{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "### 2023年度計算機演習A・B\n", "\n", "# 第7回レポート課題の解説" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習1\n", "\n", "関数 $f(x)=\\sqrt{x}+1$ に対して、区間 $[0,1]$ の分割数 $3$ の等分割に関する左リーマン和と右リーマン和を考えます。\n", "\n", "それぞれに対応する図を描画した上で値を求めるコードを書いてください。" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "分割数3の等分割に関する左リーマン和の値は 1.4646156167057838\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "分割数3の等分割に関する右リーマン和の値は 1.797948950039117\n" ] } ], "source": [ "import math\n", "import matplotlib.pyplot as plt\n", "\n", "def f(x):\n", " return math.sqrt(x)+1\n", "\n", "def draw_f():\n", " x_list = []\n", " y_list = []\n", " for i in range(201):\n", " x = i/200\n", " x_list.append(x)\n", " y_list.append(f(x))\n", " plt.plot(x_list,y_list,\"k-\")\n", "\n", "draw_f()\n", "plt.fill([0,1/3,1/3,0],[0,0,f(0),f(0)],\"r\",alpha=0.5) #長方形の頂点のx,y座標を左下、右下、右上、左上の順に並べる\n", "plt.fill([1/3,2/3,2/3,1/3],[0,0,f(1/3),f(1/3)],\"r\",alpha=0.5)\n", "plt.fill([2/3,1,1,2/3],[0,0,f(2/3),f(2/3)],\"r\",alpha=0.5)\n", "plt.show()\n", "\n", "print(\"分割数3の等分割に関する左リーマン和の値は\",f(0)*(1/3)+f(1/3)*(1/3)+f(2/3)*(1/3)) #描画した長方形の面積（縦×横）の和を計算する\n", "\n", "draw_f()\n", "plt.fill([0,1/3,1/3,0],[0,0,f(1/3),f(1/3)],\"g\",alpha=0.5)\n", "plt.fill([1/3,2/3,2/3,1/3],[0,0,f(2/3),f(2/3)],\"g\",alpha=0.5)\n", "plt.fill([2/3,1,1,2/3],[0,0,f(1),f(1)],\"g\",alpha=0.5)\n", "plt.show()\n", "\n", "print(\"分割数3の等分割に関する右リーマン和の値は\",f(1/3)*(1/3)+f(2/3)*(1/3)+f(1)*(1/3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習2\n", "\n", "関数 $f(x)=\\sqrt{x}+1$ に対して、区間 $[0,1]$ の分割数 $n$ の等分割に関する左リーマン和と右リーマン和を考えます。\n", "\n", "左リーマン和に対応する図を描画する関数`draw_left_RS(n)`、左リーマン和の値を返す関数`calc_left_RS(n)`、右リーマン和に対応する図を描画する関数`draw_right_RS(n)`、右リーマン和の値を返す関数`calc_right_RS(n)`を定義してください。\n", "\n", "$n=4$ の場合に、定義したそれぞれの関数を呼び出してください。" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "分割数 4 の等分割に関する左リーマン和の値は 1.5182830462427466\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "分割数 4 の等分割に関する右リーマン和の値は 1.7682830462427466\n" ] } ], "source": [ "import math\n", "import matplotlib.pyplot as plt\n", "\n", "def f(x):\n", " return math.sqrt(x)+1\n", "\n", "def draw_f():\n", " x_list = []\n", " y_list = []\n", " for i in range(201):\n", " x = i/200\n", " x_list.append(x)\n", " y_list.append(f(x))\n", " plt.plot(x_list,y_list,\"k-\")\n", "\n", "def draw_left_RS(n):\n", " draw_f()\n", " for i in range(n): #i=0からn-1まで繰り返す\n", " plt.fill([i/n,(i+1)/n,(i+1)/n,i/n],[0,0,f(i/n),f(i/n)],\"r\",alpha=0.5) #「(i+1)/n」と「i+1/n」は違うので注意\n", " plt.show()\n", "\n", "def calc_left_RS(n):\n", " value = 0\n", " for i in range(n):\n", " value = value+f(i/n)*(1/n)\n", " return value\n", "\n", "def draw_right_RS(n):\n", " draw_f()\n", " for i in range(n):\n", " plt.fill([i/n,(i+1)/n,(i+1)/n,i/n],[0,0,f((i+1)/n),f((i+1)/n)],\"g\",alpha=0.5)\n", " plt.show()\n", "\n", "def calc_right_RS(n):\n", " value = 0\n", " for i in range(n):\n", " value = value+f((i+1)/n)*(1/n)\n", " return value\n", "\n", "n = 4\n", "draw_left_RS(n)\n", "print(\"分割数\",n,\"の等分割に関する左リーマン和の値は\",calc_left_RS(n))\n", "draw_right_RS(n)\n", "print(\"分割数\",n,\"の等分割に関する右リーマン和の値は\",calc_right_RS(n))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 演習3（オプション）\n", "\n", "関数 $f(x)=\\sqrt{x}+1$ に対して、区間 $[0,1]$ の分割数 $n$ の等分割に関する左リーマン和と右リーマン和を考えます。\n", "\n", "右リーマン和と左リーマン和の差が $0.01$ より小さくなるような最小の $n$ を求めるコードを書いてください。\n", "\n", "また、定積分 $\\int_0^1 \\left(\\sqrt{x}+1\\right)\\,dx$ の値を手計算で求め、上記の $n$ に対する左リーマン和および右リーマン和の値と比較して考察してください。" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "右リーマン和と左リーマン和の差が0.01より小さくなるような最小のnは 100\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "分割数 100 の等分割に関する左リーマン和の値は 1.661462947103148\n" ] }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "分割数 100 の等分割に関する右リーマン和の値は 1.6714629471031477\n" ] } ], "source": [ "n = 1\n", "while calc_right_RS(n)-calc_left_RS(n) >= 0.01:\n", " n = n+1\n", "\n", "print(\"右リーマン和と左リーマン和の差が0.01より小さくなるような最小のnは\",n)\n", "\n", "draw_left_RS(n)\n", "print(\"分割数\",n,\"の等分割に関する左リーマン和の値は\",calc_left_RS(n))\n", "draw_right_RS(n)\n", "print(\"分割数\",n,\"の等分割に関する右リーマン和の値は\",calc_right_RS(n))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### （考察）\n", "\n", "定積分の値を計算すると\n", "\n", "$$\n", "\\int_0^1 \\left(\\sqrt{x}+1\\right)\\,dx=\\left[\\frac{2}{3}x^{\\frac{3}{2}}+x\\right]_0^1=\\frac{5}{3}=1.666\\cdots\n", "$$\n", "\n", "であるから、分割数 $100$ の等分割に関して\n", "\n", "$$\n", "\\text{（左リーマン和）}<\\int_0^1 \\left(\\sqrt{x}+1\\right)\\,dx<\\text{（右リーマン和）}\n", "$$\n", "\n", "が成り立つことが分かる。関数 $f(x)=\\sqrt{x}+1$ は単調増加なので、これは理論的な結果に一致している。\n", "\n", "また、次のコードから分かるように、右リーマン和の方が定積分の値に近い。関数 $f(x)=\\sqrt{x}+1$ は単調増加かつ上に凸なので、これは一般の $n$ についても成り立つと言える。上で描画した $n=4$ の場合の図を見ると、その理由が分かりやすい。" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "分割数 100 の等分割に関する右リーマン和の値と定積分の値の差は 0.004796280436480949\n", "定積分の値と分割数 100 の等分割に関する左リーマン和の値の差は 0.005203719563518838\n" ] } ], "source": [ "print(\"分割数\",n,\"の等分割に関する右リーマン和の値と定積分の値の差は\",calc_right_RS(n)-5/3)\n", "print(\"定積分の値と分割数\",n,\"の等分割に関する左リーマン和の値の差は\",5/3-calc_left_RS(n))" ] } ], "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": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 4 }