IV.11.19.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ниже приведено решение теоретической задачи 12.4(к) из раздела нелинейные уравнения."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Как было показано для исходной задачи, у нас из 4х итерационных процессов сходилось только два, причем с разной скоростью сходимости. Давайте смоделируем эту задачу на ЭВМ и проверим предположения, а заодно и провильность решения задачи."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Нам задано уравнение:\n",
    "$$ \\sin(x) - x^2 + 1 = 0 $$\n",
    "с корнем, локализованным на (1,1.5).\n",
    "\n",
    "Итерационный процесс 1:\n",
    "$$ x_{n+1} = \\arcsin(x_n^2 - 1) $$\n",
    "Итерационный процесс 2:\n",
    "$$ x_{n+1} =  \\sqrt{\\sin(x_n) + 1}$$\n",
    "Итерационный процесс 3:\n",
    "$$ x_{n+1} = \\frac{\\sin(x_n) + 1}{x_n} $$\n",
    "Итерационный процесс 4:\n",
    "$$ x_{n+1} = x_n + 0.1(\\sin(x_n) - x_n^2 + 1) $$\n",
    "\n",
    "Как было показано в решении задачи в тетради, 2 процесс сходится быстрее 4го, а о сходимости 1го и 2го процесса говорить вообще не приходится."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Исследуем процессы на сходимость с начальным приближением 1,0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "import matplotlib as mpl\n",
    "import scipy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f1(x):\n",
    "    return np.arcsin(x**2 - 1)\n",
    "def f2(x):\n",
    "    return (np.sin(x) + 1)**0.5\n",
    "def f3(x):\n",
    "    return (np.sin(x) + 1)/x\n",
    "def f4(x):\n",
    "    return x + 0.1*(np.sin(x) - x**2 + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def start_process(iterations):\n",
    "    x = 1.1\n",
    "    print 'Процесс 1:'\n",
    "    print 'Корни уравнения на каждой итерации:'\n",
    "    print x\n",
    "\n",
    "    for i in range(iterations):\n",
    "        x = f1(x)\n",
    "        print x\n",
    "\n",
    "    x = 1.1\n",
    "    print 'Процесс 2:'\n",
    "    print 'Корни уравнения на каждой итерации:'\n",
    "    print x\n",
    "\n",
    "    for i in range(iterations):\n",
    "        x = f2(x)\n",
    "        print x\n",
    "    \n",
    "    x = 1.1\n",
    "    print 'Процесс 3:'\n",
    "    print 'Корни уравнения на каждой итерации:'\n",
    "    print x\n",
    "\n",
    "    for i in range(iterations):\n",
    "        x = f3(x)\n",
    "        print x\n",
    "\n",
    "    x = 1.1\n",
    "    print 'Процесс 4:'\n",
    "    print 'Корни уравнения на каждой итерации:'\n",
    "    print x\n",
    "\n",
    "    for i in range(iterations):\n",
    "        x = f4(x)\n",
    "        print x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Запустим 10 итераций каждого процесса."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Процесс 1:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "0.21157495975809582\n",
      "-1.2704565950589028\n",
      "0.661194373129016\n",
      "-0.5977959239970859\n",
      "-0.69793906901201\n",
      "-0.5385375147206699\n",
      "-0.7894660390249377\n",
      "-0.38627810266625046\n",
      "-1.0174853126827603\n",
      "0.035283682070166175\n",
      "Процесс 2:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "1.375211750990165\n",
      "1.4074566532230295\n",
      "1.4094998077635963\n",
      "1.4096169318652243\n",
      "1.4096236014380683\n",
      "1.409623981088045\n",
      "1.4096240026982687\n",
      "1.409624003928352\n",
      "1.40962400399837\n",
      "1.4096240040023558\n",
      "Процесс 3:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "1.7192794182376685\n",
      "1.1568780520174022\n",
      "1.655794162906022\n",
      "1.2056992970100342\n",
      "1.6041222938841084\n",
      "1.2464415892571639\n",
      "1.5627339330446366\n",
      "1.279787593261684\n",
      "1.529906172512494\n",
      "1.306723346854061\n",
