{ "cells": [ { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

0.1  補間法
0.1.1  ラグランジュ補間
0.1.2  逆行列から
0.2  対数関数のニュートンの差分商補間(2014期末試験，25点)
0.2.1  差分商補間の表中の開いている箇所[ XXX ]を埋めよ．
0.2.2  ニュートンの二次多項式
0.2.3  ニュートンの三次多項式の値を求めよ．
0.2.4  補足
0.3  数値積分(I)
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 補間法\n", "次の4点\n", "```maple\n", "x y \n", "0 1 \n", "1 2\n", "2 3\n", "3 1\n", "```\n", "を通る多項式を(a)ラグランジュ補間, (b) 逆行列で求めよ．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### ラグランジュ補間" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 3 2\n", "-0.5 x + 1.5 x + 2.776e-16 x + 1\n" ] }, { "data": { "image/png": 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HscdCgwbhGu6xxG8oKID77oPLLoNzz/U7GmPCpXp16NnT3cW7c6ff0XhjiT/N\n/fSTq5172GHw17/6HY0x4dS7N/znPzAnJCuSWeJPY6pu/Z0ff4SXX4Y6dfyOyJhwOuUUt35PWIZ7\nLPGnsSefdFWE7rsPjjnG72iMCa8GDVytCkv8JtAWLoQbbnAfUa+7zu9ojAm/3r3dUM/atX5HEpsl\n/jS0eTOcf76bczxunPuIaoypnN693de33/Y3Di/sTz4NXXMNLFvmlmRo1MjvaIxJDW3bQuPG4ajK\nZYk/zbz0Ejz3nKub272739EYkzpEoEcPeO+94E/rtMSfRr7+2i3AdsIJ7oYtY0xinXKKW7ph7ly/\nIymfJf40sW0bnHce7LOPO+u3gunGJN7JJ7sz/+nT/Y6kfJb408T118OCBfDCC3DQQX5HY0xqatQI\n2rUL/ro9lvjTwCuvwFNPwfDh0KuX39EYk9pOOQU+/9zV4w0qS/wpbtkyd3dup05ujX1jTNU65RR3\ncfcf//A7krJZ4k9hW7ZA376uIHR+vlty2RhTtY4/HurWDfY4v5cKXM+JyDoRKbVsooh0E5GNIjI/\n8rgzal9PEVkmIstF5JZEBm7KpwpXXgmLF7uLuTaub0xyZGS4qdLTp7u/wyDycsY/DugZo83Hqto2\n8hgFICLVgceBXkBroJ+ItK5MsMa7Z55xZRTvvNPNNDDGJE+PHlBY6KraBVHMxK+qHwHr4/jdHYDl\nqrpCVbcD+cAZcfweU0GzZ8NVV7n/fHfc4Xc0xqSfU05xX4M63CPq4bOIiGQDU1W1TSn7ugGTgNXA\nD8AwVV0sIn2Bnqo6KNJuIHCcql5VxmvkAXkAWVlZOfn5+fH0h6KiIjIzM+N6btDE05eNGzMYPDgH\ngDFj5rD//iVVEVqFpPsxCapU6UtQ+9G//3FkZ2/mnntKHSUvVWX6kpubO1dV23tqrKoxH0A2sKiM\nffsBmZHvewPfRL7vCzwT1W4g8JiX18vJydF4FRQUxP3coKloX0pKVHv0UK1ZU3X27KqJKR7pfEyC\nLFX6EtR+DB6sWreu6o4d3p9Tmb4Ac9RDflXVys/qUdVfVLUo8v00IENEGuLO/qOrtx4U2WaqyJ13\nuhtHHnsM2nt73zfGVJGTToJNm9zQa9BUOvGLSGMRkcj3HSK/8ydgNtBKRFqKSE3gAuCNyr6eKd3E\niXDvva5u7qBBfkdjjMnNdcs3fPCB35Hszct0zpeBz4HDRWS1iFwmIkNEZEikSV9gkYgsAB4BLoh8\n8igBrgLjbUe/AAAMn0lEQVSmA0uAV1V1cdV0I70tXgz/939w3HHw+OPuP5sxxl8NGrjKdu+/73ck\ne4u5VJeq9oux/zHgsTL2TQNCUowsnDZsgLPOgsxMd9a/zz5+R2SM2e2kk+Chh1zxoyDVtLY7d0Ns\n507o1w9WroTXXoOmTf2OyBgT7cQTYccO+PhjvyP5NUv8IXbzza7az+OPQ+fOfkdjjNlT585Qs2bw\nhnss8YfU+PHwl7+4G7Xy8vyOxhhTmtq13QKJQbvAa4k