{"worksheets": [{"cells": [{"source": ["# Newton's method in 1D"], "cell_type": "markdown", "metadata": {}}, {"collapsed": false, "outputs": [], "prompt_number": 28, "cell_type": "code", "input": ["import numpy as np\n", "import matplotlib.pyplot as pt"], "metadata": {}, "language": "python"}, {"source": ["Here's a function:"], "cell_type": "markdown", "metadata": {}}, {"collapsed": false, "outputs": [], "prompt_number": 29, "cell_type": "code", "input": ["a = 17.09\n", "b = 9.79\n", "c = 0.6317\n", "d = 0.9324\n", "e = 0.4565\n", "\n", "def f(x):\n", " return a*x**4 + b*x**3 + c*x**2 + d*x + e\n", "\n", "def df(x):\n", " return 4*a*x**3 + 3*b*x**2 + 2*c*x + d\n", "\n", "def d2f(x):\n", " return 3*4*a*x**2 + 2*3*b*x + 2*c"], "metadata": {}, "language": "python"}, {"source": ["Let's plot the thing:"], "cell_type": "markdown", "metadata": {}}, {"collapsed": false, "outputs": [{"text": ["[]"], "prompt_number": 30, "output_type": "pyout", "metadata": {}}, {"png": 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SQir+IiIppOIvIpJCKv4iIimk4i8ikkIq/iIiKaTiLyKSQir+IiIppOIvIpJC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\nTkSuQmCC\n", "text": [""], "output_type": "display_data", "metadata": {}}], "prompt_number": 30, "cell_type": "code", "input": ["xmesh = np.linspace(-1, 0.5\n", " , 100)\n", "pt.ylim([-1, 3])\n", "pt.plot(xmesh, f(xmesh))"], "metadata": {}, "language": "python"}, {"source": ["Let's fix an initial guess:"], "cell_type": "markdown", "metadata": {}}, {"collapsed": false, "outputs": [], "prompt_number": 31, "cell_type": "code", "input": ["x = 0.3"], "metadata": {}, "language": "python"}, {"collapsed": false, "outputs": [{"text": ["0.14466962664624314\n"], "stream": "stdout", "output_type": "stream"}, {"png": 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"text": [""], "output_type": "display_data", "metadata": {}}], "prompt_number": 32, "cell_type": "code", "input": ["dfx = df(x)\n", "d2fx = d2f(x)\n", "\n", "# carry out the Newton step\n", "xnew = x - dfx / d2fx\n", "\n", "# plot approximate function\n", "pt.plot(xmesh, f(xmesh))\n", "pt.plot(xmesh, f(x) + dfx*(xmesh-x) + d2fx*(xmesh-x)**2/2)\n", "pt.plot(x, f(x), \"o\", color=\"red\")\n", "pt.plot(xnew, f(xnew), \"o\", color=\"green\")\n", "pt.ylim([-1, 3])\n", "\n", "# update\n", "x = xnew\n", "print(x)"], "metadata": {}, "language": "python"}, {"source": ["* What convergence order does this method achieve?"], "cell_type": "markdown", "metadata": {}}, {"collapsed": false, "outputs": [], "prompt_number": 33, "cell_type": "code", "input": ["# Quadratic, because it's just like doing 'equation-solving Newton' on f'."], "metadata": {}, "language": "python"}], "metadata": {}}], "nbformat": 3, "metadata": {"name": "", "signature": 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