{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fyrirlestur 2\n",
    "### 19. janúar\n",
    "\n",
    "Python uppsetning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import math\n",
    "from math import * # ósmekklegt en einfalt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hlutfallsleg villa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6.789e-16, 6.661338147750939e-16, 0.01880422039314497)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 1.1234567890123456789\n",
    "y = 1.123456789012345\n",
    "z = 6.789e-16 # rétt svar\n",
    "#z = 1.234567e-11\n",
    "z,x-y,(z-(x-y))/z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. stigs jafna\n",
    "\n",
    "Leysum jöfnuna $ax^2+bx+c = 0$\n",
    "með a=1,b=1e8,c=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-100000000.0\n",
      "-7.450580596923828e-09\n",
      "100000000.0\n",
      "99999999.99999999\n",
      "-1.4901161193847656e-08\n"
     ]
    }
   ],
   "source": [
    "a,b,c = 1,1e8,1\n",
    "d=b*b-4*a*c\n",
    "x1 = (-b-sqrt(d))/(2*a)\n",
    "x2 = (-b+sqrt(d))/(2*a)\n",
    "\n",
    "print(x1)\n",
    "print(x2)\n",
    "print(b)\n",
    "print(sqrt(d))\n",
    "print(-b+sqrt(d))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stingum aftur inn jöfnuna og sjáum hvað kemur út"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Lausn 1: 1.00000000\n",
      "Lausn 1 (önnur röð): 0.00000000\n",
      "Lausn 2: 0.25494194\n"
     ]
    }
   ],
   "source": [
    "print(\"Lausn 1: %.8f\"%(a*x1**2 + b*x1 + c))\n",
    "print(\"Lausn 1 (önnur röð): %.8f\"%(c + a*x1**2 + b*x1))\n",
    "print(\"Lausn 2: %.8f\"%(a*x2**2 + b*x2 + c))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notum hina formúluna"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Lausn 2, betri formúla: 0.00000000\n",
      "-7.450580596923828e-09\n",
      "-1e-08\n"
     ]
    }
   ],
   "source": [
    "x3 = -2*c/(sqrt(d)+b)\n",
    "print(\"Lausn 2, betri formúla: %.8f\"%(a*x3**2 + b*x3 + c))\n",
    "print(x2)\n",
    "print(x3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hversu rangt var hitt svarið?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2549419403076172"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x3-x2)/x3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Annað dæmi\n",
    "\n",
    "$f(x) = \\sqrt{x+1}- \\sqrt{x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sqrt(x+1)-np.sqrt(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(1e16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f2(x):\n",
