Newton.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Using matplotlib backend: MacOSX\n",
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab\n",
    "%matplotlib inline\n",
    "import mpld3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vi skal tilnærme løsningen av $\\ln(x)+cos(x)-1=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return log(x) + cos(x) - 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vi tegner for å finne $x_0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "<style>\n",
       "\n",
       "</style>\n",
       "\n",
       "<div id=\"fig_el7243044668133601579537693\"></div>\n",
       "<script>\n",
       "function mpld3_load_lib(url, callback){\n",
       "  var s = document.createElement('script');\n",
       "  s.src = url;\n",
       "  s.async = true;\n",
       "  s.onreadystatechange = s.onload = callback;\n",
       "  s.onerror = function(){console.warn(\"failed to load library \" + url);};\n",
       "  document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "}\n",
       "\n",
       "if(typeof(mpld3) !== \"undefined\" && mpld3._mpld3IsLoaded){\n",
       "   // already loaded: just create the figure\n",
       "   !function(mpld3){\n",
       "       \n",
       "       mpld3.draw_figure(\"fig_el7243044668133601579537693\", {\"data\": {\"data02\": [[1.0, 0.0], [20.0, 0.0]], \"data01\": [[1.0, -0.45969769413186023], [1.0954773869346734, -0.4511877071257602], [1.1909547738693467, -0.4544713630264825], [1.2864321608040201, -0.4675801712583847], [1.3819095477386933, -0.4887681365597666], [1.4773869346733668, -0.5164514437364028], [1.5728643216080402, -0.5491696275234845], [1.6683417085427137, -0.5855606187667927], [1.763819095477387, -0.6243450005491862], [1.8592964824120601, -0.6643165064063793], [1.9547738693467336, -0.7043368097498681], [2.050251256281407, -0.7433332825100065], [2.1457286432160805, -0.7802988013955392], [2.241206030150754, -0.8142929419805369], [2.3366834170854274, -0.8444440764128797], [2.4321608040201004, -0.8699520112182656], [2.527638190954774, -0.8900908866815207], [2.6231155778894473, -0.9042121206973023], [2.7185929648241203, -0.9117472255298995], [2.8140703517587937, -0.9122103606504227], [2.909547738693467, -0.9052005120694369], [3.0050251256281406, -0.8904032105955993], [3.100502512562814, -0.8675917197465078], [3.1959798994974875, -0.8366276396564397], [3.291457286432161, -0.7974608869812029], [3.386934673366834, -0.7501290229841818], [3.4824120603015074, -0.6947559130363244], [3.577889447236181, -0.6315497109043362], [3.6733668341708543, -0.5608001705871944], [3.7688442211055277, -0.4828752971884316], [3.8643216080402008, -0.3982173564388708], [3.959798994974874, -0.3073382700427204], [4.055276381909548, -0.2108144310209763], [4.150753768844221, -0.10928097966853612], [4.2462311557788945, -0.0034255866152690384], [4.341708542713567, 0.10601820522892802], [4.4371859296482405, 0.21827802337742774], [4.532663316582914, 0.3325500325841617], [4.628140703517587, 0.448006561812875], [4.723618090452261, 0.5638039246234161], [4.819095477386934, 0.6790903641350983], [4.914572864321608, 0.793014052298993], [5.010050251256281, 0.9047310724646564], [5.105527638190955, 1.0134133141622685], [5.201005025125628, 1.1182562096336461], [5.296482412060302, 1.2184862429248797], [5.391959798994975, 1.3133681642840935], [5.4874371859296485, 1.4022118451692336], [5.582914572864322, 1.4843787123369756], [5.6783919597989945, 1.5592877032237147], [5.773869346733668, 1.6264206891072615], [5.869346733668341, 1.6853273173127938], [5.964824120603015, 1.7356292289539867], [6.060301507537688, 1.7770236143312852], [6.155778894472362, 1.8092860740918235], [6.251256281407035, 1.8322727605341091], [6.346733668341709, 1.8459217799574934], [6.442211055276382, 1.8502538436515756], [6.5376884422110555, 1.8453721619326173], [6.633165829145728, 1.8314615825002099], [6.7286432160804015, 1.808786981244931], [6.824120603015075, 1.7776909204237707], [6.919597989949748, 1.7385905957725338], [7.015075376884422, 1.691974100582435], [7.110552763819095, 1.6383960409726157], [7.206030150753769, 1.578472542484623], [7.301507537688442, 