\n",
" jupyterでhtml.tableを使うと以前のようにcssが有効にならないので、table関連のCSSを最初に読み込みます。\n",
"
\n",
" \n",
" また、jupyterのノートブックでもSageのノートブック同様に使えるようにsage_util.pyを読み込みます。\n",
"
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%HTML\n",
""
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# ユーティリティ\n",
"load('script/sage_util.py')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\tよく使われる定数
\n",
"\t\n",
"\t\t数式でよく使われる定数をSageで使うには以下の様に表します。\n",
"\t
\n",
" \n",
" 注意）無限大は、oを2個並べたooです。\n",
"
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"
"
],
"text/plain": [
"<__main__.Table2Html instance at 0x7f9277ed0098>"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# jupyter用のdisplayメソッド\n",
"hdr=[\"円周率\", \"自然体数の底\", \"虚数単位\", \"無限大\"]\n",
"sts=[pi, e, I, oo]\n",
"# sage_util.pyのTable2Htmlを使用\n",
"Table2Html([(h, \"$%s$\"%latex(st), str(st)) for (h, st) in zip(hdr, sts)])"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"\n",
"\t基本的な計算
\n",
"\t\n",
"\t\t四則演算をはじめ、Sageで使われる基本的な計算の表現方法を以下に示します。\n",
"\t
\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"<__main__.Table2Html instance at 0x7f927768ecf8>"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# よく使われる表現\n",
"var('a b x n')\n",
"hdr=[\"積\", \"商\", \"累乗\", \"平方根\", \"n乗根\", \"絶対値\", \"自然対数\", \"階乗\"]\n",
"sts=[a*b, a/b, a^b, sqrt(x), x^(1/n), abs(x), log(x), factorial(n)]\n",
"Table2Html([(h, \"$%s$\"%latex(st), str(st)) for (h, st) in zip(hdr, sts)])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1から10までの和は55ですが、これをリストとsum関数を使って計算すると、 以下の様になります。"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"55\n",
"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n",
"55\n"
]
}
],
"source": [
"# 1から10までの和を計算\n",
"print( 1+2+3+4+5+6+7+8+9+10 )\n",
"# リスト使って計算\n",
"L = range(1,11); print L \n",
"print sum(L) # リストLの和を求める "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\tリスト内包を使うと簡単にリストの要素を変更することができます。\n",
"\t\t先ほどのリストLの要素を自乗に変えて、和を求めてみましょう。\n",
"
\n",
"\tリスト内包の書式\n",
"\t[ 式 for 変数 in リスト ]\n",
"\n",
"\t\n",
"\n",
""
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]\n",
"385\n"
]
}
],
"source": [
"L2 = [i^2 for i in L]; print L2 \n",
"print sum(L2) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\tsumと並んでリストの要素の積を求めるprodもよく使われます。\n",
"\t\t1から5までの積をprodを使って計算してみましょう。\n",
"\t
\n",
"\t\n",
"\t\t今回は、range関数の代わりにジェネレータを使って１から５までの\n",
"\t\tリストを生成し、計算してみます。\n",
"
\n",
"\t(1..5)\n",
"\n",
"\t\t\n",
"\t\n",
""
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"120\n",
"120\n"
]
}
],
"source": [
"# 1から5までの積を計算\n",
"print 1*2*3*4*5 \n",
"# リスト生成をまとめて\n",
"print prod(1..5) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t式の計算・簡素化
\n",
"\t\n",
"\t\t次に中学で習った式の因数分解や展開もSageを使うと簡単に計算することができます。\n",
"\t
\n",
"\n",
"\n",
"\n",
"\t展開
\n",
"\t\n",
"\t\t以下の関数f1をSageを使って展開してみます。展開にはexpand関数を使います。\n",
"$$\n",
"\t\tf1(x) = (x - 1)(x^2 -1)\n",
"$$\t\t\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"(x^2 - 1)*(x - 1)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"x^3 - x^2 - x + 1"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"f1 = (x - 1)*(x^2 - 1); show(f1) # f1を定義\n",
