{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\tHiroshi TAKEMOTO\n", "\t(take.pwave@gmail.com)\n", "\t

方程式の解法

\n", "\t

\n", "\t\t数式処理システムで使いたい機能の一つに、方程式の解法があります。\n", "\t

\n", "\t

\n", "\t\t方程式の解法にはsolve関数を使用します。solve関数の呼び出し形式は以下の通りです。\n", "\t\t

\n",
    "solve(解きたい方程式また方程式のリスト, 解を得る変数)\t\t\t\n",
    "\t\t
\n", "\t

\n", "\n", "\n", "\n", "\t

一次方程式の解

\n", "\t

\n", "\t\t以下の一次方程式をsolveを使って解いてみましょう。\n", "$$\n", "\t\tf(x)=2x+1\n", "$$\n", "\t\tsolverの戻り値は、解のリストです。一次方程式では1個の解、$x = −\\frac{1}{2}$\n", "\t\tが得られます。\t\t\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[x == (-1/2)]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 一次方程式の解法\n", "var('x')\n", "f = 2*x + 1\n", "sol = solve(f, x)\n", "show(sol)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

一次方程式のグラフ表現

\n", "\t

\n", "\t\t先のf(x)をyとすると、直線の式が得られます。\n", "$$\n", "\t\ty = 2x + 1\n", "$$\t\n", "\t

\n", "\t

\n", "\t\tこの直線をプロットしてみると、以下のようになります。\n", "\t\t直線とX軸が交わる点が、方程式の解になっていることがわかります。\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Jk9lrr72q5ZnVsqmrvLyc008/nQULFvD0009TVlbGmjVrAGjUqBG77LJLdXwa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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 一次方程式のグラフ\n", "plot(f, [x, -2, 2], figsize=5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

2次方程式

\n", "\t

\n", "\t\t同様に、以下の2次方程式について見てみましょう。\n", "$$\n", "\t\tf(x) = x^2 + x - 1\n", "$$\t\t\n", "\t

\n", "\t

\n", "\t\tf(x)の定義と因数分解の結果を表示します。\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "x^2 + x - 1" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f = x^2 + x - 1\n", "show(factor(f))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

2次方程式の解とグラフ表現

\n", "\t

\n", "\t\tsolve関数の解は、2個の実数が求まります。\n", "\t

\n", "\t

\n", "\t\tこの場合のグラフは、X軸で交わる点が2個存在します。\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[x == -1/2*sqrt(5) - 1/2, x == 1/2*sqrt(5) - 1/2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 2次方程式の解\n", "sol = solve(f, x)\n", "show(sol)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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EjOR27IDKlaFlS/PRoMHJA0SMX3+FKVPMzMO8eZCTY46h3X23+X2M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\nRERcQIEsIiLiAgpkERERF/h/SyVnpGaM+QwAAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 2次方程式のグラフ\n", "plot(f, [x, -2, 2], figsize=5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

重根の場合

\n", "\t

\n", "\t\t2次方程式の解が同じ値を持つ場合(重根)の場合には、X軸との交点が1個となります。\n", "\t

\n", "\t

\n", "\t\t重根の場合、解をaとすると因数分解の形は、以下のようになります。\n", "$$\n", "\t\tf(x) = (x -a)^2\n", "$$\t\t\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(x - 1)^2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 重根の場合\n", "f = x^2 - 2*x + 1\n", "show(factor(f))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[x == 1]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 2次方程式の解(重根の場合)\n", "sol = solve(f, x)\n", "show(sol)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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sbNupAEQahQy4QKVK0vvvSxMnmkVfDRqYjUUARA8KGXCJQMCMjjdulCpWNKdH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\ncIH/D9sc3ENTdWNJAAAAAElFTkSuQmCC\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 2次方程式のグラフ(重根の場合)\n", "plot(f, [x, -2, 2], figsize=5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

複素数の場合

\n", "\t

\n", "\t\t2次方程式の解が複素数の場合には、グラフはX軸と交わりません。\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "x^2 + x + 1" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 複素数\n", "f = x^2 + x + 1\n", "show(factor(f))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[x == -1/2*I*sqrt(3) - 1/2, x == 1/2*I*sqrt(3) - 1/2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 2次方程式の解\n", "sol = solve(f, x)\n", "show(sol)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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TY7ua0MWiLsDlWNSFYEpPl265RbruOmnhQrY5BRMjZADA7xo40Ow5pgFI8BHIAICL2rBB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\nSV99Jd19t+2KgNx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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 2次方程式のグラフ\n", "plot(f, [x, -2, 2], figsize=5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

多項式と数値解

\n", "\t

\n", "\t\t多項式の例として、以下の3次多項式をみてみましょう。\n", "$$\n", "\t\tf(x) = x^3 - 2 x + 4\n", "$$\n", "\t

\n", "\t

\n", "\t\tsolveの結果は、1個の実数と2個の複素数が求まります。ここから予想されるグラフの形は、X軸と交わる点が1個の3次曲線です。\n", "\t