1.5128756936551335], [7.396984924623116, 1.4423274773388526], [7.492462311557789, 1.367593245207833], [7.5879396984924625, 1.2894747950039496], [7.683417085427135, 1.2088031137299762], [7.7788944723618085, 1.126430852998714], [7.874371859296482, 1.043224605191217], [7.969849246231155, 0.960057050882023], [8.065326633165828, 0.8777990491550531], [8.160804020100503, 0.7973117429476981], [8.256281407035175, 0.7194387514178788], [8.35175879396985, 0.6449985205307536], [8.447236180904522, 0.5747769016156552], [8.542713567839197, 0.509520025562531], [8.638190954773869, 0.44992753763011084], [8.733668341708542, 0.39664625454912095], [8.829145728643216, 0.35026430175369905], [8.924623115577889, 0.31130578419719734], [9.020100502512562, 0.2802260393450693], [9.115577889447236, 0.25740751563157227], [9.21105527638191, 0.24315631396695792], [9.306532663316583, 0.2376994238395851], [9.402010050251256, 0.24118267922799275], [9.49748743718593, 0.25366945297902266], [9.592964824120603, 0.2751401015793733], [9.688442211055277, 0.30549216541071544], [9.78391959798995, 0.3445413226950351], [9.879396984924623, 0.392023088469835], [9.974874371859297, 0.4475952431447914], [10.07035175879397, 0.5108409685442559], [10.165829145728644, 0.5812726628941278], [10.261306532663315, 0.6583364000258374], [10.356783919597989, 0.7414169922008202], [10.452261306532662, 0.8298436104593185], [10.547738693467336, 0.9228959113178368], [10.64321608040201, 1.0198106140262282], [10.738693467336683, 1.1197884684902921], [10.834170854271356, 1.2220015504063388], [10.92964824120603, 1.325600817172735], [11.025125628140703, 1.4297238557672736], [11.120603015075377, 1.5335027520298348], [11.21608040201005, 1.6360720096832342], [11.311557788944723, 1.7365764469714309], [11.407035175879397, 1.8341789989975403], [11.50251256281407, 1.9280683547024577], [11.597989949748744, 2.01746635893066], [11.693467336683417, 2.101635112168985], [11.78894472361809, 2.1798837032976426], [11.884422110552764, 2.2515745140351022], [11.979899497487438, 2.31612903765952], [12.075376884422111, 2.3730331590133824], [12.170854271356784, 2.4218418477048393], [12.266331658291456, 2.4621832217639965], [12.36180904522613, 2.4937619447466175], [12.457286432160803, 2.516361925348964], [12.552763819095476, 2.5298482949506234], [12.64824120603015, 2.534168645079199], [12.743718592964823, 2.529353513531843], [12.839195979899497, 2.5155161147323053], [12.93467336683417, 2.492851316786182], [13.030150753768844, 2.4616338745586024], [13.125628140703517, 2.422215934875278], [13.22110552763819, 2.3750238365778156], [13.316582914572864, 2.320554234587157], [13.412060301507537, 2.259369583286425], [13.507537688442211, 2.1920930203702103], [13.603015075376884, 2.1194026977682863], [13.698492462311558, 2.042025611288186], [13.793969849246231, 1.9607309851870456], [13.889447236180905, 1.8763232719371281], [13.984924623115578, 1.7896348309544594], [14.080402010050252, 1.7015183529842108], [14.175879396984925, 1.6128390991531067], [14.271356783919597, 1.5244670253872088], [14.36683417085427, 1.4372688639374895], [14.462311557788944, 1.3521002341462145], [14.557788944723617, 1.269797854320621], [14.65326633165829, 1.1911719256593272], [14.748743718592964, 1.1169987576095366], [14.844221105527637, 1.0480137018338045], [14.93969849246231, 0.9849044591539011], [15.035175879396984, 0.928304820441753], [15.130653266331658, 0.8787888984745216], [15.226130653266331, 0.8368659032985963], [15.321608040201005, 0.8029755086963482], [15.417085427135678, 0.7774838519650598], [15.512562814070352, 0.7606802034485112], [15.608040201005025, 0.7527743361608454], [15.703517587939698, 0.7538946194651062], [15.798994974874372, 0.7640868541733616], [15.894472361809045, 0.783313859681632], [15.989949748743719, 0.8114558169024777], [16.085427135678394, 0.8483113638734898], [16.180904522613066, 0.893599434063705], [16.276381909547737, 0.9469618206346968], [16.371859296482413, 1.0079664433002806], [16.467336683417084, 1.0761112880287422], [16.562814070351756, 1.1508289837025623], [16.65829145728643, 1.231491974048426], [16.753768844221106, 1.3174182377279955], [16.849246231155778, 