/hD7/3CX77t1h9Gi/ozHGlOfEE2H+fPjx\nR78j+R9L/CFTWAhnnAHNmsGrr9qKm8YE3Uknua9BWqbZEn+IbNwIffq4i0VTp7rpYsaYYMvJgf33\nD9Y4v1VeDYmSEjj/fFdY5Z134Igj/I7IGONFjRrQtSsUFPgdyf/YGX8IqMI117iV/p580o0ZGmPC\nIzcXvv0WVq3yOxLHEn8IPPigS/g33WTLMRgTRrm57mtQzvot8Qfcyy+7+foXXAD33+93NMaYeBx1\nFNSvb4nfeDB//v5cfDF06QLjxkE1O1rGhFK1am6cf8YMvyNxLJUE1MKFcPvtR3HwwTB5sq3BY0zY\n5ea66diFhX5HYok/kFaudKXbatcuYfp09xHRGBNuQRrnt8QfMGvXulq5xcXwwAMLad7c74iMMYlw\n5JHQqFEwEr/N4w+QDRugZ0/417/czR7FxVv8DskYkyAi0K2bG+dX9bduhp3xB0RREfTq5YqqTJoE\nxx/vd0TGmETLzXVz+Ves8DcOLxW4nhORdSJSaql4EekvIgtF5EsR+UxEjo7aVxjZPl9E5iQy8FSy\ndSucfrqrzfnKK2583xiTeoIyzu/ljH8c0LOc/SuBrqp6FPAHYOwe+3NVta2qWvnvUhQXQ9++7uPf\nuHGumpYxJjUdfjg0bux/4vdSevEjEckuZ/9nUT/OBA6qfFjpobgYzjkHpk2DMWNgwAC/IzLGVKXd\n4/wffujvOL+oauxGLvFPVdU2MdoNA45Q1UGRn1cCG4GdwBhV3fPTQPRz84A8gKysrJz8/HyPXfi1\noqIiMjMz43puMm3fLowceSSff96Q669fxumnr9mrTVj6Ekuq9AOsL0EUtn5MmXIgDz98GBMmzOTA\nA7f9al9l+pKbmzvX88iKqsZ8ANnAohhtcoElQIOobU0jXw8AFgBdvLxeTk6OxqugoCDu5ybL1q2q\nffqoguqTT5bdLgx98SJV+qFqfQmisPVj0SL3t/+3v+29rzJ9Aeaoh/yqqomZ1SMivwOeAc5Q1Z+i\n3lR+iHxdB0wGOiTi9cKsqMitqT91Kjz1FAwZ4ndExphk+u1vXS2Njz7yL4ZKJ34RaQ5MAgaq6tdR\n2+uISN3d3wM9gFJnBqWLDRvcjJ2CAlczd/BgvyMyxiRbtWpwwgn+Jv6YF3dF5GWgG9BQRFYDdwEZ\nAKr6FHAn0AB4QtyVihJ140xZwOTIthrAS6r6ThX0IRTWrnXz9BctciUTzznH74iMMX7p0gVefx1+\n+AGaNk3+63uZ1dMvxv5BwF6rxKvqCuDovZ+Rfr75xt2R++9/w5Qp7g3AGJO+unRxXz/+2C25nmx2\n524Vmz0bOnVy9XL/8Q9L+sYYOPpoqFvXTev0gyX+KjRlirtTr04d+OwzOO44vyMyxgRBjRruhNCv\ncX5L/FVA1ZVLPOsstyLf55/DYYf5HZUxJki6doWvvoIff0z+a1viT7DiYlcXd/hwOPdctxRD48Z+\nR2WMCZrd4/yffJL817bEn0CrVrlpWs89B3fe6erl1qrld1TGmCBq3x723def4R5bjz9BPvjAXZ0v\nLnbLKttia8aY8tSs6ZZf9+MCr53xV1JJCdx1l6uadcABMGeOJX1jjDedO8OCBbBpU3Jf1xJ/JRQW\nugs0o0a5lTVnzbKLuMYY7zp3hl27YObM5L6uJf44qLq189u2dXfiTpjglmAI0QKBxpgA6NjRLeHw\n6afJfV1L/BX03XfuJqxLLoGjjoL58+HCC/2OyhgTRvvtB7/7XfJn9lji92jHDhg9Gtq0cQfpscfc\nRZmWLf2OzBgTZp07u6GekpLkvaYlfg/ef9/dYn3jjW665qJFMHSo+4hmjDGV0akTbN7sLvImi6Wu\ncsybB6eeCiefDNu3w5tvwltvQXa235EZY1JF587uazKHeyzxl2LBAlcAPSfHLbfwpz+5s/w+ffyr\nkWmMSU0HHQQtWiT3Aq8l/ohdu1zR85NOcrN1pk93d9+uXOmWX9h3X78jNMakqk6d3Bm/hxLoCREz\n8YvIcyKyTkRKrZ4lziMislxEFopIu6h9PUVkWWTfLYkMPFFWrICRI+GQQ9ywztKl7gz/++/h7rth\n//39jtAYk+o6d4Y1a2DNmuScYXo54x8H9Cxnfy+gVeSRBzwJICLVgccj+1sD/USkdWWCLdeECZCd\nTdfu3d0g/IQJpTbbtQvmznU3XR13nEv4o0bBoYfCSy/97wz/N7+pskiNMeZXOm98C4CS/k+Um78S\nxUsFro9EJLucJmcAz0eqvM8UkXoi0gTIBpZHKnEhIvmRtl9VNui9TJgAeXmwZQsCbrJ9Xh4bt2Sw\n8tjzWLnSzbefNQu++AJ+/tmN1XfoAPfe6+66bdYs4VEZY0xsEyZw5KjB7M9qPqUTF333gstnAP37\nV8lLJmKRtqbAqqifV0e2lba9akqR3HYbbNlCDnP4md9QRCabt9RhS16d/zapVs3NwT/nHDcls2dP\nt7aOMcb46rbbqLZ1M7/nMz6lk9u2ZYvLawFO/AkhInm4oSKysrKYMWOG5+d2/f57BGgd+TBRh83U\nYTNZrGXryIto0mQbzZptpVatnf99zldfuUeQFRUVVejfIahSpR9gfQmisPdjd/46m0l8QQd2IVRD\n0e+/58Oq6peqxnzghm0WlbFvDNAv6udlQBPgeGB61PYRwAgvr5eTk6MV0qKFqrsg/utHixYV+z0B\nU1BQ4HcICZEq/VC1vgRR6PuRoPwFzFEP+VVVEzKd8w3gosjsno7ARlVdA8wGWolISxGpCVwQaZt4\n99wDtWv/elvt2m67McYEmQ/5K+ZQj4i8DHQDGorIauAuIANAVZ8CpgG9geXAFuCSyL4SEbkKmA5U\nB55T1cVV0If/jYPddhv6/fdI8+buH62KxseMMSZhfMhfXmb19IuxX4GhZeybhntjqHr9+0P//nw4\nYwbdunVLyksaY0xCJDl/2Z27xhiTZizxG2NMmrHEb4wxacYSvzHGpBlL/MYYk2ZEk7UOaAWIyI/A\nd3E+vSHwnwSG46dU6Uuq9AOsL0GUKv2AyvWlhao28tIwkIm/MkRkjqq29zuOREiVvqRKP8D6EkSp\n0g9IXl9sqMcYY9KMJX5jjEkzqZj4x/odQAKlSl9SpR9gfQmiVOkHJKkvKTfGb4wxpnypeMZvjDGm\nHKFM/LGKuJdXAD5oPPSlm4hsFJH5kcedfsQZi4g8JyLrRGRRGfvDdExi9SUsx6SZiBSIyFcislhE\nri2lTSiOi8e+hOW47CsiX4jIgkhf7i6lTdUeF68L9wflgVvi+VvgYKAmsABovUeb3sDbgAAdgVl+\nx12JvnQDpvodq4e+dAHaUXb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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "from scipy import interpolate\n", "import matplotlib.pyplot as plt\n", "\n", "# もとの点\n", "x = np.array([0,1,2,3])\n", "y = np.array([1,2,3,1])\n", "for i in range(0,4):\n", " plt.plot(x[i],y[i],'o',color='r')\n", "\n", "\n", "# Lagrange補間\n", "f = interpolate.lagrange(x,y)\n", "print(f)\n", "x = np.linspace(0,3, 100)\n", "y = f(x)\n", "plt.plot(x, y, color = 'b')\n", "\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 逆行列から" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "array([[ 1., 0., 0., 0.],\n", " [ 1., 1., 1., 1.],\n", " [ 1., 2., 4., 8.],\n", " [ 1., 3., 9., 27.]])\n", "array([ 1., 2., 3., 1.])\n" ] } ], "source": [ "from pprint import pprint\n", "\n", "x = np.array([0,1,2,3])\n", "y = np.array([1,2,3,1])\n", "n = 4\n", "A = np.zeros((n,n))\n", "b = np.zeros(n)\n", "for i in range(0,n):\n", " for j in range(0,n):\n", " A[i,j] += x[i]**j\n", " b[i] = y[i]\n", "pprint(A)\n", "pprint(b)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "array([[ 1. , 0. , 