{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "code", "collapsed": false, "input": [ "from __future__ import division\n", "\n", "import numpy as np\n", "import matplotlib.pyplot as pt" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 61 }, { "cell_type": "code", "collapsed": false, "input": [ "def rk4_step(y, t, h, f):\n", " k1 = f(t, y)\n", " k2 = f(t+h/2, y + h/2*k1)\n", " k3 = f(t+h/2, y + h/2*k2)\n", " k4 = f(t+h, y + h*k3)\n", " return y + h/6*(k1 + 2*k2 + 2*k3 + k4)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 62 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Want to solve:\n", "\n", "$$w''(t)=\\frac 32w^2$$\n", "\n", "with $w(0)=4$ and $w(1)=1$. (Example due to Stoer and Bulirsch)" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def f(t, y):\n", " w, w_prime = y\n", " return np.array([w_prime, 3/2*w**2])" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 63 }, { "cell_type": "code", "collapsed": false, "input": [ "def shoot(w_prime):\n", " times = [0]\n", " y_values = [np.array([4, w_prime])]\n", " \n", " h = 1/2**7\n", " t_end = 1\n", " \n", " while times[-1] < t_end:\n", " y_values.append(rk4_step(y_values[-1], times[-1], h, f))\n", " times.append(times[-1]+h)\n", " \n", " y_values = np.array(y_values)\n", " \n", " # actually floating-point-equal due to power-of-2 h\n", " assert times[-1] == t_end\n", " \n", " print \"w'(0) = %g -> w(1)= %.5g\" % (w_prime, y_values[-1,0])\n", "\n", " pt.plot(times, y_values[:, 0], label=\"$w'(0)=%.2g$\" % w_prime)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 64 }, { "cell_type": "code", "collapsed": false, "input": [ "shoot(0)\n", "shoot(-5)\n", "shoot(-7)\n", "shoot(-7.5)\n", "\n", "pt.grid()\n", "pt.legend(loc=\"best\")" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "w'(0) = 0 -> w(1)= 87.08\n", "w'(0) = -5 -> w(1)= 12.058" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", "w'(0) = -7 -> w(1)= 3.6442\n", "w'(0) = -7.5 -> w(1)= 2.2233\n" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 56, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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fdHVPOuamf2OEEMJ9rF8Ps2dDiztaux2X1MyFEMJTNDVpg0/88Y9w003O2aec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\nr3br/LDrk1384+I/MMWa+Or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"text": [ "" ] } ], "prompt_number": 56 }, { "cell_type": "code", "collapsed": false, "input": [ "shoot(-30)\n", "\n", "pt.grid()\n", "pt.legend(loc=\"best\")" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "w'(0) = -30 -> w(1)= -1.4668\n" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 60, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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