1.4078775044866703], [16.94472361809045, 1.5020979107389012], [17.040201005025125, 1.5992730339657264], [17.1356783919598, 1.6985692418495955], [17.231155778894472, 1.7991332892045548], [17.326633165829143, 1.900100093502631], [17.42211055276382, 2.0006006181703757], [17.51758793969849, 2.0997697918478395], [17.613065326633166, 2.1967543914746663], [17.708542713567837, 2.290720817397309], [17.804020100502512, 2.3808626896751464], [17.899497487437184, 2.466408196391704], [17.99497487437186, 2.546627127036362], [18.09045226130653, 2.6208375268903916], [18.185929648241206, 2.6884119118036125], [18.281407035175878, 2.7487829867521323], [18.376884422110553, 2.80144881608786], [18.472361809045225, 2.84597739838484], [18.5678391959799, 2.8820106042110822], [18.66331658291457, 2.9092674409575667], [18.758793969849247, 2.927546614985987], [18.85427135678392, 2.936728367757447], [18.949748743718594, 2.9367755692176685], [19.045226130653266, 2.927734058479847], [19.14070351758794, 2.9097322287026977], [19.236180904522612, 2.8829798599458125], [19.331658291457288, 2.8477662106346022], [19.42713567839196, 2.804457385020459], [19.52261306532663, 2.7534930006166505], [19.618090452261306, 2.695382185967032], [19.713567839195978, 2.630698945204617], [19.809045226130653, 2.560076931624921], [19.904522613065325, 2.4842036778823475], [20.0, 2.403814335367383]]}, \"width\": 480.0, \"plugins\": [{\"type\": \"reset\"}, {\"type\": \"zoom\", \"enabled\": false, \"button\": true}, {\"type\": \"boxzoom\", \"enabled\": false, \"button\": true}], \"axes\": [{\"lines\": [{\"data\": \"data01\", \"xindex\": 0, \"yindex\": 1, \"color\": \"#0000FF\", \"linewidth\": 2.0, \"id\": \"el724304467241088\", \"zorder\": 2, \"alpha\": 1, \"dasharray\": \"none\", \"coordinates\": \"data\"}, {\"data\": \"data02\", \"xindex\": 0, \"yindex\": 1, \"color\": \"#000000\", \"linewidth\": 2.0, \"id\": \"el724304466813248\", \"zorder\": 2, \"alpha\": 1, \"dasharray\": \"none\", \"coordinates\": \"data\"}], \"id\": \"el724304466844056\", \"sharey\": [], \"markers\": [], \"bbox\": [0.125, 0.125, 0.775, 0.775], \"xlim\": [0.0, 20.0], \"images\": [], \"axes\": [{\"fontsize\": 10.0, \"grid\": {\"gridOn\": false}, \"nticks\": 5, \"position\": \"bottom\", \"scale\": \"linear\", \"tickformat\": null, \"tickvalues\": null}, {\"fontsize\": 10.0, \"grid\": {\"gridOn\": false}, \"nticks\": 9, \"position\": \"left\", \"scale\": \"linear\", \"tickformat\": null, \"tickvalues\": null}], \"texts\": [], \"xscale\": \"linear\", \"xdomain\": [0.0, 20.0], \"yscale\": \"linear\", \"collections\": [], \"ydomain\": [-1.0, 3.0], \"zoomable\": true, \"axesbg\": \"#FFFFFF\", \"ylim\": [-1.0, 3.0], \"sharex\": [], \"axesbgalpha\": null, \"paths\": []}], \"id\": \"el724304466813360\", \"height\": 320.0});\n",
       "   }(mpld3);\n",
       "}else if(typeof define === \"function\" && define.amd){\n",
       "   // require.js is available: use it to load d3/mpld3\n",
       "   require.config({paths: {d3: \"https://mpld3.github.io/js/d3.v3.min\"}});\n",
       "   require([\"d3\"], function(d3){\n",
       "      window.d3 = d3;\n",
       "      mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
       "         \n",
       "         mpld3.draw_figure(\"fig_el7243044668133601579537693\", {\"data\": {\"data02\": [[1.0, 0.0], [20.0, 0.0]], \"data01\": [[1.0, -0.45969769413186023], [1.0954773869346734, -0.4511877071257602], [1.1909547738693467, -0.4544713630264825], [1.2864321608040201, -0.4675801712583847], [1.3819095477386933, -0.4887681365597666], [1.4773869346733668, -0.5164514437364028], [1.5728643216080402, -0.5491696275234845], [1.6683417085427137, -0.5855606187667927], [1.763819095477387, -0.6243450005491862], [1.8592964824120601, -0.6643165064063793], [1.9547738693467336, -0.7043368097498681], [2.050251256281407, -0.7433332825100065], [2.1457286432160805, -0.7802988013955392], [2.241206030150754, -0.8142929419805369], [2.3366834170854274, -0.8444440764128797], [2.4321608040201004, -0.8699520112182656], [2.527638190954774, -0.8900908866815207], [2.6231155778894473, -0.9042121206973023], [2.7185929648241203, -0.9117