"f2 = expand(f1); show(f2) # f1を展開し、f2に代入"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\t多項式の係数は、coefficientsやcoefficient関数で取得できます。\n",
"\t
\n",
"\t\n",
"\t\t以下に、f2の係数をcoeeficents関数で取得する方法とxの2次の項の係数をcoefficient関数で取得する方法を例として示します。\n",
"\t
\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1, 0], [-1, 1], [-1, 2], [1, 3]]\n",
"-1\n"
]
}
],
"source": [
"# 多項式の係数\n",
"print f2.coefficients(x) # 変数の[係数, 次数]のペアのリストを返す\n",
"print f2.coefficient(x, 2) # 変数と次数を指定して係数をを返す"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t因数分解
\n",
"\t\n",
"\t\tf1を展開した結果f2を因数分解してみましょう。因数分解にはfactor関数を使用します。\n",
"\t
\n",
"\t\n",
"\t\t結果がf1と異なりますが、\n",
"$$\n",
"\t\t(x^2 - 1) = (x - 1)(x + 1)\n",
"$$\t\t\n",
"\t\tの関係から、正しい結果になっていることが分かります。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"(x + 1)*(x - 1)^2"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(factor(f2)) # f2を因数分解する"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t部分分数分解
\n",
"\t\n",
"\t\t分数で表される式の積分やラブラス逆変換では部分分数分解を利用することがあります。\n",
"\t
\n",
"\t\n",
"\t\t以下の様な関数f3を因数分解して、分数式f4に変えます。\n",
"$$\n",
"\t\tf3(x) = \\frac{1}{(x - 2)} + \\frac{1}{(x + 2)}\n",
"$$\t\t\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"1/(x + 2) + 1/(x - 2)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# 分数の因数分解\n",
"f3 = 1/(x+2)+1/(x-2)\n",
"show(f3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\t式の部分分数分解には、prtial_fraction()メソッドを使います。f4にprtial_fractionメソッドを呼び出すとf3と同じ結果となります。\n",
"\t
\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"2*x/((x + 2)*(x - 2))"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"1/(x + 2) + 1/(x - 2)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"f4 = factor(f3); show(f4) # 分数式f3の因数分解\n",
"show(f4.partial_fraction()) # 部分分数分解"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t式の整理
\n",
"\t\n",
"\t\t式を整理して簡単化する関数として、simplifyがあります。simplifyでは不要な項を消去するだけなので、\n",
"\t\tf1のような因数分解で整理されて式の場合には結果が変わりません。\n",
"\t
\n",
"\t\n",
"\t\tさらに突っ込んだ整理をする場合には、simplify_fullメソッドを使います。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"I"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"(x^2 - 1)*(x - 1)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"x^3 - x^2 - x + 1"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(simplify(I + x -x)) # 簡単化で不要な項を消去\n",
"show(simplify(f1)) # 簡単化ではf1はそのまま\n",
"show(f1.simplify_full()) # simplify_fullメソッドを使うと展開されて整理される "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t三角関数
\n",
"\t\n",
"\t\t三角関数も数式として計算するため、$ sin (\\pi/4) $も数値ではなく、\n",
"\t\t$ \\frac{1}{2} \\sqrt{2} $の式が返ってきます。\n",
"\t
\n",
"\t\n",
"\t\t値を得るには、N関数を使用します。\n",
"\t
\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1/2*sqrt(2) 0.707106781186548\n"
]
}
],
"source": [
"print sin(pi/4), N(sin(pi/4)) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\tまた三角関数への入力単位は度ではなく、ラジアンで指定します。\n",
"\t
\n",
"\t\n",
"\t\t度とラジアンの変換には、以下の様な関数rad, degを定義すると便利です。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# 度とラジアンの変換関数\n",
