace(-10, 10, 100)\n",
    "plt.plot(x, eval_poly(p, x))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Truncation Error vs. Floating Point Error\n",
    "\n",
    "Truncation error:  Errors arising from approximation of a function, truncation of a series...\n",
    "\n",
    "$$\\sin x \\approx x - \\frac{x^3}{3!} + \\frac{x^5}{5!} + O(x^7)$$\n",
    "\n",
    "Floating-point Error:  Errors arising from approximating real numbers with finite-precision numbers\n",
    "\n",
    "$$\\pi \\approx 3.14$$\n",
    "\n",
    "or $\\frac{1}{3} \\approx 0.333333333$ in decimal, results form finitely number of registers to represent each number.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Floating Point Systems\n",
    "\n",
    "Numbers in floating point systems are represented as a series of bits that represent different pieces of a number.  In *nomalized floating point systems* there are some standar conventions for what these bits are used for.  In general the numbers are stored by breaking them down into the form  \n",
    "\n",
    "$$\\hat{f} = \\pm d_1 . d_2 d_3 d_4 \\ldots d_p \\times \\beta^E$$\n",
    "\n",
    "where\n",
    "1. $\\pm$ is a single bit and of course represents the sign of the number\n",
    "2. $d_1 . d_2 d_3 d_4 \\ldots d_p$ is called the *mantissa*.  Note that technically the decimal could be moved but generally, using scientific notation, the decimal can always be placed at this location.  The digits $d_2 d_3 d_4 \\ldots d_p$ are called the *fraction* with $p$ digits of precision.  Normalized systems specifically put the decimal point in the front like we have and assume $d_1 \\neq 0$ unless the number is exactly $0$.\n",
    "3. $\\beta$ is the *base*.  For binary $\\beta = 2$, for decimal $\\beta = 10$, etc.\n",
    "4. $E$ is the *exponent*, an integer in the range $[E_{\\min}, E_{\\max}]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The important points on any floating point system is that\n",
    "1. There exist a discrete and finite set of representable numbers\n",
    "2. These representable numbers are not evenly distributed on the real line\n",
    "3. Airthmetic in floating point systems yield different results from infinite precision arithmetic (i.e. \"real\" math)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example:  Toy System\n",
    "Consider the toy 2-digit precision decimal system (normalized)\n",
    "$$f = \\pm d_1 . d_2 \\times 10^E$$\n",
    "with $E \\in [-2, 0]$.\n",
    "\n",
    "#### Number and distribution of numbers\n",
    "1. How many numbers can we represent with this system?\n",
    "\n",
    "2. What is the distribution on the real line?\n",
    "\n",
    "3. What is the underflow and overflow limits?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "How many numbers can we represent with this system?\n",
    "\n",
    "$f = \\pm d_1 . d_2 \\times 10^E$ with $E \\in [-2, 0]$.\n",
    "\n",
    "$$ 2 \\times 9 \\times 10 \\times 3 + 1 = 541$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "What is the distribution on the real line?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a179a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d_1_values = [1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
    "d_2_values = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
    "E_values = [0, -1, -2]\n",
    "\n",
    "fig = plt.figure(figsize=(10.0, 1.0))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "for E in E_values:\n",
    "    for d1 in d_1_values:\n",
    "        for d2 in d_2_values:\n",
    "            axes.plot( (d1 + d2 * 0.1) * 10**E, 0.0, 'r+', markersize=20)\n",
    "            axes.plot(-(d1 + d2 * 0.1) * 10**E, 0.0, 'r+', markersize=20)\n",
    "            \n",
    "axes.plot(0.0, 0.0, '+', markersize=20)\n",
    "axes.plot([-10.0, 10.0], [0.0, 0.0], 'k')\n",
    "\n",
    "axes.set_title(\"Distribution of Values\")\n",
    "axes.set_yticks([])\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"\")\n",
    "axes.set_xlim([-0.1, 0.1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "What is the underflow and overflow limits?\n",
    "\n",
    "Smallest number that can be represented is the underflow:  $1.0 \\times 10^{-2} = 0.01$\n",
    "Largest number that can be represented is the overflow:  $9.9 \\times 10^0 = 9.9$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Properties of Floating Point Systems\n",
    "All floating-point systems are characterized by several important numbers\n",
    " - Smalled normalized number (underflow if below)\n",
    " - Largest normalized number (overflow if above)\n",
    " - Zero\n",
    " - Machine $\\epsilon$ or $\\epsilon_{\\text{machine}}$\n",
    " - `inf` and `nan`, infinity and **N**ot **a** **N**umber respectively\n",
    " - Subnormal numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Binary Systems\n",