      "Процесс 4:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "1.1681207360061436\n",
      "1.2236717080452946\n",
      "1.2679699421519246\n",
      "1.3026449067955945\n",
      "1.3293827638518303\n",
      "1.3497570102587437\n",
      "1.3651396228488455\n",
      "1.3766717128673536\n",
      "1.385270903782956\n",
      "1.391657302617563\n"
     ]
    }
   ],
   "source": [
    "start_process(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Запустим 100 итераций каждого процесса."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Процесс 1:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "0.21157495975809582\n",
      "-1.2704565950589028\n",
      "0.661194373129016\n",
      "-0.5977959239970859\n",
      "-0.69793906901201\n",
      "-0.5385375147206699\n",
      "-0.7894660390249377\n",
      "-0.38627810266625046\n",
      "-1.0174853126827603\n",
      "0.035283682070166175\n",
      "-1.5208924869009501\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "nan\n",
      "Процесс 2:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "1.375211750990165\n",
      "1.4074566532230295\n",
      "1.4094998077635963\n",
      "1.4096169318652243\n",
      "1.4096236014380683\n",
      "1.409623981088045\n",
      "1.4096240026982687\n",
      "1.409624003928352\n",
      "1.40962400399837\n",
      "1.4096240040023558\n",
      "1.4096240040025825\n",
      "1.4096240040025956\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "1.4096240040025962\n",
      "Процесс 3:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "1.7192794182376685\n",
      "1.1568780520174022\n",
      "1.655794162906022\n",
      "1.2056992970100342\n",
      "1.6041222938841084\n",
      "1.2464415892571639\n",
      "1.5627339330446366\n",
      "1.279787593261684\n",
      "1.529906172512494\n",
      "1.306723346854061\n",
      "1.5040175765126225\n",
      "1.3282897480218907\n",
      "1.48366657698517\n",
      "1.3454549930909336\n",
      "1.4676952456261017\n",
      "1.3590626500042262\n",
      "1.455170622934255\n",
      "1.3698206663861232\n",
      "1.445351443576688\n",
      "1.3783098305947945\n",
      "1.4376532467464924\n",
      "1.3849998604407985\n",
      "1.4316170624181268\n",
      "1.3902671621801752\n",
      "1.426883172455038\n",
      "1.394411540343278\n",
      "1.4231698711638854\n",
      "1.3976708126832726\n",
      "1.420256582545956\n",
      "1.4002330987096392\n",
      "1.417970568019908\n",
      "1.4022469151371162\n",
      "1.4161765109932904\n",
      "1.4038293509395294\n",
      "1.4147683731170573\n",
      "1.4050726260026223\n",
      "1.4136630311778515\n",
      "1.4060493205258164\n",
      "1.4127953049646946\n",
      "1.4068165270363788\n",
      "1.4121140703640405\n",
      "1.4074191373295866\n",
      "1.411579219065422\n",
      "1.4078924391685035\n",
      "1.411159279046194\n",
      "1.4082641646189413\n",
      "1.4108295511694757\n",
      "1.408556104053284\n",
      "1.4105706492199621\n",
      "1.4087853768887098\n",
      "1.4103673555398524\n",
      "1.4089654314588653\n",
      "1.4102077237194262\n",
      "1.4091068313959967\n",
      "1.410082374797902\n",
      "1.4092178738918755\n",
      "1.409983945109926\n",
      "1.4093050756507668\n",
      "1.4099066530188014\n",
      "1.4093735547700736\n",
      "1.4098459588853838\n",
      "1.409427330789406\n",
      "1.4097982981758277\n",
      "1.4094695604144796\n",
      "1.4097608719556727\n",
      "1.4095027226883454\n",
      "1.4097314824183287\n",
      "1.4095287644405272\n",
      "1.409708403759555\n",
      "1.409549214528445\n",
      "1.4096902807968776\n",
      "1.4095652735630175\n",
      "1.4096760493693603\n",
      "1.409577884376964\n",
      "1.409664873836589\n",
      "1.4095877873673532\n",
      "1.4096560980025419\n",
      "1.4095955639581408\n",
      "1.4096492065800275\n",
      "1.4096016707323182\n",