472255298995], [2.8140703517587937, -0.9122103606504227], [2.909547738693467, -0.9052005120694369], [3.0050251256281406, -0.8904032105955993], [3.100502512562814, -0.8675917197465078], [3.1959798994974875, -0.8366276396564397], [3.291457286432161, -0.7974608869812029], [3.386934673366834, -0.7501290229841818], [3.4824120603015074, -0.6947559130363244], [3.577889447236181, -0.6315497109043362], [3.6733668341708543, -0.5608001705871944], [3.7688442211055277, -0.4828752971884316], [3.8643216080402008, -0.3982173564388708], [3.959798994974874, -0.3073382700427204], [4.055276381909548, -0.2108144310209763], [4.150753768844221, -0.10928097966853612], [4.2462311557788945, -0.0034255866152690384], [4.341708542713567, 0.10601820522892802], [4.4371859296482405, 0.21827802337742774], [4.532663316582914, 0.3325500325841617], [4.628140703517587, 0.448006561812875], [4.723618090452261, 0.5638039246234161], [4.819095477386934, 0.6790903641350983], [4.914572864321608, 0.793014052298993], [5.010050251256281, 0.9047310724646564], [5.105527638190955, 1.0134133141622685], [5.201005025125628, 1.1182562096336461], [5.296482412060302, 1.2184862429248797], [5.391959798994975, 1.3133681642840935], [5.4874371859296485, 1.4022118451692336], [5.582914572864322, 1.4843787123369756], [5.6783919597989945, 1.5592877032237147], [5.773869346733668, 1.6264206891072615], [5.869346733668341, 1.6853273173127938], [5.964824120603015, 1.7356292289539867], [6.060301507537688, 1.7770236143312852], [6.155778894472362, 1.8092860740918235], [6.251256281407035, 1.8322727605341091], [6.346733668341709, 1.8459217799574934], [6.442211055276382, 1.8502538436515756], [6.5376884422110555, 1.8453721619326173], [6.633165829145728, 1.8314615825002099], [6.7286432160804015, 1.808786981244931], [6.824120603015075, 1.7776909204237707], [6.919597989949748, 1.7385905957725338], [7.015075376884422, 1.691974100582435], [7.110552763819095, 1.6383960409726157], [7.206030150753769, 1.578472542484623], [7.301507537688442, 1.5128756936551335], [7.396984924623116, 1.4423274773388526], [7.492462311557789, 1.367593245207833], [7.5879396984924625, 1.2894747950039496], [7.683417085427135, 1.2088031137299762], [7.7788944723618085, 1.126430852998714], [7.874371859296482, 1.043224605191217], [7.969849246231155, 0.960057050882023], [8.065326633165828, 0.8777990491550531], [8.160804020100503, 0.7973117429476981], [8.256281407035175, 0.7194387514178788], [8.35175879396985, 0.6449985205307536], [8.447236180904522, 0.5747769016156552], [8.542713567839197, 0.509520025562531], [8.638190954773869, 0.44992753763011084], [8.733668341708542, 0.39664625454912095], [8.829145728643216, 0.35026430175369905], [8.924623115577889, 0.31130578419719734], [9.020100502512562, 0.2802260393450693], [9.115577889447236, 0.25740751563157227], [9.21105527638191, 0.24315631396695792], [9.306532663316583, 0.2376994238395851], [9.402010050251256, 0.24118267922799275], [9.49748743718593, 0.25366945297902266], [9.592964824120603, 0.2751401015793733], [9.688442211055277, 0.30549216541071544], [9.78391959798995, 0.3445413226950351], [9.879396984924623, 0.392023088469835], [9.974874371859297, 0.4475952431447914], [10.07035175879397, 0.5108409685442559], [10.165829145728644, 0.5812726628941278], [10.261306532663315, 0.6583364000258374], [10.356783919597989, 0.7414169922008202], [10.452261306532662, 0.8298436104593185], [10.547738693467336, 0.9228959113178368], [10.64321608040201, 1.0198106140262282], [10.738693467336683, 1.1197884684902921], [10.834170854271356, 1.2220015504063388], [10.92964824120603, 1.325600817172735], [11.025125628140703, 1.4297238557672736], [11.120603015075377, 1.5335027520298348], [11.21608040201005, 1.6360720096832342], [11.311557788944723, 1.7365764469714309], [11.407035175879397, 1.8341789989975403], [11.50251256281407, 1.9280683547024577], [11.597989949748744, 2.01746635893066], [11.693467336683417, 2.101635112168985], [11.78894472361809, 2.1798837032976426], [11.884422110552764, 2.2515745140351022], [11.979899497487438, 2.31612903765952], [12.075376884422111, 2.3730331590133824], [12.170854271356784, 2.4218418477048393], [12.266331658291456, 