    "Consider the 2-digit precision base 2 system:\n",
    "\n",
    "$$f=\\pm d_1 . d_2 \\times 2^E ~~~~ \\text{with} ~~~~ E \\in [-1, 1]$$\n",
    "\n",
    "#### Number and distribution of numbers\n",
    "1. How many numbers can we represent with this system?\n",
    "\n",
    "2. What is the distribution on the real line?\n",
    "\n",
    "3. What is the underflow and overflow limits?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "How many numbers can we represent with this system?\n",
    "\n",
    "$$f=\\pm d_1 . d_2 \\times 2^E ~~~~ \\text{with} ~~~~ E \\in [-1, 1]$$\n",
    "\n",
    "$$ 2 \\times 1 \\times 2 \\times 3 + 1 = 13$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "What is the distribution on the real line?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a32f310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d_1_values = [1]\n",
    "d_2_values = [0, 1]\n",
    "E_values = [1, 0, -1]\n",
    "\n",
    "fig = plt.figure(figsize=(10.0, 1.0))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "for E in E_values:\n",
    "    for d1 in d_1_values:\n",
    "        for d2 in d_2_values:\n",
    "            axes.plot( (d1 + d2 * 0.1) * 2**E, 0.0, 'r+', markersize=20)\n",
    "            axes.plot(-(d1 + d2 * 0.1) * 2**E, 0.0, 'r+', markersize=20)\n",
    "            \n",
    "axes.plot(0.0, 0.0, 'r+', markersize=20)\n",
    "axes.plot([-4.5, 4.5], [0.0, 0.0], 'k')\n",
    "\n",
    "axes.set_title(\"Distribution of Values\")\n",
    "axes.set_yticks([])\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"\")\n",
    "axes.set_xlim([-3, 3])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Smallest number that can be represented is the underflow:  $1.0 \\times 2^{-1} = 0.25$\n",
    "Largest number that can be represented is the overflow:  $1.1 \\times 2^1 = 2.2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Real Systems - IEEE 754 Binary Floating Point Systems\n",
    "\n",
    "### Single Precision\n",
    " - Total storage alloted is 32 bits\n",
    " - Exponent is 8 bits $\\Rightarrow E \\in [-126, 127]$\n",
    " - Fraction 23 bits ($p = 24$)\n",
    " \n",
    "```\n",
    "s EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF\n",
    "0 1      8 9                     31\n",
    "```\n",
    "Overflow $= 2^{127} \\approx 3.4 \\times 10^{38}$\n",
    "\n",
    "Underflow $= 2^{-126} \\approx 1.2 \\times 10^{-38}$\n",
    "\n",
    "$\\epsilon_{\\text{machine}} = 2^{-23} \\approx 1.2 \\times 10^7$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Double Precision\n",
    " - Total storage alloted is 64 bits\n",
    " - Exponent is 11 bits $\\Rightarrow E \\in [-1022, 1024]$\n",
    " - Fraction 52 bits ($p = 53$)\n",
    " \n",
    "```\n",
    "s EEEEEEEEEE FFFFFFFFFF FFFFFFFFFF FFFFFFFFFF FFFFFFFFFF FFFFFFFFFF FF\n",
    "0 1       11 12                                                      63\n",
    "```\n",
    "Overflow $= 2^{1024} \\approx 1.8 \\times 10^{308}$\n",
    "\n",
    "Underflow $= 2^{-1022} \\approx 2.2 \\times 10^{-308}$\n",
    "\n",
    "$\\epsilon_{\\text{machine}} = 2^{-52} \\approx 2.2 \\times 10^{-16}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Python Access to IEEE Numbers\n",
    "\n",
    "Access many important parameters, such as machine epsilon:\n",
    "\n",
    "```python\n",
    "import numpy\n",
    "numpy.finfo(float).eps\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.22044604925e-16\n",
      "1.79769313486e+308\n",
      "-1.79769313486e+308\n",
      "52\n",
      "15\n",
      "2.22507385851e-308\n"
     ]
    }
   ],
   "source": [
    "print numpy.finfo(float).eps\n",
    "print numpy.finfo(float).max\n",
    "print numpy.finfo(float).min\n",
    "print numpy.finfo(float).nmant\n",
    "print numpy.finfo(float).precision\n",
    "print numpy.finfo(float).tiny"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Why should we care about this?\n",
    "\n",
    " - Floating point arithmetic is not commutative or associative\n",
    " - Floating point errors compound, do not assume even double precision is enough!\n",
    " - Mixing precision is very dangerous"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    " ### Examples\n",
    " \n",
    "Simple arithmetic $\\delta < \\epsilon_{\\text{machine}}$\n",
    "\n",
    "   $$(1+\\delta) - 1 = 1 - 1 = 0$$\n",
    "\n",
    "   $$1 - 1 + \\delta = \\delta$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Catastrophic cancelation (assume $x + y \\neq 0$):\n",
    "\n",
    "$$x(1 + \\epsilon_x) + y (1 + \\epsilon_y) = x + y + x \\epsilon_x + y \\epsilon_y =(x+y) \\left( 1 + \\frac{x \\epsilon_x + y \\epsilon_y}{x+y} \\right ) \\Rightarrow$$\n",
    "\n",
    "$$\\epsilon_{x+y} = \\frac{x}{x + y} \\epsilon_x  + \\frac{y}{x+y} \\epsilon_y$$\n",
    "\n",