0. , -0. ],\n", " [-1.83333333, 3. , -1.5 , 0.33333333],\n", " [ 1. , -2.5 , 2. , -0.5 ],\n", " [-0.16666667, 0.5 , -0.5 , 0.16666667]])\n" ] }, { "data": { "text/plain": [ "array([ 1.00000000e+00, -9.99200722e-16, 1.50000000e+00,\n", " -5.00000000e-01])" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import scipy.linalg as linalg # SciPy Linear Algebra Library\n", "\n", "inv_A = linalg.inv(A)\n", "pprint(inv_A)\n", "np.dot(inv_A,b)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 対数関数のニュートンの差分商補間(2014期末試験，25点) \n", "\n", "2を底とする対数関数(Mapleでは$\\log[2](x)$)の$F(9.2)=2.219203$をニュートンの差分商補間を用いて求める．ニュートンの内挿公式は，\n", "\n", "$$\n", "\\begin{array}{rc}\n", "F (x )&=F (x _{0})+\n", "(x -x _{0})f _{1}\\lfloor x_0,x_1\\rfloor+\n", "(x -x _{0})(x -x _{1})\n", "f _{2}\\lfloor x_0,x_1,x_2\\rfloor + \\\\\n", "& \\cdots + \\prod_{i=0}^{n-1} (x-x_i) \\, \n", "f_n \\lfloor x_0,x_1,\\cdots,x_n \\rfloor\n", "\\end{array}\n", "$$\n", "である．ここで$f_i \\lfloor\\, \\rfloor$ は次のような関数を意味していて，\n", "$$\n", "\\begin{array}{rc}\n", "f _{1}\\lfloor x_0,x_1\\rfloor &=& \\frac{y_1-y_0}{x_1-x_0} \\\\\n", "f _{2}\\lfloor x_0,x_1,x_2\\rfloor &=& \\frac{f _{1}\\lfloor x_1,x_2\\rfloor-\n", "f _{1}\\lfloor x_0,x_1\\rfloor}{x_2-x_0} \\\\\n", "\\vdots & \\\\\n", "f _{n}\\lfloor x_0,x_1,\\cdots,x_n\\rfloor &=& \\frac{f_{n-1}\\lfloor x_1,x_2\\cdots,x_{n}\\rfloor-\n", "f _{n-1}\\lfloor x_0,x_1,\\cdots,x_{n-1}\\rfloor}{x_n-x_0} \n", "\\end{array}\n", "$$\n", "差分商と呼ばれる．$x_k=8.0,9.0,10.0,11.0$をそれぞれ選ぶと，差分商補間のそれぞれの項は以下の通りとなる．\n", "\n", "$$\n", "\\begin{array}{ccl|lll}\n", "\\hline\n", "k & x_k & y_k=F_0( x_k) &f_1\\lfloor x_k,x_{k+1}\\rfloor & f_2\\lfloor x_k,x_{k+1},x_{k+2}\\rfloor & f_3\\lfloor x_k,x_{k+1},x_{k+2},x_{k+3}\\rfloor \\\\\n", "\\hline\n", "0 & 8.0 & 2.079442 & & &\\\\\n", "& & & 0.117783 & &\\\\ \n", "1 & 9.0 & 2.197225 & & [ XXX ] &\\\\\n", "& & & 0.105360 & & 0.0003955000 \\\\\n", "2 & 10.0 & 2.302585 & & -0.0050250 &\\\\ \n", "& & & 0.095310 & &\\\\ \n", "3 & 11.0 & 2.397895 & & &\\\\ \n", "\\hline\n", "\\end{array}\n", "$$\n", "それぞれの項は，例えば，\n", "\n", "$$\n", "f_2\\lfloor x_1,x_2,x_3 \\rfloor =\\frac{0.095310-0.105360}{11.0-9.0}=-0.0050250\n", "$$\n", "で求められる．ニュートンの差分商の一次多項式の値はx=9.2で\n", "\n", "$$\n", "F(x)=F_0(8.0)+(x-x_0)f_1\\lfloor x_0,x_1\\rfloor =2.079442+0.117783(9.2-8.0)=2.220782\n", "$$\n", "となる．