report_III-B.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jaeqy3WhYbK3"
      },
      "source": [
        "# 数値解析2020 レポート\n",
        "\n",
        "**III-B.** (2) 2分法により $f(x)$ の根を求めるプログラムを作成して問1で求めた回数繰り返し、繰り返し操作後の解の存在する $x$ の区間の両端の値を10進11桁以降を切り捨てて求めよ。またその区間の両端の値の平均値を求め根の近似値として10進11桁以降を切り捨てて求めよ。作成したプログラムも提出すること。プログラミング言語は問わない。\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6oMty-RaogOy"
      },
      "source": [
        "以下Python3を用いたプログラム例を示す。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mMdicca6YOfm",
        "outputId": "66032f89-119c-407e-9370-186e102ae907",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "# Pythonのバージョン確認\n",
        "import sys\n",
        "print(sys.version)"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "3.9.16 (main, Dec  7 2022, 01:11:51) \n",
            "[GCC 9.4.0]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8Z5uhXiGnrc6"
      },
      "source": [
        "import math\n",
        "def f(x):\n",
        "    return 2 - math.exp(x)"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TVvZbsGpHJU7",
        "outputId": "5f528879-f48a-4162-eae6-837653733bed",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "a = 0; b = 1\n",
        "for i in range(1,27):\n",
        "    #c = (a+b)/2\n",
        "    c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "    if f(a) * f(c) > 0:\n",
        "        a = c\n",
        "    else:\n",
        "        b = c\n",
        "\n",
        "print()\n",
        "print(\"解の存在する区間幅 [{:.15f}, {:.15f}]\".format(a, b))\n",
        "c = a + (b-a)/2\n",
        "print(\"根の近似値 {:.15f}\".format(c))\n",
        "print(\"残差 {:.15f}\".format(abs(f(c))))"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\n",
            "解の存在する区間幅 [0.693147167563438, 0.693147182464600]\n",
            "根の近似値 0.693147175014019\n",
            "残差 0.000000011091853\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Z44mY5RYTg3Y",
        "outputId": "c1caf1b8-350c-47e4-a33a-1171f7f7e779",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "a = 0; b = 1\n",
        "i = 0\n",
        "c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "print(\"(i = {:2d}) [{:.15f}, {:.15f}] 残差 {:18.15f}\".format(i, a, b, abs(f(c))))\n",
        "for i in range(1,28):\n",
        "    #c = (a+b)/2\n",
        "    c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "    if f(a) * f(c) > 0:\n",
        "        a = c\n",
        "    else:\n",
        "        b = c\n",
        "    c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "    print(\"(i = {:2d}) [{:.15f}, {:.15f}] 残差 {:18.15f}\".format(i, a, b, abs(f(c))))\n",
        "\n",
        "print()\n",
        "print(\"解の存在する区間幅 [{:.15f}, {:.15f}]\".format(a, b))\n",
        "c = a + (b-a)/2\n",
        "print(\"根の近似値 {:.15f}\".format(c))\n",
        "print(\"残差 {:.15f}\".format(f(c)))"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(i =  0) [0.000000000000000, 1.000000000000000] 残差  0.351278729299872\n",
            "(i =  1) [0.500000000000000, 1.000000000000000] 残差  0.117000016612675\n",
            "(i =  2) [0.500000000000000, 0.750000000000000] 残差  0.131754042567778\n",
            "(i =  3) [0.625000000000000, 0.750000000000000] 残差  0.011262530417708\n",
            "(i =  4) [0.687500000000000, 0.750000000000000] 残差  0.051866773487977\n",
            "(i =  5) [0.687500000000000, 0.718750000000000] 残差  0.020055527708696\n",
            "(i =  6) [0.687500000000000, 0.703125000000000] 残差  0.004335330874331\n",
            "(i =  7) [0.687500000000000, 0.695312500000000] 残差  0.003478832038738\n",
            "(i =  8) [0.691406250000000, 0.695312500000000] 残差  0.000424433909775\n",
            "(i =  9) [0.691406250000000, 0.693359375000000] 残差  0.001528152010194\n",
            "(i = 10) [0.692382812500000, 0.693359375000000] 残差  0.000552097402978\n",
            "(i = 11) [0.692871093750000, 0.693359375000000] 残差  0.000063891349343\n",