\n", "\t

\n", "\t\tこのように関数の解を求めたり、グラフと解の関係を理解する上で、Sageはとても有効です。\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(x^2 - 2*x + 2)*(x + 2)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 3次方程式\n", "var('x')\n", "f = x^3-2*x+4\n", "show(factor(f))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[x == (-I + 1), x == (I + 1), x == -2]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 3次方程式の解\n", "sol = solve(f, x)\n", "show(sol)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 3次方程式のグラフ\n", "plot(f, -2.5, 2.5, figsize=5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

数値解法

\n", "\t

\n", "\t\tsolveが解析的に解を求めたのに対し、与えられた範囲で解となる点を数値解法で求める\n", "\t\t関数が、find_rootです。\n", "\t

\n", "\t

\n", "\t\tfind_rootの呼び出し形式は以下の通りです。fには1変数の関数を与えます。\n", "

\n",
    "\tfind_root(f, 計算開始点, 計算終了点)\n",
    "
\t\t\n", "\t

\n", "\t

\n", "\t\t先の関数f(x)をfind_rootで求めると、解析解(x = 2)に非常に近い値が求まります。\n", "$$\n", "\t\tf(x) = x^3 - 2 x + 4\n", "$$\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1.999999999999995" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# グラフから-3から3の範囲で数値解を求める\n", "find_root(f, -3, 3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

連立方程式

\n", "\t

\n", "\t\t連立方程式の解とグラフの関係をSageを使ってみてみましょう。\n", "\t

\n", "\t

\n", "\t\t先に解いた一次方程式と2次方程式を1つの連立方程式にして解いてみましょう。\n", "$$\n", "\t\t\\left\\{ \\begin{eqnarray}\n", "\t\t\ty & = & 2 x + 1 \\\\\n", "\t\t\ty & = & x^2 + x - 1\n", "\t\t\\end{eqnarray} \\right.\n", "$$\n", "\t

\n", "\t

\n", "\t\t方程式のリストをsolve関数に与えることで、連立方程式の解を得ることができます。\n", "\t

\n", "\t

\n", "\t\tグラフから1次曲線と2次曲線の交点(-1, -1), (2, 5)が、連立方程式の解となっていることが分かります。\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[[x == -1, y == -1], [x == 2, y == 5]]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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UjM3bb9sS9fDhcNddrqMROTNVyBIx2rRpQ3x8PAsWLHAdis9MnWpV8ZEjtkzd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\nXLlcR3Z6cXFx1KxZkyJFihA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xpel337Uq+frrXUclmTFu3Djy5MlDnjx5yJs3LwsWLPjr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\nEgSUkEVERIKAErKIiEgQUEIWEREJAv8HZHIYY91iIMAAAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "var('x y')\n", "eq = [y == 2*x + 1, y == x^2 + x - 1]\n", "# 解を求める\n", "show(solve(eq, [x, y]))\n", "# グラフから解を求める\n", "p1 = plot(x^2 + x - 1, [x, -2, 3])\n", "p2 = plot(2*x + 1, [x, -2, 3])\n", "(p1+p2).show(ymin=-3, ymax=8, figsize=5) # 不等号の例と描画範囲を合せた" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

不等式とグラフ

\n", "\t

\n", "\t\tregion_plot関数を使うことで不等式の範囲をグラフで表示することができます。\n", "\t

\n", "\t

\n", "\t\tregion_plotの引数には、xとy2変数の関係式または2変数を引数に持つ関数とx変数とy変数の\n", "\t\t描画範囲を指定します。\n", "\t

\n", "\t

\n", "\t\t以下に1次不等式の例を示します。\n", "$$\n", "\t\ty \\le 2 x + 1\n", "$$\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(x, y) = var('x y')\n", "f(x, y) = y <= 2*x + 1\n", "region_plot(f(x, y), [x, -2, 3], [y, -3, 8])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

\n", "\t\tincol, outcolで領域内、領域外の色を指定したり、bordercolで境界線の色を指定することができます。\n", "\t

\n", "\t

\n", "\t\t以下では、2次不等式にincolに薄い青、境界線に灰色を指定して表示した例を示します。\n", "$$\n", "\t\ty > x^2 + x -1\n", "$$\t\t\n", "\t

\n", "" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(x, y) = y > x^2 + x - 1\n", "region_plot(g(x, y), [x, -2, 3], [y, -3, 8], incol='lightblue', bordercol='gray')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\t

連立不等式のグラフ

\n", "\t

\n", "\t\tregion_plotに複数の式または関数をリストで与えることによって、連立不等式の\n", "\t\t範囲を表示することができます。\n", "\t

\n", "\t

\n", "\t\t以下に、先の2つの不等式を組み合わせた連立不等式の例を示します。\n", "$$\n", "\t\t\\left\\{ \\begin{eqnarray}\n", "\t\t\ty & \\le & 2 x + 1 \\\\\n", "\t\t\ty & > & x^2 + x - 1\n", "\t\t\\end{eqnarray} \\right.\n", "$$\n", "\t\t\n", "" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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