"rad(x) = x*pi/180\n",
"deg(x) = x*180/pi"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1/2*sqrt(2) 1/2*sqrt(2)\n",
"1/4*pi 45\n"
]
}
],
"source": [
"print sin(rad(45)), sin(pi/4)\n",
"print atan(1), deg(atan(1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t三角関数の簡単化
\n",
"\t\n",
"\t\t三角関数や指数関数を含む式の簡単化には simplify_fullメソッドを使います。\n",
"\t
\n",
"\t\n",
"\t\t例として、sin関数の倍角公式をsimplify_fullメソッドで求めてみます。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"sin(2*x)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"2*cos(x)*sin(x)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# 倍角公式\n",
"fs = sin(2*x)\n",
"show(fs)\n",
"show(fs.simplify_full())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\t更に、simplify_fullでは三角関数の公式を活用して式を整理します。\n",
"\t
\n",
"\t\n",
"\t\t以下の例では、以下の三角関数の公式を使って簡単化しています。\n",
"$$\n",
"\t\tcos^2 x + sin^2 x = 1\n",
"$$\t\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"cos(x)^2 - sin(x)^2"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
"2*cos(x)^2 - 1"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"f4 = cos(x)^2-sin(x)^2\n",
"show(f4) # cos(x)^2 + sin(x)^2 = 1を使って\n",
"show(f4.simplify_full()) # 簡素化する"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t導関数
\n",
"\t\n",
"\t\t高校で習った、導関数をSageを使って導いてみましょう。\n",
"\t\t関数f(x)と平均変化率g(x)を以下の様に定義します。\n",
"$$\n",
"\t\tf(x) = \\frac{1}{2} x^3\n",
"$$\n",
"$$\n",
"\t\tg(x)= \\frac{f(x+h)−f(x)}{h}\n",
"$$\n",
"\t
\n",
"\t\n",
"\t\t変数x, hと関数f, gをSageで定義します。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"x |--> 1/2*x^3"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x, h = var('x h')\n",
"f(x) = x^3/2\n",
"show(f)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"1/2*((h + x)^3 - x^3)/h"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# 平均変化率\n",
"g =(f(x + h) - f(x))/h\n",
"show(g)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\tgを展開して整理すると以下の様になり、h→0の極値（limit）をとった時、hが掛からないxの2次の項のみが残ります。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"1/2*h^2 + 3/2*h*x + 3/2*x^2"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# 展開して整理すると\n",
"g1 = g.simplify_full()\n",
"show(g1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\t求める導関数は、以下の様になります。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"3/2*x^2"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# h→0の極値を取ると導関数が求まる\n",
"g2 = limit(g1, h=0)\n",
"show(g2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t特殊な関数のリミット
\n",
"\t\n",
"\t\t次のような特殊な関数fを考えます。\n",
"$$\n",
"\t\tf(x) = \\frac{sin(x)}{x}\n",
"$$\t\t\n",
"\t
\n",
"\t\n",
"\t\tこの関数にx=0を代入するとゼロ割のエラーとなりますが、limitを使ってx→0を求めると1となります。\n",
"\t
\n",
"\t\n",
"\t\tSageではグラフの表示でも極値をきちんと計算しているので、上記のf(x)もきちんと表示することができます。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"ename": "ValueError",