            "(i = 12) [0.693115234375000, 0.693359375000000] 残差  0.000180256377712\n",
            "(i = 13) [0.693115234375000, 0.693237304687500] 残差  0.000058178788786\n",
            "(i = 14) [0.693115234375000, 0.693176269531250] 残差  0.000002857211600\n",
            "(i = 15) [0.693145751953125, 0.693176269531250] 残差  0.000027660555759\n",
            "(i = 16) [0.693145751953125, 0.693161010742188] 残差  0.000012401613872\n",
            "(i = 17) [0.693145751953125, 0.693153381347656] 残差  0.000004772186584\n",
            "(i = 18) [0.693145751953125, 0.693149566650391] 残差  0.000000957483854\n",
            "(i = 19) [0.693145751953125, 0.693147659301758] 残差  0.000000949864782\n",
            "(i = 20) [0.693146705627441, 0.693147659301758] 残差  0.000000003809308\n",
            "(i = 21) [0.693146705627441, 0.693147182464600] 残差  0.000000473027794\n",
            "(i = 22) [0.693146944046021, 0.693147182464600] 残差  0.000000234609257\n",
            "(i = 23) [0.693147063255310, 0.693147182464600] 残差  0.000000115399978\n",
            "(i = 24) [0.693147122859955, 0.693147182464600] 残差  0.000000055795335\n",
            "(i = 25) [0.693147152662277, 0.693147182464600] 残差  0.000000025993014\n",
            "(i = 26) [0.693147167563438, 0.693147182464600] 残差  0.000000011091853\n",
            "(i = 27) [0.693147175014019, 0.693147182464600] 残差  0.000000003641272\n",
            "\n",
            "解の存在する区間幅 [0.693147175014019, 0.693147182464600]\n",
            "根の近似値 0.693147178739309\n",
            "残差 0.000000003641272\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8UbkV8lSpLox",
        "outputId": "647b16c5-ade5-4faa-a4e6-53fe9bf6c16e",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "a = 0; b = 1\n",
        "alpha = 0.6931471805599453\n",
        "for i in range(27):\n",
        "    #c = (a+b)/2\n",
        "    c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "    if f(a) * f(c) > 0:\n",
        "        a = c\n",
        "    else:\n",
        "        b = c\n",
        "    print(\"(i = {:d}) [{:.15f}, {:.15f}] {:.15f}\".format(i, a, b, abs(alpha - c)))\n",
        "\n",
        "print()\n",
        "c = a + (b-a)/2\n",
        "print(\"解の存在する区間幅 [{:.15f}, {:.15f}] 絶対誤差 {:.15f}\".format(a, b, abs(alpha - c)))\n",
        "print(\"根の近似値 {:.15f} 絶対誤差 {:.15f}\".format(c, abs(alpha - c)))"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(i = 0) [0.500000000000000, 1.000000000000000] 0.193147180559945\n",
            "(i = 1) [0.500000000000000, 0.750000000000000] 0.056852819440055\n",
            "(i = 2) [0.625000000000000, 0.750000000000000] 0.068147180559945\n",
            "(i = 3) [0.687500000000000, 0.750000000000000] 0.005647180559945\n",
            "(i = 4) [0.687500000000000, 0.718750000000000] 0.025602819440055\n",
            "(i = 5) [0.687500000000000, 0.703125000000000] 0.009977819440055\n",
            "(i = 6) [0.687500000000000, 0.695312500000000] 0.002165319440055\n",
            "(i = 7) [0.691406250000000, 0.695312500000000] 0.001740930559945\n",
            "(i = 8) [0.691406250000000, 0.693359375000000] 0.000212194440055\n",
            "(i = 9) [0.692382812500000, 0.693359375000000] 0.000764368059945\n",
            "(i = 10) [0.692871093750000, 0.693359375000000] 0.000276086809945\n",
            "(i = 11) [0.693115234375000, 0.693359375000000] 0.000031946184945\n",
            "(i = 12) [0.693115234375000, 0.693237304687500] 0.000090124127555\n",
            "(i = 13) [0.693115234375000, 0.693176269531250] 0.000029088971305\n",
            "(i = 14) [0.693145751953125, 0.693176269531250] 0.000001428606820\n",
            "(i = 15) [0.693145751953125, 0.693161010742188] 0.000013830182242\n",
            "(i = 16) [0.693145751953125, 0.693153381347656] 0.000006200787711\n",
            "(i = 17) [0.693145751953125, 0.693149566650391] 0.000002386090445\n",
            "(i = 18) [0.693145751953125, 0.693147659301758] 0.000000478741813\n",