\n", "\n", "### 差分商補間の表中の開いている箇所[ XXX ]を埋めよ．\n", "### ニュートンの二次多項式\n", "\n", "$$\n", "F (x )=F (x _{0})+(x -x _{0})f _{1}\\lfloor x_0,x_1\\rfloor+(x -x _{0})(x -x _{1})\n", "f _{2}\\lfloor x_0,x_1,x_2\\rfloor\n", "$$\n", "の値を求めよ．\n", "### ニュートンの三次多項式の値を求めよ．\n", "ただし，ここでは有効数字7桁程度はとるように．(E.クライツィグ著「数値解析」(培風館,2003), p.31, 例4改)" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.2192034840549946" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.log(9.2)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-0.005024999999999995\n", "-0.006211500000000002\n" ] } ], "source": [ "x_3 = 11\n", "x_1 = 9\n", "f1_23 = 0.095310\n", "f1_12 = 0.105360\n", "f2_123 = (f1_32-f1_21)/(x_3-x_1)\n", "print(f2_123)\n", "f1_01 = 0.117783\n", "x_0 = 8\n", "x_2 = 10\n", "f2_012 = (f1_12-f1_01)/(x_2-x_0)\n", "print((0.105360-0.117783)/(10-8))" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.2192908399999998" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = 9.2\n", "F_0 = 2.079442\n", "F_0 + (x-x_0)*(f1_01)+(x-x_0)*(x-x_1)*f2_012" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.2192149039999998" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f3_0123 = 0.0003955000\n", "F_0 + (x-x_0)*(f1_01)+(x-x_0)*(x-x_1)*f2_012+(x-x_0)*(x-x_1)*(x-x_2)*f3_0123" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 補足\n", "\n", "テキストに紹介したコードによって求められるNewton差分商補間の様子を以下に示しておく．" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# https://stackoverflow.com/questions/14823891/newton-s-interpolating-polynomial-python\n", "# by Khalil Al Hooti (stackoverflow)\n", "\n", "\n", "def _poly_newton_coefficient(x,y):\n", " \"\"\"\n", " x: list or np array contanining x data points\n", " y: list or np array contanining y data points\n", " \"\"\"\n", "\n", " m = len(x)\n", "\n", " x = np.copy(x)\n", " a = np.copy(y)\n", " for k in range(1,m):\n", " a[k:m] = (a[k:m] - a[k-1])/(x[k:m] - x[k-1])\n", "\n", " return a\n", "\n", "def newton_polynomial(x_data, y_data, x):\n", " \"\"\"\n", " x_data: data points at x\n", " y_data: data points at y\n", " x: evaluation point(s)\n", " \"\"\"\n", " a = _poly_newton_coefficient(x_data, y_data)\n", " n = len(x_data) - 1 # Degree of polynomial\n", " p = a[n]\n", " for k in range(1,n+1):\n", " p = a[n-k] + (x -x_data[n-k])*p\n", " return p" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 4. -6. 