"evalue": "power::eval(): division by zero",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 1\u001b[0m \u001b[1;31m# 特殊な関数のリミット\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2\u001b[0m \u001b[0mf\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0msin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m/\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 3\u001b[1;33m \u001b[1;32mprint\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mInteger\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[1;32m/usr/lib/sagemath/src/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression.__call__ (/usr/lib/sagemath//src/build/cythonized/sage/symbolic/expression.cpp:28301)\u001b[1;34m()\u001b[0m\n\u001b[0;32m 4974\u001b[0m \u001b[0mz\u001b[0m\u001b[1;33m^\u001b[0m\u001b[1;36m2\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m^\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 4975\u001b[0m \"\"\"\n\u001b[1;32m-> 4976\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_parent\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_call_element_\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 4977\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 4978\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mvariables\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32m/usr/lib/sagemath/src/sage/symbolic/ring.pyx\u001b[0m in \u001b[0;36msage.symbolic.ring.SymbolicRing._call_element_ (/usr/lib/sagemath//src/build/cythonized/sage/symbolic/ring.cpp:10250)\u001b[1;34m()\u001b[0m\n\u001b[0;32m 867\u001b[0m \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"the number of arguments must be less than or equal to %s\"\u001b[0m\u001b[1;33m%\u001b[0m\u001b[0mlen\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mvars\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 868\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 869\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0m_the_element\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0md\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 870\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 871\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0msubring\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;32m/usr/lib/sagemath/src/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression.substitute (/usr/lib/sagemath//src/build/cythonized/sage/symbolic/expression.cpp:27844)\u001b[1;34m()\u001b[0m\n\u001b[0;32m 4883\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 4884\u001b[0m return new_Expression_from_GEx(self._parent,\n\u001b[1;32m-> 4885\u001b[1;33m self._gobj.subs_map(smap, 0))\n\u001b[0m\u001b[0;32m 4886\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 4887\u001b[0m \u001b[0msubs\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0msubstitute\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;31mValueError\u001b[0m: power::eval(): division by zero"
]
}
],
"source": [
"# 特殊な関数のリミット\n",
"f = sin(x)/x\n",
"print f(x=0)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"1"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"limit(f, x=0) "