    "    return 1.0/(np.sqrt(x+1)+np.sqrt(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hversu vitlaust er svarið?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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fMtYF/r316uoXERHJlKwF/mnVLvDXNSrjFxERyZTsXc5XVgHAgWYFfhERkUzJ2t358vPy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\nwS9JUo/8BbG+3r8ivcImAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1088cf7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(1e14,1e16,1e14)\n",
    "\n",
    "plt.plot(x, f(x))\n",
    "plt.plot(x, f2(x))\n",
    "plt.title('$f(x)$ vs. $f_2(x)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hlutfallsleg villa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108bea4a8>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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XNuIiXUOjiNmzgb//e+CLX9SJn6tWAd//PvD61wN33qnLlBuxccop2tl45BH3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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108a9aac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,(f2(x)-f(x))/f2(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Enn annað dæmi\n",
    "\n",
    "Nú er $h(x) = \\log(1+x) \\approx x - x^2$, þegar $x$ er nálægt 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108e1fac8>]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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X4Uc/8s/bYtWLQlbaIQyzgpMop0v6vwLbD5X0ZJ3bXlUKkyh33bV0gJg61coe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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107f6a7b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def h(x):\n",
    "    return np.log(1+x)\n",
    "\n",
    "def h2(x):\n",
    "    return x-x**2\n",
    "\n",
    "x = np.arange(1e-17,5e-15,1e-17)\n",
    "\n",
    "# np.log1p er fall sem er sérhannað til að reikna út log(1+x) fyrir lítil gildi á x\n",
    "plt.plot(x, (np.log1p(x)-h(x))/np.log1p(x))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ef við notum Taylor nálgunina (athugið skalann á y-ás)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108f38a90>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HgqYqBCKdVVv1ITBgtpl9YWbPmNmubfS5ItLFLFgAN98MhxwC3/42rFnT+P7L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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108ade2e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, (np.log1p(x)-h2(x))/np.log1p(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Summur"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000000004e+16"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = [1e16,1,2,3]\n",
    "sum(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000000006e+16"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(t[::-1]) # t[::-1] er listinn aftur á bak"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t2 = [1e16,1,2,3,-1e16]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4.0, 8.0)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(t2), sum(t2[::-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kahan summa\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def kahan_sum(l):\n",
