{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "
\n", "数値計算2017 試験解答例\n", "
\n", "
\n", "
\n", "cc by 西谷@関西学院大・理工 2017 \n", "
\n" ] }, { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

1  簡単な行列計算
2  誤差(2次方程式)
3  積分
4  最小2乗法(自動車の加速性能)
5  常微分方程式(ボールの自由落下)
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 簡単な行列計算" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sympy import *\n", "import numpy as np" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 4. -1. -1.]\n", " [ 1. 2. -1.]\n", " [ 3. -1. 0.]]\n", "[ 1.+0.j 3.+0.j 2.+0.j]\n", "[[-0.40825 -0.70711 -0.57735]\n", " [-0.40825 -0. -0.57735]\n", " [-0.8165 -0.70711 -0.57735]]\n" ] } ], "source": [ "import scipy.linalg # SciPy Linear Algebra Library\n", "\n", "np.set_printoptions(precision=5, suppress=True)\n", "\n", "dA = np.diag([1,2,3])\n", "P = np.array([[1,1,2],[1,1,1],[1,0,1]]).transpose()\n", "iP = scipy.linalg.inv(P)\n", "A = P.dot(dA).dot(iP)\n", "\n", "print(A)\n", "l, PP = scipy.linalg.eig(A)\n", "print(l)\n", "print(PP)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 誤差(2次方程式)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from decimal import *\n", "from numpy import sqrt\n", "def solve_normal_formula(a,b,c):\n", " x0=(-b-sqrt(b**2-4*a*c))/(2*a)\n", " x1=(-b+sqrt(b**2-4*a*c))/(2*a)\n", " return (x0,x1)\n", "\n", "def solve_precise_formula(a,b,c):\n", " x0=(-b-sqrt(b**2-4*a*c))/(2*a)\n", " x1=c/(a*x0)\n", " return (x0,x1)\n" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(Decimal('-39.95'), Decimal('-0.05'))\n", "(Decimal('-39.95'), Decimal('-0.05006'))\n" ] } ], "source": [ "getcontext().prec = 4\n", "\n", "print(solve_normal_formula(Decimal('1'),\n", " Decimal('40'),\n", " Decimal('2')))\n", "print(solve_precise_formula(Decimal('1'),\n", " Decimal('40'),\n", " Decimal('2')))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 積分" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1/3" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "x = symbols('x')\n", "integrate(x**2,(x,0,1))" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 1 0.5000000000 1.66667e-01\n", " 2 0.3750000000 4.16667e-02\n", " 4 0.3437500000 1.04167e-02\n", " 8 0.3359375000 2.60417e-03\n", " 16 0.3339843750 6.51042e-04\n", " 32 0.3334960938 1.62760e-04\n", " 64 0.3333740234 4.06901e-05\n", " 128 0.3333435059 1.01725e-05\n", " 256 0.3333358765 2.54313e-06\n", " 512 0.3333339691 6.35783e-07\n" ] } ], "source": [ "def func(x):\n", " return 1.0*x**2\n", "\n", "x, y = [], []\n", "for j in range(0, 10):\n", " N, x0, xn = 2**j, 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", "\n", " S = S + func(xn)/2.0\n", " print(\"%4d %15.10f %10.5e\" % (N, h*S, h*S-1/3))\n", " x.append(N)\n", " y.append(abs(h*S-1/3))\n" ] }, { "cell_type": "code", "execution_count": 17, "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.yscale('log')\n", "plt.xscale('linear') # or log\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "必要な分点の数は,計算結果によると512ぐらい,プロットからは450ぐらい." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 最小2乗法(自動車の加速性能)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 17.34679]\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.optimize import curve_fit\n", "\n", "def f(x, a):\n", " return a*x**2\n", "\n", "xdata = np.array([0,0.751,1.113,1.504])\n", "ydata = np.array([0,10,20,40])\n", "plt.plot(xdata,ydata, 'o', color='r')\n", "\n", "params, cov = curve_fit(f, xdata, ydata)\n", "print(params)\n", "\n", "x =np.linspace(0,5,20)\n", "a = params[0]\n", "y = f(x,a)\n", "plt.plot(x,y, color='b')\n", "\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[-2.40099080206130, 2.40099080206130]\n" ] } ], "source": [ "from sympy import *\n", "t = symbols('t')\n", "s1 = solve(f(t,a)-100,t)\n", "print(s1)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "149.938100425429" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a*s1[1]*60*60/1000" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[-4.80198160412261, 4.80198160412261]\n" ] }, { "data": { "text/plain": [ "299.876200850858" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s2 = solve(f(t,a)-400,t)\n", "print(s2)\n", "a*s2[1]*60*60/1000" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 常微分方程式(ボールの自由落下)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def euler(x0, v0):\n", " v1 = v0 - g * dt\n", " x1 = x0 + v0 * dt\n", " return x1, v1" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "\n", "def my_plot(xx, vv, tt):\n", " plt.plot(tt, xx, color = 'b')\n", " plt.plot(tt, vv, color = 'r')\n", " plt.grid()\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g, dt=9.8, 0.01\n", "tt,xx,vv=[0.0],[0.0],[19.6]\n", "t = 0.0\n", "\n", "for i in range(0,500):\n", " t += dt\n", " x, v = euler(xx[-1],vv[-1])\n", " tt.append(t)\n", " xx.append(x)\n", " vv.append(v)\n", "\n", "\n", "my_plot(xx, vv, tt)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(tt, xx, color = 'b')\n", "plt.plot(tt, vv, color = 'r')\n", "plt.xlim(3.98,4.02)\n", "plt.ylim(-1,1)\n", "\n", "plt.grid()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "この条件では4.01秒後となります.解析的に得られる解4.00秒を数値解で得るには,Euler法の小時間刻みを細かくするか,Runge-Kutta法を使う必要があります." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "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": "3.6.1" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 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