2.4621832217639965], [12.36180904522613, 2.4937619447466175], [12.457286432160803, 2.516361925348964], [12.552763819095476, 2.5298482949506234], [12.64824120603015, 2.534168645079199], [12.743718592964823, 2.529353513531843], [12.839195979899497, 2.5155161147323053], [12.93467336683417, 2.492851316786182], [13.030150753768844, 2.4616338745586024], [13.125628140703517, 2.422215934875278], [13.22110552763819, 2.3750238365778156], [13.316582914572864, 2.320554234587157], [13.412060301507537, 2.259369583286425], [13.507537688442211, 2.1920930203702103], [13.603015075376884, 2.1194026977682863], [13.698492462311558, 2.042025611288186], [13.793969849246231, 1.9607309851870456], [13.889447236180905, 1.8763232719371281], [13.984924623115578, 1.7896348309544594], [14.080402010050252, 1.7015183529842108], [14.175879396984925, 1.6128390991531067], [14.271356783919597, 1.5244670253872088], [14.36683417085427, 1.4372688639374895], [14.462311557788944, 1.3521002341462145], [14.557788944723617, 1.269797854320621], [14.65326633165829, 1.1911719256593272], [14.748743718592964, 1.1169987576095366], [14.844221105527637, 1.0480137018338045], [14.93969849246231, 0.9849044591539011], [15.035175879396984, 0.928304820441753], [15.130653266331658, 0.8787888984745216], [15.226130653266331, 0.8368659032985963], [15.321608040201005, 0.8029755086963482], [15.417085427135678, 0.7774838519650598], [15.512562814070352, 0.7606802034485112], [15.608040201005025, 0.7527743361608454], [15.703517587939698, 0.7538946194651062], [15.798994974874372, 0.7640868541733616], [15.894472361809045, 0.783313859681632], [15.989949748743719, 0.8114558169024777], [16.085427135678394, 0.8483113638734898], [16.180904522613066, 0.893599434063705], [16.276381909547737, 0.9469618206346968], [16.371859296482413, 1.0079664433002806], [16.467336683417084, 1.0761112880287422], [16.562814070351756, 1.1508289837025623], [16.65829145728643, 1.231491974048426], [16.753768844221106, 1.3174182377279955], [16.849246231155778, 1.4078775044866703], [16.94472361809045, 1.5020979107389012], [17.040201005025125, 1.5992730339657264], [17.1356783919598, 1.6985692418495955], [17.231155778894472, 1.7991332892045548], [17.326633165829143, 1.900100093502631], [17.42211055276382, 2.0006006181703757], [17.51758793969849, 2.0997697918478395], [17.613065326633166, 2.1967543914746663], [17.708542713567837, 2.290720817397309], [17.804020100502512, 2.3808626896751464], [17.899497487437184, 2.466408196391704], [17.99497487437186, 2.546627127036362], [18.09045226130653, 2.6208375268903916], [18.185929648241206, 2.6884119118036125], [18.281407035175878, 2.7487829867521323], [18.376884422110553, 2.80144881608786], [18.472361809045225, 2.84597739838484], [18.5678391959799, 2.8820106042110822], [18.66331658291457, 2.9092674409575667], [18.758793969849247, 2.927546614985987], [18.85427135678392, 2.936728367757447], [18.949748743718594, 2.9367755692176685], [19.045226130653266, 2.927734058479847], [19.14070351758794, 2.9097322287026977], [19.236180904522612, 2.8829798599458125], [19.331658291457288, 2.8477662106346022], [19.42713567839196, 2.804457385020459], [19.52261306532663, 2.7534930006166505], [19.618090452261306, 2.695382185967032], [19.713567839195978, 2.630698945204617], [19.809045226130653, 2.560076931624921], [19.904522613065325, 2.4842036778823475], [20.0, 2.403814335367383]]}, \"width\": 480.0, \"plugins\": [{\"type\": \"reset\"}, {\"type\": \"zoom\", \"enabled\": false, \"button\": true}, {\"type\": \"boxzoom\", \"enabled\": false, \"button\": true}], \"axes\": [{\"lines\": [{\"data\": \"data01\", \"xindex\": 0, \"yindex\": 1, \"color\": \"#0000FF\", \"linewidth\": 2.0, \"id\": \"el724304467241088\", \"zorder\": 2, \"alpha\": 1, \"dasharray\": \"none\", \"coordinates\": \"data\"}, {\"data\": \"data02\", \"xindex\": 0, \"yindex\": 1, \"color\": \"#000000\", \"linewidth\": 2.0, \"id\": \"el724304466813248\", \"zorder\": 2, \"alpha\": 1, \"dasharray\": \"none\", \"coordinates\": \"data\"}], \"id\": \"el724304466844056\", \"sharey\": [], \"markers\": [], \"bbox\": [0.125, 0.125, 0.775, 0.775], \"xlim\": [0.0, 20.0], \"images\": [], \"axes\": [{\"fontsize\": 