    "  sum=0.0\n",
    "  y,t,c = 0.0, 0.0, 0.0\n",
    "  for x in l:\n",
    "    y = x-c\n",
    "    t = sum + y\n",
    "    c = (t-sum)-y\n",
    "    sum = t\n",
    "  return sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1e+16, 1, 2, 3]\t10000000000000004.0\t10000000000000006.0\n",
      "[3, 2, 1, 1e+16]\t10000000000000006.0\t10000000000000006.0\n",
      "[1e+16, 1, 2, 3, -1e+16]\t4.0\t6.0\n",
      "[-1e+16, 3, 2, 1, 1e+16]\t8.0\t6.0\n"
     ]
    }
   ],
   "source": [
    "for x in t,t[::-1],t2,t2[::-1]:\n",
    "    print(\"%s\\t%.1f\\t%.1f\"%(x,sum(x),kahan_sum(x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Staðalfrávik\n",
    "\n",
    "Listarnir $l$ og $l2$ hafa sama staðalfrávik en mismunandi meðaltöl.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "l = [10000000000.0, 10000000001.0, 10000000002.0, 10000000003.0, 10000000004.0, 10000000005.0, 10000000006.0, 10000000007.0, 10000000008.0, 10000000009.0]\n",
      "l2 = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
      "bad(l)   = 0.000000\n",
      "good(l) = 2.872281\n",
      "bad(l2)   = 2.872281\n",
      "good(l2) = 2.872281\n"
     ]
    }
   ],
   "source": [
    "def bad_stdev(l):\n",
    "    n = len(l)\n",
    "    x2=sum(x**2 for x in l)\n",
    "    return sqrt((x2 - sum(l)**2/n)/n)\n",
    "\n",
    "def good_stdev(l):\n",
    "    n = len(l)\n",
    "    mean = sum(l)/n\n",
    "    return sqrt(sum((x-mean)**2 for x in l)/n)\n",
    "\n",
    "l = [1e10+x for x in range(10)]\n",
    "l2 = list(range(10))\n",
    "print(\"l = %s\"%(l))\n",
    "print(\"l2 = %s\"%(l2))\n",
    "print(\"bad(l)   = %.6f\"%(bad_stdev(l)))\n",
    "print(\"good(l) = %.6f\"%(good_stdev(l)))\n",
    "print(\"bad(l2)   = %.6f\"%(bad_stdev(l2)))\n",
    "print(\"good(l2) = %.6f\"%(good_stdev(l2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Taylor raðir"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nálgun á $\\sin$ á bilinu $[-\\frac{\\pi}{4},\\frac{\\pi}{4}]$ með Taylor margliðum\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x109071eb8>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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BgsLy5fDkk3lXIknSZuvWpUUBDQo5c0CjJKkaPfQQrF9f/TMeoM6Dwj77wF572f0gSaou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\nFCRJUkkGBUmSVJJBQZIklWRQkCRJJf1/JpMgw4CIhw4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108f57978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(-np.pi/4,np.pi/4,1e-2)\n",