    "This implies that the error made from doing the actual sum can be arbitrarily large!  This is not too bad unless $x$ and $y$ have opposite sign, then if $x+y ~ 0$ then $\\frac{1}{x+y}$ can be huge.  This is refered to as *catastrophic* cancelation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Matrix vector multiplication\n",
    "\n",
    "   $$ \\begin{bmatrix} \\epsilon & 1 \\\\ 1 & 1 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} \n",
    "= \\begin{bmatrix} \\epsilon + 1 \\\\ 1 + 1 \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Solve $A x = b$ where\n",
    "\n",
    "   $$A = \\begin{bmatrix} 10 & 1 \\\\ 3 & 0.3 \\end{bmatrix}~~~~~~\\text{and}~~~~~~b=\\begin{bmatrix} 11 \\\\ 3.3 \\end{bmatrix}$$\n",
    "\n",
    "   and we know that\n",
    "\n",
    "   $$x = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}$$\n",
    "\n",
    "   When used in a solver we get\n",
    "\n",
    "   $$x = \\begin{bmatrix}-0.5 \\\\ 16 \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Evaluation of a polynomial\n",
    "\n",
    "   $$f(x) = x^7 - 7x^6 + 21 x^5 - 35 x^4 + 35x^3-21x^2 + 7x - 1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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jQwIXm1O76tjeqfm8krYByP3kPFMcosJzxBWH1tagmw+IF1ZE3h2pGopD1pg1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\nQghpFP8PzkJMeoTf8NsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a362cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = numpy.linspace(0.988, 1.012, 1000)\n",
    "y = x**7 - 7.0 * x**6 + 21.0 * x**5 - 35.0 * x**4 + 35.0 * x**3 - 21.0 * x**2 + 7.0 * x - 1.0\n",
    "\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(x, y, 'r')\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"y\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For more examples see the \"[100-digit Challenge](http://bookstore.siam.org/ot86/)\" book."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Combination of Error\n",
    "\n",
    "In general we need to concern ourselves with the combination of both truncation error and floating point error.\n",
    "\n",
    "### Example\n",
    "\n",
    "Consider the finite difference approximation where $f(x) = e^x$ and we are evaluating at $x=1$?\n",
    "\n",
    "$$f'(x) \\approx \\frac{f(x + \\Delta x) - f(x)}{\\Delta x}$$\n",
    "\n",
    "Compare the error between decreasing $\\Delta x$ and the true solution $f'(1) = e$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a362050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "delta_x = numpy.linspace(1e-20, 5.0, 1000)\n",
    "x = 1.0\n",
    "f_hat = (numpy.exp(x + delta_x) - numpy.exp(x)) / (delta_x)\n",
    "\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.loglog(delta_x, numpy.abs(f_hat - numpy.exp(1)) / numpy.exp(1))\n",
    "axes.set_xlabel(\"$\\Delta x$\")\n",
    "axes.set_ylabel(\"Relative Error\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example 2\n",
    "\n",
    "Evaluate $e^x$ with its taylor series.\n",
    "\n",
    "$$e^x = \\sum^\\infty_{n=0} \\frac{x^n}{n!}$$\n",
    "\n",
    "Can we pick $N < \\infty$ that can approximate $e^x$ over a give range $x \\in [a,b]$ such that the relative error $E$ satisfies $E < 8 \\cdot \\varepsilon_{\\text{machine}}$?\n",
    "\n",
    "What might be a better way than simply evaluating the Taylor polynomial directly for various $N$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "22\n"
     ]
    },
    {
     "data": {
      "image/png": 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RkV9FFQiGDWMgICLyq2ADgXmHrwIBS0NERP4UbCCwygiGDAEOHZL5BERE5E1RBYK+fXni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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a1f1ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.misc\n",
    "\n",
    "def my_exp(x, N=10):\n",
    "    value = 0.0\n",
    "    for n in xrange(N + 1):\n",
    "        value += x**n / scipy.misc.factorial(n)\n",
    "        \n",
    "    return value\n",
    "\n",
    "x = numpy.linspace(-2, 2, 100)\n",
    "for N in range(1, 50):\n",
    "    error = numpy.abs((numpy.exp(x) - my_exp(x, N=N)) / numpy.exp(x))\n",
    "    if numpy.all(error < 100.0 * numpy.finfo(float).eps):\n",
    "        break\n",
    "\n",
    "print N\n",
    "\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(x, error)\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"Relative Error\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some other links that might be helpful regarding IEEE Floating Point:\n",
    " - [What Every Computer Scientist Should Know About Floating-Point Arithmetic](http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html)\n",
    " - [IEEE 754 Floating Point Calculator](http://babbage.cs.qc.edu/courses/cs341/IEEE-754.html)\n",
    " - [Numerical Computing with IEEE Floating Point Arithmetic](http://epubs.siam.org/doi/book/10.1137/1.9780898718072)"
   ]
  }
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