10.0, \"grid\": {\"gridOn\": false}, \"nticks\": 5, \"position\": \"bottom\", \"scale\": \"linear\", \"tickformat\": null, \"tickvalues\": null}, {\"fontsize\": 10.0, \"grid\": {\"gridOn\": false}, \"nticks\": 9, \"position\": \"left\", \"scale\": \"linear\", \"tickformat\": null, \"tickvalues\": null}], \"texts\": [], \"xscale\": \"linear\", \"xdomain\": [0.0, 20.0], \"yscale\": \"linear\", \"collections\": [], \"ydomain\": [-1.0, 3.0], \"zoomable\": true, \"axesbg\": \"#FFFFFF\", \"ylim\": [-1.0, 3.0], \"sharex\": [], \"axesbgalpha\": null, \"paths\": []}], \"id\": \"el724304466813360\", \"height\": 320.0});\n",
       "      });\n",
       "    });\n",
       "}else{\n",
       "    // require.js not available: dynamically load d3 & mpld3\n",
       "    mpld3_load_lib(\"https://mpld3.github.io/js/d3.v3.min.js\", function(){\n",
       "         mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
       "                 \n",
       "                 mpld3.draw_figure(\"fig_el7243044668133601579537693\", {\"data\": {\"data02\": [[1.0, 0.0], [20.0, 0.0]], \"data01\": [[1.0, -0.45969769413186023], [1.0954773869346734, -0.4511877071257602], [1.1909547738693467, -0.4544713630264825], [1.2864321608040201, -0.4675801712583847], [1.3819095477386933, -0.4887681365597666], [1.4773869346733668, -0.5164514437364028], [1.5728643216080402, -0.5491696275234845], [1.6683417085427137, -0.5855606187667927], [1.763819095477387, -0.6243450005491862], [1.8592964824120601, -0.6643165064063793], [1.9547738693467336, -0.7043368097498681], [2.050251256281407, -0.7433332825100065], [2.1457286432160805, -0.7802988013955392], [2.241206030150754, -0.8142929419805369], [2.3366834170854274, -0.8444440764128797], [2.4321608040201004, -0.8699520112182656], [2.527638190954774, -0.8900908866815207], [2.6231155778894473, -0.9042121206973023], [2.7185929648241203, -0.9117472255298995], [2.8140703517587937, -0.9122103606504227], [2.909547738693467, -0.9052005120694369], [3.0050251256281406, -0.8904032105955993], [3.100502512562814, -0.8675917197465078], [3.1959798994974875, -0.8366276396564397], [3.291457286432161, -0.7974608869812029], [3.386934673366834, -0.7501290229841818], [3.4824120603015074, -0.6947559130363244], [3.577889447236181, -0.6315497109043362], [3.6733668341708543, -0.5608001705871944], [3.7688442211055277, -0.4828752971884316], [3.8643216080402008, -0.3982173564388708], [3.959798994974874, -0.3073382700427204], [4.055276381909548, -0.2108144310209763], [4.150753768844221, -0.10928097966853612], [4.2462311557788945, -0.0034255866152690384], [4.341708542713567, 0.10601820522892802], [4.4371859296482405, 0.21827802337742774], [4.532663316582914, 0.3325500325841617], [4.628140703517587, 0.448006561812875], [4.723618090452261, 0.5638039246234161], [4.819095477386934, 0.6790903641350983], [4.914572864321608, 0.793014052298993], [5.010050251256281, 0.9047310724646564], [5.105527638190955, 1.0134133141622685], [5.201005025125628, 1.1182562096336461], [5.296482412060302, 1.2184862429248797], [5.391959798994975, 1.3133681642840935], [5.4874371859296485, 1.4022118451692336], [5.582914572864322, 1.4843787123369756], [5.6783919597989945, 1.5592877032237147], [5.773869346733668, 1.6264206891072615], [5.869346733668341, 1.6853273173127938], [5.964824120603015, 1.7356292289539867], [6.060301507537688, 1.7770236143312852], [6.155778894472362, 1.8092860740918235], [6.251256281407035, 1.8322727605341091], [6.346733668341709, 1.8459217799574934], [6.442211055276382, 1.8502538436515756], [6.5376884422110555, 1.8453721619326173], [6.633165829145728, 1.8314615825002099], [6.7286432160804015, 1.808786981244931], [6.824120603015075, 1.7776909204237707], [6.919597989949748, 1.7385905957725338], [7.015075376884422, 1.691974100582435], [7.110552763819095, 1.6383960409726157], [7.206030150753769, 1.578472542484623], [7.301507537688442, 1.5128756936551335], [7.396984924623116, 1.4423274773388526], [7.492462311557789, 1.367593245207833], [7.5879396984924625, 1.2894747950039496], [7.683417085427135, 1.2088031137299762], [7.7788944723618085, 1.126430852998714], [7.874371859296482, 