    "#x = np.arange(-math.pi,math.pi,1e-2)\n",
    "#plt.plot(x,np.sin(x)-(x- x**3/6))\n",
    "#plt.plot(x,np.sin(x))\n",
    "#plt.plot(x,(x-x**3/6))\n",
    "#plt.plot(x,(x-x**3/6 + x**5/120))\n",
    "#plt.plot(x,np.sin(x) - (x-x**3/6+x**5/120-x**7/(7*6*120)))\n",
    "plt.plot(x,(np.sin(x) - (x-x**3/6+x**5/120-x**7/(7*6*120)))/np.sin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hversu marga liði þarf til að nálga $\\sin$ með 15 aukastöfum?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n = 0, R_(n+1) = 8e-01f\n",
      "n = 1, R_(n+1) = 3e-01f\n",
      "n = 2, R_(n+1) = 8e-02f\n",
      "n = 3, R_(n+1) = 2e-02f\n",
      "n = 4, R_(n+1) = 2e-03f\n",
      "n = 5, R_(n+1) = 3e-04f\n",
      "n = 6, R_(n+1) = 4e-05f\n",
      "n = 7, R_(n+1) = 4e-06f\n",
      "n = 8, R_(n+1) = 3e-07f\n",
      "n = 9, R_(n+1) = 2e-08f\n",
      "n = 10, R_(n+1) = 2e-09f\n",
      "n = 11, R_(n+1) = 1e-10f\n",
      "n = 12, R_(n+1) = 7e-12f\n",
      "n = 13, R_(n+1) = 4e-13f\n"
     ]
    }
   ],
   "source": [
    "def fact(n):\n",
    "    r = 1.0\n",
    "    for x in range(1,n+1):\n",
    "        r *= x\n",
    "    return r\n",
    "\n",
    "for n in range(14):\n",
    "    R = (math.pi/4)**(n+1)/fact(n+1)\n",
    "    print(\"n = %d, R_(n+1) = %.ef\"%(n,R))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nálgun á $ln(x)$ á bilinu $[\\frac{\\sqrt{2}}{2},\\sqrt{2}]$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x109198748>]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ERUREctYmm8BRR8HEiaH3W/KHEhQREclpP/85vP8+PPlk3JFIOilBERGRnLbL\nLjBwoFaWzTdKUEREJKeZhSnH06bBRx/FHY2kixIUERHJeaNGQefOMGlS3JFIuihBERGRnLfRRnDM\nMTB5MlRXxx2NpIMSFBERyQtjx8J//xt2oZfcpwRFRETywuDB0L9/6EWR3KcERURE8oJZ6EV5+GFY\ntizuaKSllKCIiEjeGD0a1q2D++6LOxJpKSUoIiKSNzbbDPbdF26/Pe5IpKWUoIiISF4ZMwZmz4b5\n8+OORFpCCYqIiOSVkSNh443Vi5LrlKCIiEhe6dAhrIly551hPIrkJiUoIiKSd8aMgcpKmD497kik\nuZSgiIhI3hk0CHbYQY95cpkSFBERyTtmoRflb3+DL76IOxppDiUoIiKSl445BmpqoKIi7kikOZSg\niIhIXurZEw44QEvf5yolKCIikrfGjoW5c+GNN+KORFKlBEVERPLW/vvDppuGKceSW5SgiIhI3mrX\nDo48Eu6/P4xHkdyhBEVERPJaWRl8/DG8+GLckUgqMpqgmFl3M7vHzKrMbKmZ3WpmnRup08HMJpjZ\nYjNbYWZTzKxnUpktzewxM1tpZpVmdrWZFSVc39XMXozu8ZWZzTezMzPVThERyV5Dh8J3vgP33ht3\nJJKKTPeg3Av0B0qBA4BhwMRG6lwXlT08Kt8HeLD2YpSIPA60BXYBjgPGAJck3GMlcAOwO9APuBS4\nzMxOaGmDREQktxQVwdFHw5QpsHZt3NFIU2UsQTGzfsAIYJy7v+buM4HTgKPNrHc9dboAxwPl7v6c\nu88DxgK7mtmQqNgIQtJxjLu/6e7TgQuAU82sLYC7/9Pd73f3+e7+kbvfC0wnJCwiIlJgysrCgm1P\nPRV3JNJUmexBGQosjZKMWjMAB3aup04JoWfk6doT7v4O8FF0Pwi9Jm+6++KEetOBrsCAum5qZoOi\n+s+m3AoREcl5O+wA/ftr0bZckskEpTewKPGEu1cDS6Jr9dVZ4+7Lk85/llCnd/Q6+Tok3dfMFprZ\nKuAVYIK7a7keEZECZBZ6UaZOha++ijsaaYq2qVYwsyuAcxoo4oRxJ/XeIiqT0ts2sU5ymd2ADQm9\nLleZ2b/d/f6GblBeXk7Xrl3XO1dWVkZZWVkK4YqISLYpK4MLL4THHoOf/CTuaHJXRUUFFUldUVVV\nVWl/n5QTFOAaoLGeiPeASiB59k0boDvf7gGpVQm0N7MuSb0oPRPqVAKDk+r1ij6ud193/zD69K1o\n3MvFQIPMRL7LAAAYyUlEQVQJyvjx4ykuLm6oiIiI5KBttoHBg8NsHiUozVfXH+1z586lpKQkre+T\n8iMed//C3d9t5FgHzAK6ReM/apUSekNm13P7OcC6qBwAZtYX2AqYGZ2aBfzQzHok1NsHqALebiD0\nNkCHFJoqIiJ5pqwMHn8cli2LOxJpTMbGoLj7AsLg1UlmNtjMdiVM/a1w90oAM+sTrVGyU1RnOXAb\ncK2Z7WlmJYTempfc/dXo1k8SEpG7zGwHMxtBmEZ8o7uvje57ipkdaGbbRMc44JfAXZlqr4iIZL+j\njgpTjR9+OO5IpDGZXgdlFLCAMHtnGvA8cFLC9XZAX6BTwrnyqOwUwqybTwhrogDg7jXAgUA1oVfl\nTuB24KKEexQBVwDzgFeBk4Ffu3tiGRERKTB9+sAee2g2Ty5ozhiUJnP3ZcDoBq5/SHj0knhuNWG9\nlNMaqLeQkKTUd/1G4MZU4xURkfxXVgYnnwyffQa9ejVeXuKhvXhERKSgHH54WF32r3+NOxJpiBIU\nEREpKJtsAiNGaG+ebKcERURECs6oUTBrFnzwQdyRSH2UoIiISME56CDo2DFsICjZSQmKiIgUnA03\nhH33hQcfjDsSqY8SFBERKUiHHQYvvwwffxx3JFIXJSgiIlKQRo6Edu3CBoKSfZSgiIhIQerWDUpL\n9ZgnWylBERGRgnXYYfD88/D553FHIsmUoIiISME65JDw8ZFH4o1Dvk0JioiIFKxNN4Vhw/SYJxsp\nQRERkYJ22GHw9NOwbFnckUgiJSgiIlLQDj0U1q6FadPijkQSKUEREZGCtsUWsPPO8NBDcUciiZSg\niIhIwTv8cHjiCVi5Mu5IpJYSFBERKXiHHQZffx2SFMkOSlBERKTgff/7MHCgZvNkEyUoIiIihMc8\n06bB6tVxRyKgBEVERAQIj3lWrIAZM+KOREAJioiICADbbw/bbafHPNlCCYqIiAhgFh7zPPJIWBdF\n4qUERUREJHLoobBkCbz0UtyRiBIUERGRSHExbLYZPPpo3JGIEhQREZFIUREceKASlGygBEVERCTB\nyJHwr3/Bu+/GHUlhU4IiIiKSoLQUOnZUL0rclKCIiIgk6NQpJClKUOKlBEVERCTJyJHw4ouwdGnc\nkRQuJSgiIiJJDjgAqqu1eWCclKCIiIgk2WILGDRIj3nipARFRESkDiNHwt//rlVl46IERUREpA4j\nR8KyZTBzZtyRFCYlKCIiInUoLobevfWYJy5KUEREROqgVWXjpQRFRESkHiNHhhVltaps61OCIiIi\nUo/SUujQAaZNizuSwqMERUREpB6dO2tV2bgoQREREWnAyJHwwgtaVba1KUERERFpgFaVjYcSFBER\nkQZsuSXsuKPGobQ2JSgiIiKNqF1Vdt26uCMpHBlNUMysu5ndY2ZVZrbUzG41s86N1OlgZhPMbLGZ\nrTCzKWbWM6nMlmb2mJmtNLNKM7vazOpsi5ntamZrzWxuOtsmIiKFY7/9whiUV1+NO5LCkekelHuB\n/kApcAAwDJjYSJ3rorKHR+X7AA/WXowSkceBtsAuwHHAGOCS5BuZWRfgDmBGy5ohIiKFbPBg6NYN\npk+PO5LCkbEExcz6ASOAce7+mrvPBE4Djjaz3vXU6QIcD5S7+3PuPg8YC+xqZkOiYiOAfsAx7v6m\nu08HLgBONbO2SbecCNwDvJzu9omISOFo2xb22ksJSmvKZA/KUGBplGTUmgE4sHM9dUoIPSNP155w\n93eAj6L7Qeg1edPdFyfUmw50BQbUnjCzscD3gN+1rBkiIiKw777wyiuwZEnckRSGTCYovYFFiSfc\nvRpYEl2rr84ad1+edP6zhDq9o9fJ12uvYWbbAr8HRrl7TbOiFxERSTBiBNTUwAwNGmgVyY9EGmVm\nVwDnNFDECeNO6r1FVCalt21iHY/GqNwDXOTu/0mo3yTl5eV07dp1vXNlZWWUlZU19RYiIpKHttgC\ntt8+POY58si4o4lPRUUFFRUV652rqqpK+/uknKAA1wCTGynzHlAJJM++aQN059s9ILUqgfZm1iWp\nF6VnQp1KYHBSvV7Rx8+AjYCdgB3NbEJ0vii8va0B9nH3Z+sLfPz48RQXFzfQNBERKVT77gv33Qfu\nYE3+0ze/1PVH+9y5cykpKUnr+6T8iMfdv3D3dxs51gGzgG5mNiiheimhN2N2PbefA6yLygFgZn2B\nrYCZ0alZwA/NrEdCvX2AKuBtYDnwA2BHYGB0/BlYEH1e33uLiIg0aMQI+OQTeOutuCPJfxkbg+Lu\nCwiDVyeZ2WAz2xW4Aahw90oAM+tjZvPNbKeoznLgNuBaM9vTzEoIvTUvuXvt7PMnCYnIXWa2g5mN\nAC4FbnT3tR68nXgQxsKscvf57v51ptosIiL5bffdoWNHzeZpDZleB2UUoediBjANeB44KeF6O6Av\n0CnhXHlUdgrwLPAJYU0UAKJBrwcC1YRelTuB24GLMtMEERGRYIMNYM89laC0huaMQWkyd18GjG7g\n+odAm6RzqwnrpZzWQL2FhCSlqXH8Dk