gwtaylor/RootFinding.ipynb

## RootFinding.ipynb
{
 "metadata": {
  "name": "RootFinding"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we need to import some tools."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "from scipy import optimize"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we define our function. Note that we use `deg2rad` to convert the angle to radians (the numpy trig functions expect radians)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def f(x):\n",
      "    y = np.deg2rad(x)\n",
      "    return 5*np.tan(y) - 6*np.sin(y) - 0.4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's make a plot of our function on the interval [0, 45] degrees."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.arange(0, 45, 0.01)\n",
      "y = f(x)\n",
      "plt.plot(x,y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107da76d0>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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V/aKiokS9evVEkyZNxNtvv211HwDcuHHjxk3DVlV1UYGBAwciIyOjzM8XLFgA\nmdUVy8zMxKRJk3Ds2DE0b94cw4YNw86dOxEYGFhqv8q8FhER2a7C0I+Oji73d+vXr0dKSgq8vb2R\nkpICHx+fMvtYLBb4+fmhc+fOAIBhw4YhLi6uTOgTEVHt0Nyn7+vri4iICOTn5yMiIgJ+fn5l9unb\nty/279+Pa9eu4ffff8euXbsQEBBgU8FERKSd5tCfNGkSzp07h27duuHChQuYOHEiAODixYslLfmm\nTZti9uzZePbZZ9GnTx94enri8ccfr57KiYio6qp8FaCaxcbGCldXV9G5c2excuVK1eUIIYRwcXER\nDz/8sPDy8hI+Pj5KahgzZoxo3bq16N69e8nPsrKyxJAhQ8RDDz0kgoKCRHZ2tl3UNWfOHPHggw8K\nLy8v4eXlJXbt2lWrNZ07d06YTCbh7u4u+vfvLzZv3iyEUHu+yqtJ5bnKz88XPXv2FJ6ensLX11d8\n9NFHQgi156m8mlT/mxJCiBs3bggvLy/xzDPPCCHs4+/vzpq0nCfloe/l5SViY2PFmTNnRLdu3URm\nZqbqkkT79u3F1atXldYQFxcnDh48WCpcFy9eLKZMmSIKCgrE5MmTxdKlS+2irrlz54ply5bVei23\nXLp0SRw6dEgIIURmZqbo0KGDyMrKUnq+yqtJ9bnKzc0VQghRUFAgPDw8RFpamvJ/V9ZqUn2ehBBi\n2bJlYuTIkWLw4MFCCPv4+7uzJi3nSekyDNevXwcA9OvXDy4uLggICLA63l8FoXhEUd++fcvMaajM\n3AgVdQFqz1ebNm3g5eUFAGjZsiU8PDyQlJSk9HyVVxOg9lw1atQIAJCTk4MbN26gfv36yv9dWasJ\nUHuezp8/j6ioKIwfP76kDtXnyVpNQjbcq/Q6SkM/KSkJrq6uJY/d3d2RkJCgsCLJYDBgwIABGDp0\nKCIjI1WXU+L28+Xq6gqLxaK4oj+FhYXBz88PixcvRnZ2trI6Tpw4geTkZPTs2dNuztetmnx9fQGo\nPVfFxcXw9PTEAw88gClTpsBoNCo/T9ZqAtSep+nTp2Pp0qWoU+fPiFR9nqzVZDAYqnyeuOCaFfv2\n7cPhw4exaNEivPHGG1bnKqig+tNHeSZNmoTTp09j9+7dOHnyJMLDw5XUkZ2djRdffBHLly9HkyZN\n7OJ83V5T48aNlZ+rOnXq4PDhwzhx4gRWr16NQ4cOKT9P1mpSeZ6+/fZbtG7dGt7e3qXOjcrzVF5N\nWs6T0tCcqFCPAAACCklEQVT38fFBampqyePk5GSrQz9rW9u2bQEAbm5uGDJkCL755hvFFUk+Pj5I\nSUkBgHLnRqjQunVrGAwGNGvWDJMnT8aOHTtqvYbCwkIEBwdj9OjRCAoKAqD+fFmryR7OFQC0b98e\nTz/9NBITE5WfJ2s1qTxP8fHxiIyMRIcOHTBixAjs2bMHo0ePVnqerNUUEhKi6TwpDf1mzZoBAOLi\n4nDmzBlER0eXfARWJS8vr+QjUmZmJnbv3o1BgwYpremWysyNUOHSpUsAgBs3bmDLli14+umna/X4\nQgiMGzcO3bt3L7WSq8rzVV5NKs/VlStX8NtvvwEArl69iu+//x5BQUFKz1N5Nak8TwsXLkR6ejpO\nnz6Nbdu2YcCAAdi4caPS82Stpg0bNmg7T9VxRdkWZrNZuLq6ik6dOokVK1aoLkecOnVKeHp6Ck9P\nTzFgwADx2WefKalj+PDhom3btuKee+4R7dq1ExEREXYxZOxWXfXq1RPt2rUTn332mRg9erR4+OGH\nRY8ePcT06dNrfeTT3r17hcFgEJ6enqWGrqk8X9ZqioqKUnqujhw5Iry9vcUjjzwiAgICxPr164UQ\naocilleT6n9Tt5jN5pKRMvbw9yeEEP/+979Laho1alSVz5NBCDvo+CQiolrBC7lERDrC0Cci0hGG\nPhGRjjD0iYh0hKFPRKQjDH0iIh35fzpFKqJ5gUVAAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We use the `brentq` method to find a root of a function in a given interval."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_opt = optimize.brentq(f, 0, 45)\n",
      "print x_opt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "40.8633818305\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}
	{
	"metadata": {
	"name": "RootFinding"
	},
	"nbformat": 3,
	"nbformat_minor": 0,
	"worksheets": [
	{
	"cells": [
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"First we need to import some tools."
	]
	},
	{
	"cell_type": "code",
	"collapsed": true,
	"input": [
	"import numpy as np\n",
	"import matplotlib.pyplot as plt\n",
	"from scipy import optimize"
	],
	"language": "python",
	"metadata": {},
	"outputs": [],
	"prompt_number": 6
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Now we define our function. Note that we use `deg2rad` to convert the angle to radians (the numpy trig functions expect radians)."
	]
	},
	{
	"cell_type": "code",
	"collapsed": true,
	"input": [
	"def f(x):\n",
	" y = np.deg2rad(x)\n",
	" return 5np.tan(y) - 6np.sin(y) - 0.4"
	],
	"language": "python",
	"metadata": {},
	"outputs": [],
	"prompt_number": 7
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Let's make a plot of our function on the interval [0, 45] degrees."
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"x = np.arange(0, 45, 0.01)\n",
	"y = f(x)\n",
	"plt.plot(x,y)"
	],
	"language": "python",
	"metadata": {},
	"outputs": [
	{
	"output_type": "pyout",
	"prompt_number": 8,
	"text": [
	"[<matplotlib.lines.Line2D at 0x107da76d0>]"
	]
	},
	{
	"output_type": "display_data",