1.043224605191217], [7.969849246231155, 0.960057050882023], [8.065326633165828, 0.8777990491550531], [8.160804020100503, 0.7973117429476981], [8.256281407035175, 0.7194387514178788], [8.35175879396985, 0.6449985205307536], [8.447236180904522, 0.5747769016156552], [8.542713567839197, 0.509520025562531], [8.638190954773869, 0.44992753763011084], [8.733668341708542, 0.39664625454912095], [8.829145728643216, 0.35026430175369905], [8.924623115577889, 0.31130578419719734], [9.020100502512562, 0.2802260393450693], [9.115577889447236, 0.25740751563157227], [9.21105527638191, 0.24315631396695792], [9.306532663316583, 0.2376994238395851], [9.402010050251256, 0.24118267922799275], [9.49748743718593, 0.25366945297902266], [9.592964824120603, 0.2751401015793733], [9.688442211055277, 0.30549216541071544], [9.78391959798995, 0.3445413226950351], [9.879396984924623, 0.392023088469835], [9.974874371859297, 0.4475952431447914], [10.07035175879397, 0.5108409685442559], [10.165829145728644, 0.5812726628941278], [10.261306532663315, 0.6583364000258374], [10.356783919597989, 0.7414169922008202], [10.452261306532662, 0.8298436104593185], [10.547738693467336, 0.9228959113178368], [10.64321608040201, 1.0198106140262282], [10.738693467336683, 1.1197884684902921], [10.834170854271356, 1.2220015504063388], [10.92964824120603, 1.325600817172735], [11.025125628140703, 1.4297238557672736], [11.120603015075377, 1.5335027520298348], [11.21608040201005, 1.6360720096832342], [11.311557788944723, 1.7365764469714309], [11.407035175879397, 1.8341789989975403], [11.50251256281407, 1.9280683547024577], [11.597989949748744, 2.01746635893066], [11.693467336683417, 2.101635112168985], [11.78894472361809, 2.1798837032976426], [11.884422110552764, 2.2515745140351022], [11.979899497487438, 2.31612903765952], [12.075376884422111, 2.3730331590133824], [12.170854271356784, 2.4218418477048393], [12.266331658291456, 2.4621832217639965], [12.36180904522613, 2.4937619447466175], [12.457286432160803, 2.516361925348964], [12.552763819095476, 2.5298482949506234], [12.64824120603015, 2.534168645079199], [12.743718592964823, 2.529353513531843], [12.839195979899497, 2.5155161147323053], [12.93467336683417, 2.492851316786182], [13.030150753768844, 2.4616338745586024], [13.125628140703517, 2.422215934875278], [13.22110552763819, 2.3750238365778156], [13.316582914572864, 2.320554234587157], [13.412060301507537, 2.259369583286425], [13.507537688442211, 2.1920930203702103], [13.603015075376884, 2.1194026977682863], [13.698492462311558, 2.042025611288186], [13.793969849246231, 1.9607309851870456], [13.889447236180905, 1.8763232719371281], [13.984924623115578, 1.7896348309544594], [14.080402010050252, 1.7015183529842108], [14.175879396984925, 1.6128390991531067], [14.271356783919597, 1.5244670253872088], [14.36683417085427, 1.4372688639374895], [14.462311557788944, 1.3521002341462145], [14.557788944723617, 1.269797854320621], [14.65326633165829, 1.1911719256593272], [14.748743718592964, 1.1169987576095366], [14.844221105527637, 1.0480137018338045], [14.93969849246231, 0.9849044591539011], [15.035175879396984, 0.928304820441753], [15.130653266331658, 0.8787888984745216], [15.226130653266331, 0.8368659032985963], [15.321608040201005, 0.8029755086963482], [15.417085427135678, 0.7774838519650598], [15.512562814070352, 0.7606802034485112], [15.608040201005025, 0.7527743361608454], [15.703517587939698, 0.7538946194651062], [15.798994974874372, 0.7640868541733616], [15.894472361809045, 0.783313859681632], [15.989949748743719, 0.8114558169024777], [16.085427135678394, 0.8483113638734898], [16.180904522613066, 0.893599434063705], [16.276381909547737, 0.9469618206346968], [16.371859296482413, 1.0079664433002806], [16.467336683417084, 1.0761112880287422], [16.562814070351756, 1.1508289837025623], [16.65829145728643, 1.231491974048426], [16.753768844221106, 1.3174182377279955], [16.849246231155778, 1.4078775044866703], [16.94472361809045, 1.5020979107389012], [17.040201005025125, 1.5992730339657264], [17.1356783919598, 1.6985692418495955], [17.231155778894472, 1.7991332892045548], [17.326633165829143, 1.900100093502631], [17.42211055276382, 