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_downloads/01d26ddfe66a9569b4cf79881141dece/cross_product_error.ipynb

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+{
+ "metadata": {
+  "name": "cross_product_error"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The error in a cross product calculation with (3,) tuples"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from sympy import Symbol, symarray, Matrix, matrices\n",
+      "u = Matrix(symarray('u', (3, 1)))\n",
+      "v = Matrix(symarray('v', (3, 1)))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "c = u.cross(v)\n",
+      "c"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "pyout",
+       "prompt_number": 2,
+       "text": [
+        "[u_1_0*v_2_0 - u_2_0*v_1_0, -u_0_0*v_2_0 + u_2_0*v_0_0, u_0_0*v_1_0 - u_1_0*v_0_0]"
+       ]
+      }
+     ],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Assuming same error $\\delta$ for both vectors:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "d = Symbol('d')\n",
+      "e = matrices.ones((3,1)) * d"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Same calculation as above, with error"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "ue = u + e\n",
+      "ve = v + e\n",
+      "ce = ue.cross(ve)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Calculate absolute error by subtracting true result from result with error"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "cce = ce - c\n",
+      "cce.simplify()\n",
+      "cce"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "pyout",
+       "prompt_number": 5,
+       "text": [
+        "[d*(u_1_0 - u_2_0 - v_1_0 + v_2_0), d*(-u_0_0 + u_2_0 + v_0_0 - v_2_0), d*(u_0_0 - u_1_0 - v_0_0 + v_1_0)]"
+       ]
+      }
+     ],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Floating point calculation error given by operations on elements:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "c"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "pyout",
+       "prompt_number": 6,
+       "text": [
+        "[u_1_0*v_2_0 - u_2_0*v_1_0, -u_0_0*v_2_0 + u_2_0*v_0_0, u_0_0*v_1_0 - u_1_0*v_0_0]"
+       ]
+      }
+     ],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Each element has two products and one subtraction; The input values are $\\le 1$.  Calculation error per element then $3 \\epsilon / 2$"
+     ]
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}