	"png": 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V/aKiokS9evVEkyZNxNtvv211HwDcuHHjxk3DVlV1UYGBAwciIyOjzM8XLFgA\nmdUVy8zMxKRJk3Ds2DE0b94cw4YNw86dOxEYGFhqv8q8FhER2a7C0I+Oji73d+vXr0dKSgq8vb2R\nkpICHx+fMvtYLBb4+fmhc+fOAIBhw4YhLi6uTOgTEVHt0Nyn7+vri4iICOTn5yMiIgJ+fn5l9unb\nty/279+Pa9eu4ffff8euXbsQEBBgU8FERKSd5tCfNGkSzp07h27duuHChQuYOHEiAODixYslLfmm\nTZti9uzZePbZZ9GnTx94enri8ccfr57KiYio6qp8FaCaxcbGCldXV9G5c2excuVK1eUIIYRwcXER\nDz/8sPDy8hI+Pj5KahgzZoxo3bq16N69e8nPsrKyxJAhQ8RDDz0kgoKCRHZ2tl3UNWfOHPHggw8K\nLy8v4eXlJXbt2lWrNZ07d06YTCbh7u4u+vfvLzZv3iyEUHu+yqtJ5bnKz88XPXv2FJ6ensLX11d8\n9NFHQgi156m8mlT/mxJCiBs3bggvLy/xzDPPCCHs4+/vzpq0nCfloe/l5SViY2PFmTNnRLdu3URm\nZqbqkkT79u3F1atXldYQFxcnDh48WCpcFy9eLKZMmSIKCgrE5MmTxdKlS+2irrlz54ply5bVei23\nXLp0SRw6dEgIIURmZqbo0KGDyMrKUnq+yqtJ9bnKzc0VQghRUFAgPDw8RFpamvJ/V9ZqUn2ehBBi\n2bJlYuTIkWLw4MFCCPv4+7uzJi3nSekyDNevXwcA9OvXDy4uLggICLA63l8FoXhEUd++fcvMaajM\n3AgVdQFqz1ebNm3g5eUFAGjZsiU8PDyQlJSk9HyVVxOg9lw1atQIAJCTk4MbN26gfv36yv9dWasJ\nUHuezp8/j6ioKIwfP76kDtXnyVpNQjbcq/Q6SkM/KSkJrq6uJY/d3d2RkJCgsCLJYDBgwIABGDp0\nKCIjI1WXU+L28+Xq6gqLxaK4oj+FhYXBz88PixcvRnZ2trI6Tpw4geTkZPTs2dNuztetmnx9fQGo\nPVfFxcXw9PTEAw88gClTpsBoNCo/T9ZqAtSep+nTp2Pp0qWoU+fPiFR9nqzVZDAYqnyeuOCaFfv2\n7cPhw4exaNEivPHGG1bnKqig+tNHeSZNmoTTp09j9+7dOHnyJMLDw5XUkZ2djRdffBHLly9HkyZN\n7OJ83V5T48aNlZ+rOnXq4PDhwzhx4gRWr16NQ4cOKT9P1mpSeZ6+/fZbtG7dGt7e3qXOjcrzVF5N\nWs6T0tCcqFCPAAACCklEQVT38fFBampqyePk5GSrQz9rW9u2bQEAbm5uGDJkCL755hvFFUk+Pj5I\nSUkBgHLnRqjQunVrGAwGNGvWDJMnT8aOHTtqvYbCwkIEBwdj9OjRCAoKAqD+fFmryR7OFQC0b98e\nTz/9NBITE5WfJ2s1qTxP8fHxiIyMRIcOHTBixAjs2bMHo0ePVnqerNUUEhKi6TwpDf1mzZoBAOLi\n4nDmzBlER0eXfARWJS8vr+QjUmZmJnbv3o1BgwYpremWysyNUOHSpUsAgBs3bmDLli14+umna/X4\nQgiMGzcO3bt3L7WSq8rzVV5NKs/VlStX8NtvvwEArl69iu+//x5BQUFKz1N5Nak8TwsXLkR6ejpO\nnz6Nbdu2YcCAAdi4caPS82Stpg0bNmg7T9VxRdkWZrNZuLq6ik6dOokVK1aoLkecOnVKeHp6Ck9P\nTzFgwADx2WefKalj+PDhom3btuKee+4R7dq1ExEREXYxZOxWXfXq1RPt2rUTn332mRg9erR4+OGH\nRY8ePcT06dNrfeTT3r17hcFgEJ6enqWGrqk8X9ZqioqKUnqujhw5Iry9vcUjjzwiAgICxPr164UQ\naocilleT6n9Tt5jN5pKRMvbw9yeEEP/+979Laho1alSVz5NBCDvo+CQiolrBC7lERDrC0Cci0hGG\nPhGRjjD0iYh0hKFPRKQjDH0iIh35fzpFKqJ5gUVAAAAAAElFTkSuQmCC\n"
	}
	],
	"prompt_number": 8
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"We use the `brentq` method to find a root of a function in a given interval."
	]
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [
	"x_opt = optimize.brentq(f, 0, 45)\n",
	"print x_opt"
	],
	"language": "python",
	"metadata": {},
	"outputs": [
	{
	"output_type": "stream",
	"stream": "stdout",
	"text": [
	"40.8633818305\n"
	]
	}
	],
	"prompt_number": 9
	},
	{
	"cell_type": "code",
	"collapsed": false,
	"input": [],
	"language": "python",
	"metadata": {},
	"outputs": []
	}
	],
	"metadata": {}
	}
	]
	}