3.4]\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "from scipy import interpolate\n", "import matplotlib.pyplot as plt\n", "\n", "x = [-1, 0, 1]\n", "y = [4., -2., -1.2]\n", "\n", "for i in range(0,3):\n", " plt.plot(x[i],y[i],'o',color='r')\n", "\n", "\n", "print(_poly_newton_coefficient(x,y))\n", "\n", "xx = np.linspace(-1,4, 100)\n", "yy = newton_polynomial(x, y, xx)\n", "plt.plot(xx, yy, color = 'b')\n", "\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 4. -6. 3.4]\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = [-1, 0, 1]\n", "y = [4., -2., -1.2]\n", "\n", "for i in range(0,3):\n", " plt.plot(x[i],y[i],'o',color='r')\n", "\n", "\n", "print(_poly_newton_coefficient(x,y))\n", "\n", "xx = np.linspace(-1,4, 100)\n", "yy = newton_polynomial(x, y, xx)\n", "plt.plot(xx, yy, color = 'b')\n", "\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 数値積分(I)\n", "次の関数\n", "\n", "$$\n", "f(x) = \\frac{4}{1+x^2}\n", "$$\n", "を$x = 0..1$で数値積分する．\n", "1. $N$を2,4,8,…256ととり，$N$個の等間隔な区間にわけて中点法で求めよ．(15)\n", "1. 小数点以下10桁まで求めた値3.141592654との差をdXとする．dXと分割数Nとを両対数プロット(loglogplot)して比較せよ(10)\n", "(2008年度期末試験）" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np\n", "\n", "def func(x):\n", " return 4.0/(1.0+x**2)\n", "\n", "def mid(N):\n", " x0, xn = 0.0, 1.0\n", "\n", " h = (xn-x0)/N\n", " S = 0.0\n", " for i in range(0, N):\n", " xi = x0 + (i+0.5)*h\n", " dS = h * func(xi)\n", " S = S + dS\n", " return S\n", "\n", "def trape(N):\n", " x0, xn = 0.0, 1.0\n", "\n", " h = (xn-x0)/N\n", " S = func(x0)/2.0\n", " for i in range(1, N):\n", " xi = x0 + i*h\n", " dS = func(xi)\n", " S = S + dS\n", " S = S + func(xn)/2.0\n", " return h*S\n", "\n", "x, y = [], []\n", "for i in range(1,8):\n", " x.append(2**i)\n", " y.append(abs(mid(2**i)-np.pi))\n", "\n", "x2, y2 = [], []\n", "for i in range(1,8):\n", " x2.append(2**i)\n", " y2.append(abs(trape(2**i)-np.pi))\n", "\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "plt.plot(x, y, color = 'r')\n", "plt.plot(x2, y2, color = 'b')\n", "plt.yscale('log')\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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