textbook/GettingStart.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## 新規ノートブックの作成\n",
    "新規にノートブックを作成するには、「Files」タグの右端にある「New」プルダウンメニューから「SageMath 7.2」を選択します。\n",
    "\n",
    "![New Button](images/New_Menu.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## セルを評価してみよう\n",
    "In[ ]: と表示されている入力用セルに式を入力して、その値を評価してみましょう。\n",
    "\n",
    "以下の2行を入力して、シフトキーを押さえながらリターンキーを押してください（以下shift-returnと表します）。または既に入力されているIn [n]のセルの式をクリックして選択した後、shift-returnを押します。\n",
    "\n",
    "5/6と数値ではなく、分数で返ってくるところが数式システムならではの芸当です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5/6"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 1/2 + 1/3\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "複雑な数式を入力すると、テキストベースの結果では分かりづらいです。そんな時には、show関数を使って表示すると数式がきれいに表示されます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{5}{6}</script></html>"
      ],
      "text/plain": [
       "5/6"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 多項式\n",
    "中学の数学に出てきた多項式をSageで処理してみましょう。\n",
    "\n",
    "以下の様な3次多項式を持つ関数$f(x)$をSageで定義します。\n",
    "$$\n",
    "f(x) = x^3 - x^2 - 2x\n",
    "$$\n",
    "\n",
    "最初に変数xをvar関数で定義します。次に上記の多項式を変数fにセットします。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} - x^{2} - 2 \\, x</script></html>"
      ],
      "text/plain": [
       "x^3 - x^2 - 2*x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = var('x')\n",
    "f(x) = x^3 - x^2 - 2*x\n",
    "show(f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "多項式の因数分解には、factor関数を使います。\n",
    "\n",
    "因数分解の結果から、関数fはx=-1, x=0, x=2でX軸と交わります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x + 1)*(x - 2)*x"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 多項式のグラフ\n",
    "先ほどの3次多項式$f(x)$をプロットして、X軸と交差する位置を確認してみましょう。\n",
    "\n",
    "plot関数には、表示したい関数とその範囲を指定します。ここではx=-2.5からx=2.5の範囲を指定します。\n",
    "\n",
    "Sageの図化機能を使うことで簡単に$f(x)$の特徴を理解することができます。\n",
    "\n",
    "一番最初は、以下の様なワーニングがでますが、2回目以降はでません。\n",
    "```\n",
    "/usr/lib/sagemath/local/lib/python2.7/site-packages/matplotlib-1.5.1-py2.7-linux-x86_64.egg/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.\n",
    "  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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Oq/SPnBzp/PNtGcynn0rdu5ft6w73fUbw1a0r3XGHvVmbM8dOoXrsMTvBq3dve+PG8ZDY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\nTWwUAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(f, [x, -2.5, 2.5], figsize=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 関数の極\n",
    "3次多項式$f(x)$の極は、関数の接線の傾きが０（傾きがX軸と平行）の場所です。 関数の極を求めるには$f(x)$を微分し、その値が０となるｘを求めます。\n",
    "\n",
    "関数の微分には、diff関数を使います。diff関数には、微分したい関数とその変数を引数とします。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, x^{2} - 2 \\, x - 2</script></html>"
      ],
      "text/plain": [
       "3*x^2 - 2*x - 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df(x) = diff(f, x)\n",
    "show(df(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 関数の解\n",
    "関数solverは関数が０となる変数の値を求めます。solverの引数は、解を求めたい関数とその変数を指定します。\n",
    "\n",
    "関数fとそれを微分した関数dfのグラフを比べると関数dfがX軸と交わる点で、 関数fの接線の傾きが0となっていることが見て取れます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[x = -\\frac{1}{3} \\, \\sqrt{7} + \\frac{1}{3}, x = \\frac{1}{3} \\, \\sqrt{7} + \\frac{1}{3}\\right]</script></html>"
      ],
      "text/plain": [
       "[x == -1/3*sqrt(7) + 1/3, x == 1/3*sqrt(7) + 1/3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ZVcgnQaNkEZHItnkzDBsGZ5wBPl9wnlOFfBIOHSVv2WI7jYiIhNuQIaaIe/cO3nOqkE9S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\nXeI0gQDccos5WW7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      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = solve(df, x)\n",
    "show(sol)\n",
    "plot(df, [x, -2.5, 2.5], figsize=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 数値解\n",
    "Sageは数式処理システムなので、関数solverの結果が数式で返ってきます。 数値解が欲しい場合にはfind_root関数を使います。\n",
    "\n",
    "このようにSageを使って関数fをプロットしたり、解を求めることによって関数fの理解を深めることができます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-0.5485837703548542, 1.2152504370215302)\n"
     ]
    }
   ],
   "source": [
    "print(  find_root(df, -2, 0), find_root(df, 0, 2) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 7.2",
   "language": "",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}