      "1.4096437949313312\n",
      "1.4096064662363295\n",
      "1.4096395453073332\n",
      "1.4096102320297055\n",
      "1.4096362081893161\n",
      "1.409613189215305\n",
      "1.4096335876375483\n",
      "1.4096155114204\n",
      "1.4096315297861337\n",
      "1.4096173349905834\n",
      "1.4096299138084587\n",
      "1.4096187669948994\n",
      "1.4096286448226754\n",
      "1.4096198915121467\n",
      "1.4096276483206138\n",
      "1.4096207745673928\n",
      "1.4096268657929825\n",
      "1.4096214680085235\n",
      "1.4096262512939701\n",
      "1.4096220125504157\n",
      "Процесс 4:\n",
      "Корни уравнения на каждой итерации:\n",
      "1.1\n",
      "1.1681207360061436\n",
      "1.2236717080452946\n",
      "1.2679699421519246\n",
      "1.3026449067955945\n",
      "1.3293827638518303\n",
      "1.3497570102587437\n",
      "1.3651396228488455\n",
      "1.3766717128673536\n",
      "1.385270903782956\n",
      "1.391657302617563\n",
      "1.3963860446498686\n",
      "1.3998795503154433\n",
      "1.4024561998114313\n",
      "1.4043542835990857\n",
      "1.405751235297722\n",
      "1.4067786765396166\n",
      "1.4075339753233171\n",
      "1.4080890140361195\n",
      "1.4084967810641147\n",
      "1.4087962942386183\n",
      "1.4090162611195556\n",
      "1.409177790967011\n",
      "1.4092963991279461\n",
      "1.4093834857840373\n",
      "1.4094474254682279\n",
      "1.409494369027674\n",
      "1.4095288335092269\n",
      "1.409554135826134\n",
      "1.409572711456118\n",
      "1.409586348584706\n",
      "1.4095963600921255\n",
      "1.4096037098650598\n",
      "1.409609105553154\n",
      "1.4096130666788629\n",
      "1.409615974646621\n",
      "1.409618109460128\n",
      "1.4096196766797169\n",
      "1.4096208272136541\n",
      "1.4096216718480743\n",
      "1.4096222919141073\n",
      "1.4096227471190592\n",
      "1.4096230812955695\n",
      "1.4096233266222549\n",
      "1.409623506722213\n",
      "1.40962363893772\n",
      "1.4096237360001476\n",
      "1.4096238072558926\n",
      "1.4096238595663606\n",
      "1.4096238979686675\n",
      "1.4096239261606758\n",
      "1.409623946857071\n",
      "1.409623962050766\n",
      "1.4096239732048033\n",
      "1.4096239813932359\n",
      "1.4096239874045504\n",
      "1.4096239918175932\n",
      "1.4096239950573084\n",
      "1.409623997435657\n",
      "1.4096239991816568\n",
      "1.409624000463435\n",
      "1.4096240014044175\n",
      "1.4096240020952142\n",
      "1.4096240026023437\n",
      "1.409624002974639\n",
      "1.4096240032479495\n",
      "1.409624003448593\n",
      "1.4096240035958898\n",
      "1.4096240037040237\n",
      "1.4096240037834074\n",
      "1.4096240038416847\n",
      "1.4096240038844674\n",
      "1.4096240039158752\n",
      "1.4096240039389323\n",
      "1.4096240039558592\n",
      "1.4096240039682855\n",
      "1.409624003977408\n",
      "1.409624003984105\n",
      "1.4096240039890213\n",
      "1.4096240039926307\n",
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    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/vladislav/anaconda2/lib/python2.7/site-packages/ipykernel_launcher.py:2: RuntimeWarning: invalid value encountered in arcsin\n",
      "  \n"
     ]
    }
   ],
   "source": [
    "start_process(100)"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Выводы:\n",
    "\n",
    "### 1 итерационный процесс не сходится, как и предполагалось.\n",
    "### 2 итерационный процесс cходится значительно быстрее 4го (на 4 итерации достигнуто 5 значащих цифр), как и предполагалось.\n",
    "### 3 итерационный процесс сходится!!! хотя и очень медленно. Предполагалось, что он не сойдется, но вероятно был неправильно оценен модуль производной, либо ошибка в том, что воспользовался достаточным условием вместо критерия."
   ]
  }
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