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pomo-mondreganto/BHW 4.ipynb

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Task 23 (5) from BHW 4
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BHW 4.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$f(x) = \\sqrt{1 - x^2} \\quad [a, b] = [0, 1]$\n",
"\n",
"$x_0 = 0, x_i = \\frac{i}{n}$\n",
"\n",
"$S_i = f\\left(\\frac{x_{i - 1} + x_i}{2}\\right) \\frac{1}{n}$\n",
"\n",
"$I = \\int \\limits_0^1 \\sqrt{1 - x^2} = \\frac{S_{\\text{circle}}}{4} = \\frac{\\pi}{4} \\approx 0.78539816$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def f(x):\n",
" return np.sqrt(1 - x * x)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def get(n):\n",
" d = 1. / n\n",
" result = 0.\n",
" \n",
" for i in range(n):\n",
" x_i = d * i\n",
" x_ii = d * (i + 1)\n",
" result += f((x_i + x_ii) / 2.) / n\n",
" return result"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>real</th>\n",
" <th>approx</th>\n",
" <th>err</th>\n",
" <th>s</th>\n",
" </tr>\n",
" <tr>\n",
" <th>n</th>\n",
" <th></th>\n",
" <th></th>\n",
" <th></th>\n",
" <th></th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>8</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.789171732824577</td>\n",
" <td>0.003773569427129</td>\n",
" <td>2.683284822374950</td>\n",
" </tr>\n",
" <tr>\n",
" <th>16</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.786737951981630</td>\n",
" <td>0.001339788584182</td>\n",
" <td>2.385944730064856</td>\n",
" </tr>\n",
" <tr>\n",
" <th>32</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785872850670559</td>\n",
" <td>0.000474687273111</td>\n",
" <td>2.208147001901588</td>\n",
" </tr>\n",
" <tr>\n",
" <th>64</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785566168003386</td>\n",
" <td>0.000168004605938</td>\n",
" <td>2.089868598954976</td>\n",
" </tr>\n",
" <tr>\n",
" <th>128</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785457593379461</td>\n",
" <td>0.000059429982013</td>\n",
" <td>2.005492790019204</td>\n",
" </tr>\n",
" <tr>\n",
" <th>256</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785419180620520</td>\n",
" <td>0.000021017223072</td>\n",
" <td>1.942258551314332</td>\n",
" </tr>\n",
" <tr>\n",
" <th>512</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785405595089595</td>\n",
" <td>0.000007431692147</td>\n",
" <td>1.893097536565201</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1024</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785400791070969</td>\n",
" <td>0.000002627673521</td>\n",
" <td>1.853778253215027</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2048</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785399092451015</td>\n",
" <td>0.000000929053567</td>\n",
" <td>1.821612262099438</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4096</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398491872911</td>\n",
" <td>0.000000328475462</td>\n",
" <td>1.794809255095244</td>\n",
" </tr>\n",
" <tr>\n",
" <th>8192</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398279532022</td>\n",
" <td>0.000000116134574</td>\n",
" <td>1.772130702027616</td>\n",
" </tr>\n",
" <tr>\n",
" <th>16384</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398204457387</td>\n",
" <td>0.000000041059939</td>\n",
" <td>1.752692377329566</td>\n",
" </tr>\n",
" <tr>\n",
" <th>32768</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398177914360</td>\n",
" <td>0.000000014516912</td>\n",
" <td>1.735846010594087</td>\n",
" </tr>\n",
" <tr>\n",
" <th>65536</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398168529956</td>\n",
" <td>0.000000005132508</td>\n",
" <td>1.721105551736002</td>\n",
" </tr>\n",
" <tr>\n",
" <th>131072</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398165212066</td>\n",
" <td>0.000000001814618</td>\n",
" <td>1.708099249120909</td>\n",
" </tr>\n",
" <tr>\n",
" <th>262144</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398164039013</td>\n",
" <td>0.000000000641565</td>\n",
" <td>1.696538080966664</td>\n",
" </tr>\n",
" <tr>\n",
" <th>524288</th>\n",
" <td>0.785398163397448</td>\n",
" <td>0.785398163624275</td>\n",
" <td>0.000000000226827</td>\n",
" <td>1.686194097522441</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" real approx err \\\n",
"n \n",
"8 0.785398163397448 0.789171732824577 0.003773569427129 \n",
"16 0.785398163397448 0.786737951981630 0.001339788584182 \n",
"32 0.785398163397448 0.785872850670559 0.000474687273111 \n",
"64 0.785398163397448 0.785566168003386 0.000168004605938 \n",
"128 0.785398163397448 0.785457593379461 0.000059429982013 \n",
"256 0.785398163397448 0.785419180620520 0.000021017223072 \n",
"512 0.785398163397448 0.785405595089595 0.000007431692147 \n",
"1024 0.785398163397448 0.785400791070969 0.000002627673521 \n",
"2048 0.785398163397448 0.785399092451015 0.000000929053567 \n",
"4096 0.785398163397448 0.785398491872911 0.000000328475462 \n",
"8192 0.785398163397448 0.785398279532022 0.000000116134574 \n",
"16384 0.785398163397448 0.785398204457387 0.000000041059939 \n",
"32768 0.785398163397448 0.785398177914360 0.000000014516912 \n",
"65536 0.785398163397448 0.785398168529956 0.000000005132508 \n",
"131072 0.785398163397448 0.785398165212066 0.000000001814618 \n",
"262144 0.785398163397448 0.785398164039013 0.000000000641565 \n",
"524288 0.785398163397448 0.785398163624275 0.000000000226827 \n",
"\n",
" s \n",
"n \n",
"8 2.683284822374950 \n",
"16 2.385944730064856 \n",
"32 2.208147001901588 \n",
"64 2.089868598954976 \n",
"128 2.005492790019204 \n",
"256 1.942258551314332 \n",
"512 1.893097536565201 \n",
"1024 1.853778253215027 \n",
"2048 1.821612262099438 \n",
"4096 1.794809255095244 \n",
"8192 1.772130702027616 \n",
"16384 1.752692377329566 \n",
"32768 1.735846010594087 \n",
"65536 1.721105551736002 \n",
"131072 1.708099249120909 \n",
"262144 1.696538080966664 \n",
"524288 1.686194097522441 "
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"interesting = [2 ** x for x in range(3, 20)]\n",
"\n",
"results = []\n",
"\n",
"I = np.pi / 4.\n",
"\n",
"for n in interesting:\n",
" approx = get(n)\n",
" err = np.abs(approx - I)\n",
" s = - np.log(err) / np.log(n)\n",
" results.append([I, approx, err, s])\n",
"\n",
"pd.set_option(\"display.precision\", 15)\n",
"df = pd.DataFrame(index=interesting, columns=['real', 'approx', 'err', 's'], data=results)\n",
"df.index.name = 'n'\n",
"df"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Сходится со скоростью $\\frac{1}{n^{1.5}}$"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3.7",
"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": "3.7.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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