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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X7kiIfAeTMFExbNnCrUlXatlStmxxSprIdUzCRG46cwbYuVO25dAlFSoAN93EJEzkDiZh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\n7vjqq6+wdetWdO7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/ANdTowYwerQk4WsVl1jlTcFd338vU9E33GDsda/ka9f1FF/7e5g/\nf/7FdWEjjza0yr9bQoKsh770krHXLSlvX/epp+Q43AEDpMGLTgHuPFgphWwX5knPnz+PLDd2Rv/4\noxQkPf64fEop6qnuXtdVZrvu8OFSlPbii9LS0shrX4+Zr5ubK9NI48df+h0xc7zeuq7zer4Sr6ev\nW7FiFmrUAL75Buja1bjr+trfw5XXPXlStiMNGABUqVK85hVW+HtwevNNoFMn6UPxzTfuHd3oarxB\nQUFwXGd/rVu9o509aomIiOjaXOnn7lYSdnUk7Krz54HevaXhuvNsWBInT8rBBHFxsoeYZFbgnXfk\n3FO7bKtwRVZWFqpXr45Dhw7Z+gCHgj75RGaU9u2To0Pt7vhx2e75wAPFn4q2KufvyttvS2GskQwf\nCRttzBiZEli9mltOCvPSSzJ9tHcvEBmpOxr9OnWSI+vY0ORyPEXpakeOyGtm/nz5IGt3zz0ne4L3\n75epaLrcww8DH38sDZOaN/fuvbUVZs2aJad3TJ/OBFyU0aOlr2tR68J2cvq0vEC6dNEdCfmCqlVl\n5Pf117oj0e/ECRnsPPIIE3BR3noLaNRIWiWfOOHde2tJwj/+CAwbBgwdKu0pqXAVKkirtZkzeTzb\n6tVSmBUTozsS8hUxMVJwY/d+7P/+t3Ths+ohOEYIDJStsSdPAvfe692uhV5PwgcOSDXarbfKpw8r\nH8xghJEjgfLl7bmxvqCvv5ata/Xr646EfEWPHnJ6zk8/6Y5EnxMnJAk/8ghrbq6nTh05q2DtWmDU\nKO99ePNqEs7OBmJjZYS3eDF7/7qiQgXgySelleWhQ7qj0UMp4KuvgJ49+aGNXHfbbbKcY+cp6Tfe\nkBkkjoJd06mTbA+dMUMKtbyhxEl46dKliImJQWhoKPz8/LBjx46rHnPu3DkMH/4oQkND8dNPQahb\n9x7k5/9Z0lvbxogRQFCQdHnxlMGDB8PPz++yP3feeafnbuiGPXukyrVHD92RkC8pXVreVFeu1B3J\n5SZOnHjVa61x48aG3+fkSRkFDxsGhIcbfnnLevBBKRoePfoHtGsXi8jISPj5+WF5Ie21JkyYgKpV\nq6Js2bK44447sGfPHrfvV+IkfPr0aXTo0AGvvvpqkaXYjz32GD75ZAXy8j7Fm2+uw6lTR9C3b9+S\n3to2goLkk+wHHwC//+65+/Ts2RPp6elIS0tDWlqaaboEffWVdMhiv2hyV48eUoNy6pTuSC7XpEmT\ny15rP/74o+H3ePNN4Nw5Ob6P3PPqq0CrVqeRktIcTz/9TqG5bcqUKXj77bcxc+ZMJCUloVy5coiJ\niUFubq57N1MGOXDggHI4HGr79u2XfT8zM1MFBNyggCVq2jT53q5du5TD4VCbNm0y6vaWl52tVGio\nUkOGeOb6gwYNUnfddZdnLl5CPXsq1bWr7ijMKzMzUwFQmZmZukMxnd27lQKU+vxz3ZFc8sILL6io\nqCiP3uPkSaUqVlRq9GiP3sbSMjOVuuUWpapVU8rhcKjPPvvssv8eERGhpjmTmpLXYWBgoFq4cKFb\n9/H4mvDUqck4f/48Rozoiscfl+/ddNNNqFGjBhITEz19e8soX14+0c6aJcVtnrBmzRqEhYWhYcOG\nGD58OI4fP+6ZG7nhzBlgzRpZDyZyV/36QK1a5puSTk1NRWRkJOrWrYv7778fhwwu+HjzTXntcBRc\nfBUqSD2Bv7/UpRTsU7V//36kpaWha4G+qBUqVEDbtm3dzmseTcIrVwKTJqXB3/8GvPHG5U0EwsLC\nkHatEwroKsOHAyEhwP/9n/HX7tmzJ+bMmYPvv/8er776KtauXYs777wTSvP+jnXr5M2E68FUHA6H\n/O6YKQm3a9cOs2fPxsqVKzFjxgzs378ft99+O06fPm3I9bOypP/CkCGyX5qKr2pV2eYGSC9/Z7vo\ntLQ0OBwOhF1Rcl6cvOZWEp43bx6CgoIQFBSEChUqYP369UU+9vvvZStS06ZSBe1nmvOafFe5cvLJ\n9qOPpPNNcRX279ivXz/87W9/w80334zY2Fh88cUXSEpKwpo1awyLvzi+/hqoXh3wQN0K2URMDJCa\nqv+0HKeYmBj07dsXTZo0wR133IEvv/wSJ06cwKJFiwy5/ltvSXObceMMuZztNWggH+YOH5YZOaPP\nencrNfbp0wfbt2/H9u3bsW3bNrRq1arQx61aBfztb9IJa9KkcOTm5l514kR6ejrCWbLntkcekdFw\nSSqlXfl3rF27NkJDQ4tV7Wekr76SkQy3JlFxdekCBASYazRcUHBwMBo0aGDIay0rS9pTDhnCVrdG\nmzhRzqm+804gKCgcSimkp6df9pji5DW3knC5cuVQp06di39Kly592X93OBzYtEkScMeOcj5wdHRL\nBAQEYNWqVRcft3v3bhw8eBDt27d3K1iSfY9jxwKzZxd/bfh6/44AcPjwYRw7dgwRERElirck9u8H\ndu/merCr4uLiEBsba5qqdrOoUAFo3968SfjUqVPYu3evIa+1t9+WSnCOgo3XoIFMTW/fDgwfXhth\nYeGX5bWsrCxs2rQJ0dHR7l24pBVkx48fV9u2bVMrVqxQDodDTZ68UPXsuU0dOJB28TGPPPKIqlWr\nllq9erXasmWLio6OVrfddltJb21bp04pVbmyUg8/bNT1TqmnnnpKbdy4UR04cEB99913qmXLlqph\nw4YqNzfXmJsUwzvvKBUQIFWKVDRWR1/fyy8rVb68UufO6Y5EqSeffFKtXbtWHThwQK1fv15169ZN\nValSRWVkZJToullZSoWEKDV8uEGB2typU6fUtm3bVEpKinI4HGr69Olq27ZtaunSg2rgQKUmTZqi\nQkJC1PLly9WOHTtUnz59VL169dQ5N3/JSpyEZ8+erRwOh/Lz87vsz8SJEy8+5uzZs2rEiBGqUqVK\nqnz58uqee+5R6enpJb21rb36qiSoAwdKfq0zZ86omJgYFRYWpkqXLq1q166thg0bpv7888+SX7wE\nYmOV6thRawg+gUn4+pKTZavSqlW6I1EqLi5ORUZGqsDAQFW9enUVHx+v9u3bV+LrvvKKUqVKKXXw\noAFBklqzZk2huW3w4MEXH/P888+riIgIVaZMGdW9e3eVmprq9n20HmVIxXf6tGy96NtXWqxZzdmz\nQGgo8OyzwNNP647G3HiU4fUpBdSoIcWib76pOxrjnTol7wf9+smZ2+Q7WLPso8qVky5as2YBBw/q\njsZ4330nHzT69NEdCVmBwwH8/e/AsmXWPFUpIUGKsviB1fcwCfuw4cOB4GBrnrC0ZAnQsKGc8Ulk\nhLvukkNQkpN1R2KsU6eAqVOBBx6Q0T75FiZhH1a+vJyw9MEH1hoNnz8vlfV33607ErKS22+X7X1L\nl+qOxFjvvgtkZgLjx+uOhIqDSdjHPfqobMGYPFl3JMZZtw44fpxJmIwVEAD07m2tJHz6NPDaa8Dg\nwUDNmrqjoeJgEvZxBUfDVjlveMkSmVZr0UJ3JGQ1d90F/Pqr7D+3gnffBU6c4CjYlzEJW8Cjj0oy\ntsJo+MIFScJ3380uWWS87t2l4Y0VRsM5OTIKHjRIKqPJNzEJW0BQkIyG339f+pv6sk2bgKNHORVN\nnlGmjLRBtUISTkiQZZt//Ut3JFQSTMIWMWKEjIanTNEdScksWQKEhQHudn4jclXfvkBSkueOBPWG\nrCyZ+XrwQaB2bd3RUEkwCVtEUBAwZgwwcybwxx+6oymeCxeAhQtl3c7fX3c0ZFWxsTIlvWCB7kiK\nb/p0Kcp69lndkVBJMQlbyMiR0sTDV0fDa9ZIcdmAAboj8U08wME15ctLE5hPPtEdSfEcOwa8/rr0\nCahWTXc0VFJsW2kx//d/8mffPt870HvgQCAxUSpXWZTlOratdN8XX8h2pR07gFtu0R2Ne55+Wk5L\n2rcPqFJFdzRUUhwJW8zIkTLV5muV0qdOAZ9+KqNgJmDytO7dpXHHvHm6I3FPWpr0vn7sMSZgq2AS\ntpjgYOCJJ4D//Me3umh9+qmscf3zn7ojITu44QY57GDePKlF8BWTJgGlS8tuCLIGJmELGj0aqFgR\neOEF3ZG47qOPgM6d2fWHvKd/f/mgun697khcc/CgfLh+6il5fZM1MAlbUPnywDPPSGL79Vfd0Vzf\n778Dq1fLmjCRt9x6K1Cnjuyv9wUvvigzXaNG6Y6EjMQkbFFDhwLVqwPPPac7kuubO1equvv21R0J\n2YmfH/Dww8CiRdL60cxSU4HZs6U9ZfnyuqMhIzEJW1Tp0sDEibLWunGj7miKphQwZ44kYL65kLcN\nHiynds2dqzuSa3v+eSA8HHjkEd2RkNGYhC3s/vuBZs2kktKsG9ESE+VTPvcGkw5hYdIc5j//Me9r\nJCVFGos89xwQGKg7GjIak7CF+ftLZ51NmwCz9m94911pPt+5s+5IyK6GDgV27jRngZZSstvhppuk\nRSVZD5OwxXXuLJ/0x42TU1fM5MgRaVM5cqSszxHp0LkzUK+ejIbN5osvpGjxtdfkPGSyHr712cBr\nrwHp6cDUqbojudw778jaNT/hk07OAq3//hfIyNAdzSW5ubIdqWtXoFcv3dGQpzAJ20DdurIuPGWK\neY46PHMGmDFDEnBwsO5oyO4efFCWbxISdEdyydSpwJ49sqTELnLWxSRsE888I9uAzHL26Ny5chbq\nyJG6IyECKlUCHngAeOstcyzb7N8PvPQS8PjjvtfbmtzDAxxsZOZMKUJJTATatdMXh1LAzTdLsYkV\nDlfXzXmAQ8+ePREQEID4+HjEx8frDsvn7N8va8NvvSUnFOmi1KXDJXbu5NY9q2MStpH8fKBtW1lr\nSk4GSpXSE8fy5XKU3Lp1QIcOemKwEp6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"text/plain": [
"Graphics object consisting of 1 graphics primitive"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(f, [x, -10, 10], figsize=5) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t関係式
\n",
"\t\n",
"\t\tSageでの関係式の扱い方を以下の関係式eqを例に示します。\n",
"$$\n",
"\t\tx + 2 \\le 0\n",
"$$\n",
"\t
\n",
"\t\n",
"\t\tこれをSageで表現すると以下の様になります。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"eq = x + 2 <= 0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\t\n",
"\t\t関係式から左辺、等号、不等号のオペレータ、左辺を取得するには、それぞれlhs, operator, rhs関数を使用します。\n",
"\t
\n",
""
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x + 2\n",
"\n",
"0\n"
]
}
],
"source": [
"print eq.lhs() # 左辺\n",
"print eq.operator() # 等号、不等号などのオペレータ\n",
"print eq.rhs() # 右辺"
]
},
{
"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
}