03FhGRNBgxAn7zG/jqK+jUqfHy0jzai0dERCQFI0bA6tXw\n3HNxR5LflKCIiIikoF8/2GorPebJNCUoIiIiKTALvShPPBF3JPlNCYqIiEiKRoyAd96BDz+MO5L8\npQRFREQkRaWl0KaNHvNkkhIUERGRFHXrBrvsosc8maQERUREpBlGjICnn4a1a+OOJD8pQREREWmG\nffeF5cthtjZQyQglKCIiIs1QXAybbKJxKJmiBEVERKQZ2rSBvfdWgpIpSlBERESaaa+9YM4cWLYs\n7kjyjxIUERGRZho+HGpqtOx9JihBERERaaattw7HM8/EHUn+UYIiIiLSAsOHh+nGkl5KUERERFqg\ntBTeegs++yzuSPKLEhQREZEW+PGPw0c95kkvJSgiIiIt0Ls3DBigBCXdlKCIiIi0kMahpJ8SFBER\nkRYqLYX33w+HpIcSFBERkRbaYw8oKtJjnnRSgiIiItJC3bqFvXmUoKSPEhQREZE0KC0NCYp73JHk\nByUoIiIiaTB8OFRWwvz5cUeSH5SgiIiIpMFuu0G7dprNky5KUERERNKgUycYOlTjUNJFCYqIiEia\nlJbCs89CdXXckeQ+JSgiIiJpMnw4LFsG8+bFHUnuU4IiIiKSJkOGQOfOGoeSDkpQRERE0qR9e9h9\nd41DSQclKCIiImlUWgovvACrV8cdSW5TgiIiIpJGw4fD11/D7NlxR5LblKCIiIik0Y47QvfuGofS\nUkpQRERE0qioKGwe+PzzcUeS25SgiIiIpNmwYfDyyxqH0hJKUERERNJsjz1g1Sp49dW4I8ldSlBE\nRETSbOBA2GgjPeZpCSUoIiIiadamTdg8UAlK8ylBERERyYBhw+Cll2DdurgjyU1KUERERDJg2DD4\n8kv45z/jjiQ3KUERERHJgJ12go4d9ZinuTKaoJhZdzO7x8yqzGypmd1qZp0bqdPBzCaY2WIzW2Fm\nU8ysZ1KZLc3sMTNbaWaVZna1mRUlXN/DzGqSjurk+4iIiGRK+/YwdKgSlObKdA/KvUB/oBQ4ABgG\nTGykznVR2cOj8n2AB2svRonI40BbYBfgOGAMcEnSfRzYFugdHZu5+6IWtUZERCQFw4aFfXlqauKO\nJPdkLEExs37ACGCcu7/m7jOB04Cjzax3PXW6AMcD5e7+nLvPA8YCu5rZkKjYCKAfcIy7v+nu04EL\ngFPNrG3SLT9390W1R/pbKSIiUr9hw2DJEnj77bgjyT2Z7EEZCiyNkoxaMwg9GzvXU6eE0DPyvx0M\n3P0d4KPofhB6Td5098UJ9aYDXYEBCecM+KeZfWJmT5rZj1rSGBERkVTtsgu0a6fHPM2RyQSlN7Be\nr4W7VwNLomv11Vnj7suTzn+WUKd39Dr5OgllPgVOIjwmOgxYCDxrZjum2AYREZFm69QJBg+G556L\nO5Lck3KCYmZX1DEANXkwat+GbkHoRUnpbZtYxwHc/V13n+Tu89z9ZXcfB8wEylN8XxERkRYZNiz0\noHiqv/kKXPKYjaa4BpjcSJn3gEogefZNG6A73+4BqVUJtDezLkm9KD0T6lQCg5Pq9Yo+1ndfgFeA\nXRuJm/Lycrp27breubKyMsrKyhqrKiIi8i3DhsGVV8K//w3bbht3NC1XUVFBRUXFeueqqqrS/j7m\nGUrpokGybwE71Y5DMbN9CDNwtnD3yjrqdAE+B45294ejc32BBcDO7v6qme0LPEqYlbM4KnMicBXQ\n093X1hPPk8Bydz+inuvFwJw5c+ZQXFzckqaLiIj8T1UVbLwx3HILjBsXdzSZMXfuXEpKSgBK3H1u\nOu6ZsTEo7r6AMHh1kpkNNrNdgRuAitrkxMz6mNl8M9spqrMcuA241sz2NLMSQm/NS+5euyfkk8Db\nwF1mtoOZjQAuBW6sTU7M7AwzO8jMvm9mA8zsOuDHwI2Zaq+IiEhdunaFHXfUQNlUNecRTypGEZKC\nGUANMAU4I+F6O6Av0CnhXDlQHZXtADwBnFp70d1rzOxA4GbCuJKVwO3ARQn3aA/8kbCGylfAG0Cp\nu+u/h4iItLphw2Dq1LijyC0ZTVDcfRkwuoHrHwJtks6tJqyXcloD9RYCBzZw/Q/AH1KNV0REJBOG\nDYPrroOPPoKttoo7mtygvXhEREQybPfdw8cXXog3jlyiBEVERCTDevSAAQM0DiUVSlBERERaQe16\nKNI0SlBERERawbBhsGABLNLOcE2iBEVERKQV1I5DUS9K0yhBERERaQWbbw6XXJIfq8m2hkyvgyIi\nIiKRCy6IO4LcoR4UERERyTpKUERERCTrKEERERGRrKMERURERLKOEhQRERHJOkpQREREJOsoQRER\nEZGsowRFREREso4SFBEREck6SlBEREQk6yhBERERkayjBEVERESyjhIUERERyTpKUERERCTrKEER\nERGRrKMERURERLKOEhQRERHJOkpQREREJOsoQREREZGsowRFREREso4SFBEREck6SlBEREQk6yhB\nERERkayjBEVERESyjhIUERERyTpKUERERCTrKEERERGRrKMERURERLKOEhQRERHJOkpQREREJOso\nQREREZGsowRFREREso4SFBEREck6SlAKUEVFRdwhtAq1M7+onfmlUNoJhdXWdMpYgmJm3c3sHjOr\nMrOlZnarmXVupE4HM5tgZovNbIWZTTGznklltjSzx8xspZlVmtnVZlaUVKa9mV1uZh+Y2Soze8/M\nxmSgmTmpUL5Z1M78onbml0JpJxRWW9OpbQbvfS/QCygF2gO3AxOB0Q3UuQ7YDzgcWA5MAB4EdgeI\nEpHHgU+AXYA+wF3AGuD8hPv8FdgUGAv8B9gM9RaJiIjkjIwkKGbWDxgBlLj7vOjcacBjZvYrd6+s\no04X4HjgaHd/Ljo3FphvZkPc/ZXonv2AH7v7YuBNM7sAuNLMLnb3dWa2LyGh+Z67L4tu/1Em2iki\nIiKZkalehaHA0trkJDIDcGDneuqUEBKmp2tPuPs7hORiaHRqF+DNKDmpNR3oCgyIXo8EXgPOMbOP\nzewdM/uDmXVsYZtERESklWTqEU9vYFHiCXevNrMl0bX66qxx9+VJ5z9LqNM7ep18vfba68D3CD0o\nq4BDgB7AzUB34IQGYu4IMH/+/AaK5Ieqqirmzp0bdxgZp3bmF7UzvxRKO6Ew2prwuzN9nQHu3uQD\nuAKoaeCoBvoCvwXm11F/EXBiPfcuA76u4/wrwO+jzycCf0+6vkH03vtEr6cDK4ENE8ocCqwDOjTQ\ntlGEHh4dOnTo0KFDR/OOUankFQ0dqfagXANMbqTMe0AlkDz7pg2hFyO5B6RWJdDezLok9aL0TKhT\nCQxOqtcr+lhb5lPgv+7+ZUKZ+YABWxAGzdZlOnAM8AGh90VERESapiPwXcLv0rRIKUFx9y+ALxor\nZ2azgG5mNihhHEopIUmYXU+1OYRejlLg4eg+fYGtgJlRmVnAuWbWI2Ecyj5AFfB29Pol4Agz6+Tu\nX0XntiP0snzcSNvubaxtIiIiUqeZjRdpOoseb6SdmT1O6P04mTDN+C/AK+5+bHS9D2FA7LHu/lp0\n7ibCNOOxwArgeqDG3ROnGc8jTDM+hzB9+E7gFne/ICrTmZCsvAxcTJhuPAn4h7v/PCONFRERkbTK\n5Nogo4AFhNk704DngZMSrrcjjFfplHCuPCo7BXiWkIgcXnvR3WuAAwljXWYSkpPbgYsSyqwE9ga6\nAa8S1kl5BDgjfU0TERGRTMpYD4qIiIhIc2l1VREREck6SlBEREQk6xRsgtLMzQz/bGb/NrOvzGyR\nmU01s+1aK+bmSrWtUfnrzWxBtCnjh2b2p2g7gqzVzK/pz8zsH1Gdmmxso5mdambvm9nXZvaymSVP\ntU8u/xMzmx+Vf93M9mutWFsilXaa2fbRZqLvR1+301sz1pZIsZ0nmNnzZrYkOp5q7OufLVJs56Fm\n9mr0ffulmc0zs4b2bcsaqX5/JtQ7Ovq/+1CmY0yXFL+mx0Xtq44+1pjZV/WVr0vBJiiEKcX9CdOa\nDwCGERaCa8hrwBjCfkD7EKZNTzczy1yYaZFqW/sQZkidBfwAOA7YF7g1s2G2WHO+phsAfwcuJywy\nlFXM7Cjgj4SB4IMIqyVPN7Me9ZQfSvh3mATsCEwFpprZ9q0TcfOk2k7C4Pr/EGbzfdoqQaZBM9q5\nB+HruSdhq4+FwJNmtlnmo22+ZrTzC+AyQht/SFhva7KZ7d0K4TZbM9pZW+87wB8Ik0dyQjPbWkVY\n5b32+E5Kb5quFd9y6SAkGDXAoIRzIwjrsPRO4T4/JMwo2jruNrVCW48AvgaK4m5TJtpJ+EVQDXSJ\nuy1Jcb0M/CnhtRHW8zm7nvL3AX9LOjcLuCnutqSznUl13wdOj7sNmW5nVL4o+qE/Ou62ZLKdUZ05\nwO/ibku62xl9DV8gLKcxGXgo7nZkoq2EP2yXtOQ9C7UHpTmbGa4nenRwPGHl3IVpjzB9WtzWSDdg\nuYep3tkoXe3MGmbWjrCJZuIGmk5o19B6qg2Nriea3kD52DWznTknTe3sTFiiYUnaA0yTdLTTzEoJ\ny1A8l4kY06EF7bwIWOTuja3KnjVa0NYNzewDM/soGhKRUk9uoSYodW5mSPimr28zQwDM7GQzW0FY\nSG4fwh5A6zIVaBo0u621oi6882n8cUmcWtzOLNQDaEPdG2Q2tOlmKuWzQXPamYvS0c6rgP/y7SQ0\nmzSrnWbWxcxWmNka4FHgNHd/JnNhtljK7TSzXQk9Jw1tXJuNmvM1fYfwR/xBhG1kioCZZrZ5U980\nrxIUM7siYTBOXUe1heXz670FjY9DuJvwbH8Y8C/gr2bWPk1NaLJWaitmthHwGPB/wO/SFH6TtVY7\nc0yqbcrVf4NcjTtVTf1e/A1wJHCIu6/JeFTp11g7VwADgZ2A84DxZjasNQJLszrbaWYbEhYO/Zm7\nL231qDKj3q+pu7/s7ne7+xvu/gJwGPA5cGJTb57qZoHZLpObGQLg7rW9J/8xs9nAUsJuyfc3M+bm\nynhbo2+o6cAy4LCoR6K1ZbydWWwxYVxMr6TziRtoJqtMsXw2aE47c1Gz22lmvwLOBkrd/a3MhJc2\nzWpn9MjgvejlG9HjgN+SvQNJU23n9wmDRB9NmFhRBBD1Gm3n7u9nKNaWavH3qLuvM7N5wDZNfdO8\n6kFx9y/c/d1GjnWEQYPdzGxQQvXGNjOsS1FUp0P6WtE0mW5r1HPyJGFg7EFx/cUWw9c0a7j7WsJA\nwdLac9EPtlLq35RrVmL5yN7R+azUzHbmnOa208x+TehRGJE0xiorpfHrWUQMP1ubqhntnE+YWLEj\noadoIPA34Jno86wdy5iOr6mFvfR+QCqz7uIeGRzXATxOmDY8GNiV8LzsroTrfQj/oXaKXm8N/AYo\nBrYEfkT4z/U50CPu9qS5rRsSRmz/M2p3r4QjK2fxNKed0blehB8OJxBmAe0Wve4ed3ui+I4kJIk/\nJcxUmkiYkrlpdP1O4PcJ5YcCawhTxLcjbJi5Ctg+7rakuZ3toq/TjoQxGVdFr78fd1vS3M6zo6/f\noUnfh53jbkua2/kbYK/o500/4JfAamBs3G1JZzvrqJ9Ls3hS/ZpeQPjjaGvCtOQKYCXQr8nvGXej\nY/zH7kYYT1JFeEwzCeiUcP07hC6tYdHrzQhjMT6NfmB8SHieuG3cbclAW2un3CYeNdHHreJuT7ra\nGZ27KKFticdP425PQoynAB9EPxxmsX6C9Qzwl6TyhxM26vwaeIPwl3fs7UhnO6OvZV1ft2fibkea\n2/l+HW2sBi6Mux1pbuelhD8oVhIeJ7wIHBF3G9Ldzjrq5kyC0oyv6bXR/9+vCRv/PgrskMr7abNA\nERERyTp5NQZFRERE8oMSFBEREck6SlBEREQk6yhBERERkayjBEVERESyjhIUERERyTpKUERERCTr\nKEERERGRrKMERURERLKOEhQRERHJOkpQREREJOv8P2HQIe4ldQb0AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108f7df60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(math.sqrt(2)/2,math.sqrt(2),1e-2)-1\n",
    "#plt.plot(x,np.log(1+x))\n",
    "#plt.plot(x,x)\n",
    "#plt.plot(x,(x-x**2/2))\n",
    "#plt.plot(x,(x-x**2/2 + x**3/3))\n",
    "plt.plot(x,np.log(1+x)-(x-x**2/2 + x**3/3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n = 0, R_(n+1) = 4e-01f\n",
      "n = 1, R_(n+1) = 9e-02f\n",
      "n = 2, R_(n+1) = 2e-02f\n",
      "n = 3, R_(n+1) = 7e-03f\n",
      "n = 4, R_(n+1) = 2e-03f\n",
      "n = 5, R_(n+1) = 8e-04f\n",
      "n = 6, R_(n+1) = 3e-04f\n",
      "n = 7, R_(n+1) = 1e-04f\n",
      "n = 8, R_(n+1) = 4e-05f\n",
      "n = 9, R_(n+1) = 1e-05f\n",
      "n = 10, R_(n+1) = 6e-06f\n",
      "n = 11, R_(n+1) = 2e-06f\n",
      "n = 12, R_(n+1) = 8e-07f\n",
      "n = 13, R_(n+1) = 3e-07f\n",
      "n = 14, R_(n+1) = 1e-07f\n",
      "n = 15, R_(n+1) = 5e-08f\n",
      "n = 16, R_(n+1) = 2e-08f\n",
      "n = 17, R_(n+1) = 7e-09f\n",
      "n = 18, R_(n+1) = 3e-09f\n",
      "n = 19, R_(n+1) = 1e-09f\n",
      "n = 20, R_(n+1) = 4e-10f\n",
      "n = 21, R_(n+1) = 2e-10f\n",
      "n = 22, R_(n+1) = 7e-11f\n",
      "n = 23, R_(n+1) = 3e-11f\n",
      "n = 24, R_(n+1) = 1e-11f\n",
      "n = 25, R_(n+1) = 4e-12f\n",
      "n = 26, R_(n+1) = 2e-12f\n",
      "n = 27, R_(n+1) = 7e-13f\n",
      "n = 28, R_(n+1) = 3e-13f\n",
      "n = 29, R_(n+1) = 1e-13f\n",
      "n = 30, R_(n+1) = 4e-14f\n",
      "n = 31, R_(n+1) = 2e-14f\n",
      "n = 32, R_(n+1) = 7e-15f\n",
      "n = 33, R_(n+1) = 3e-15f\n",
      "n = 34, R_(n+1) = 1e-15f\n",
      "n = 35, R_(n+1) = 5e-16f\n",
      "n = 36, R_(n+1) = 2e-16f\n",
      "n = 37, R_(n+1) = 7e-17f\n",
      "n = 38, R_(n+1) = 3e-17f\n",
      "n = 39, R_(n+1) = 1e-17f\n"
     ]
    }
   ],
   "source": [
    "for n in range(40):\n",
    "    R = (math.sqrt(2)-1)**(n+1)/(n+1)\n",
    "    print(\"n = %d, R_(n+1) = %.ef\"%(n,R))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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