2.0006006181703757], [17.51758793969849, 2.0997697918478395], [17.613065326633166, 2.1967543914746663], [17.708542713567837, 2.290720817397309], [17.804020100502512, 2.3808626896751464], [17.899497487437184, 2.466408196391704], [17.99497487437186, 2.546627127036362], [18.09045226130653, 2.6208375268903916], [18.185929648241206, 2.6884119118036125], [18.281407035175878, 2.7487829867521323], [18.376884422110553, 2.80144881608786], [18.472361809045225, 2.84597739838484], [18.5678391959799, 2.8820106042110822], [18.66331658291457, 2.9092674409575667], [18.758793969849247, 2.927546614985987], [18.85427135678392, 2.936728367757447], [18.949748743718594, 2.9367755692176685], [19.045226130653266, 2.927734058479847], [19.14070351758794, 2.9097322287026977], [19.236180904522612, 2.8829798599458125], [19.331658291457288, 2.8477662106346022], [19.42713567839196, 2.804457385020459], [19.52261306532663, 2.7534930006166505], [19.618090452261306, 2.695382185967032], [19.713567839195978, 2.630698945204617], [19.809045226130653, 2.560076931624921], [19.904522613065325, 2.4842036778823475], [20.0, 2.403814335367383]]}, \"width\": 480.0, \"plugins\": [{\"type\": \"reset\"}, {\"type\": \"zoom\", \"enabled\": false, \"button\": true}, {\"type\": \"boxzoom\", \"enabled\": false, \"button\": true}], \"axes\": [{\"lines\": [{\"data\": \"data01\", \"xindex\": 0, \"yindex\": 1, \"color\": \"#0000FF\", \"linewidth\": 2.0, \"id\": \"el724304467241088\", \"zorder\": 2, \"alpha\": 1, \"dasharray\": \"none\", \"coordinates\": \"data\"}, {\"data\": \"data02\", \"xindex\": 0, \"yindex\": 1, \"color\": \"#000000\", \"linewidth\": 2.0, \"id\": \"el724304466813248\", \"zorder\": 2, \"alpha\": 1, \"dasharray\": \"none\", \"coordinates\": \"data\"}], \"id\": \"el724304466844056\", \"sharey\": [], \"markers\": [], \"bbox\": [0.125, 0.125, 0.775, 0.775], \"xlim\": [0.0, 20.0], \"images\": [], \"axes\": [{\"fontsize\": 10.0, \"grid\": {\"gridOn\": false}, \"nticks\": 5, \"position\": \"bottom\", \"scale\": \"linear\", \"tickformat\": null, \"tickvalues\": null}, {\"fontsize\": 10.0, \"grid\": {\"gridOn\": false}, \"nticks\": 9, \"position\": \"left\", \"scale\": \"linear\", \"tickformat\": null, \"tickvalues\": null}], \"texts\": [], \"xscale\": \"linear\", \"xdomain\": [0.0, 20.0], \"yscale\": \"linear\", \"collections\": [], \"ydomain\": [-1.0, 3.0], \"zoomable\": true, \"axesbg\": \"#FFFFFF\", \"ylim\": [-1.0, 3.0], \"sharex\": [], \"axesbgalpha\": null, \"paths\": []}], \"id\": \"el724304466813360\", \"height\": 320.0});\n",
       "            })\n",
       "         });\n",
       "}\n",
       "</script>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = linspace(1,20,200)\n",
    "y = f(x)\n",
    "plot(x,y, lw=2)\n",
    "plot([1,20],[0,0], lw=2, color='k')\n",
    "ax = gca()\n",
    "mpld3.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vi ser en løsning nær $4$, så vi velger $x_0=4$.\n",
    "Vi deriverer og finner at $f'(x)=1/x-\\sin(x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fder(x):\n",
    "    return 1/x - sin(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vi kan beregne det en om en:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.26554290538\n",
      "4.24930879016\n",
      "4.24926425674\n"
     ]
    }
   ],
   "source": [
    "x0=4\n",
    "x1=x0- f(x0)/fder(x0)\n",
    "print(x1)\n",
    "x2=x1-f(x1)/fder(x1)\n",
    "print(x2)\n",
    "x3=x2-f(x2)/fder(x2)\n",
    "print(x3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Når stopper vi? Vi kan sjekke for `x3`: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.8805558766341619e-10"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Liten nok? Da kan vi stoppe!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ellers kan Python vurdere selv når vi kan stoppe, f.eks. når $|f(x_n)|$ er liten... "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.24930879016\n"
     ]
    }
   ],
   "source": [
    "x=4\n",
    "for i in range(100):\n",
    "    x = x-f(x)/fder(x)\n",
    "    if abs(f(x)) < 0.001:\n",
    "        break\n",
    "print(x)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}