            "(i = 19) [0.693146705627441, 0.693147659301758] 0.000000474932504\n",
            "(i = 20) [0.693146705627441, 0.693147182464600] 0.000000001904654\n",
            "(i = 21) [0.693146944046021, 0.693147182464600] 0.000000236513925\n",
            "(i = 22) [0.693147063255310, 0.693147182464600] 0.000000117304635\n",
            "(i = 23) [0.693147122859955, 0.693147182464600] 0.000000057699990\n",
            "(i = 24) [0.693147152662277, 0.693147182464600] 0.000000027897668\n",
            "(i = 25) [0.693147167563438, 0.693147182464600] 0.000000012996507\n",
            "(i = 26) [0.693147175014019, 0.693147182464600] 0.000000005545926\n",
            "\n",
            "解の存在する区間幅 [0.693147175014019, 0.693147182464600] 絶対誤差 0.000000001820636\n",
            "根の近似値 0.693147178739309 絶対誤差 0.000000001820636\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dMSjO2G_HbrV",
        "outputId": "3474a46a-6884-4af1-d6e5-1348d2e6d56f",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "a = 0; b = 1\n",
        "alpha = 0.6931471805599453\n",
        "i = 0\n",
        "c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "print(\"{:2d},{:.15f},{:.15f},{:.15f},{:18.15f},{:18.15f},{:18.15f}\".format(i, a, b, c, alpha - c, abs(alpha-c) ,abs(f(c))))\n",
        "for i in range(1,28):\n",
        "    #c = (a+b)/2\n",
        "    c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "    if f(a) * f(c) > 0:\n",
        "        a = c\n",
        "    else:\n",
        "        b = c\n",
        "    c = a + (b-a)/2 # 平均値の計算を旨くする\n",
        "    print(\"{:2d},{:.15f},{:.15f},{:.15f},{:18.15f},{:18.15f},{:18.15f}\".format(i, a, b, c, alpha - c, abs(alpha-c) ,abs(f(c))))"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            " 0,0.000000000000000,1.000000000000000,0.500000000000000, 0.193147180559945, 0.193147180559945, 0.351278729299872\n",
            " 1,0.500000000000000,1.000000000000000,0.750000000000000,-0.056852819440055, 0.056852819440055, 0.117000016612675\n",
            " 2,0.500000000000000,0.750000000000000,0.625000000000000, 0.068147180559945, 0.068147180559945, 0.131754042567778\n",
            " 3,0.625000000000000,0.750000000000000,0.687500000000000, 0.005647180559945, 0.005647180559945, 0.011262530417708\n",
            " 4,0.687500000000000,0.750000000000000,0.718750000000000,-0.025602819440055, 0.025602819440055, 0.051866773487977\n",
            " 5,0.687500000000000,0.718750000000000,0.703125000000000,-0.009977819440055, 0.009977819440055, 0.020055527708696\n",
            " 6,0.687500000000000,0.703125000000000,0.695312500000000,-0.002165319440055, 0.002165319440055, 0.004335330874331\n",
            " 7,0.687500000000000,0.695312500000000,0.691406250000000, 0.001740930559945, 0.001740930559945, 0.003478832038738\n",
            " 8,0.691406250000000,0.695312500000000,0.693359375000000,-0.000212194440055, 0.000212194440055, 0.000424433909775\n",
            " 9,0.691406250000000,0.693359375000000,0.692382812500000, 0.000764368059945, 0.000764368059945, 0.001528152010194\n",
            "10,0.692382812500000,0.693359375000000,0.692871093750000, 0.000276086809945, 0.000276086809945, 0.000552097402978\n",
            "11,0.692871093750000,0.693359375000000,0.693115234375000, 0.000031946184945, 0.000031946184945, 0.000063891349343\n",
            "12,0.693115234375000,0.693359375000000,0.693237304687500,-0.000090124127555, 0.000090124127555, 0.000180256377712\n",
            "13,0.693115234375000,0.693237304687500,0.693176269531250,-0.000029088971305, 0.000029088971305, 0.000058178788786\n",
            "14,0.693115234375000,0.693176269531250,0.693145751953125, 0.000001428606820, 0.000001428606820, 0.000002857211600\n",
            "15,0.693145751953125,0.693176269531250,0.693161010742188,-0.000013830182242, 0.000013830182242, 0.000027660555759\n",
            "16,0.693145751953125,0.693161010742188,0.693153381347656,-0.000006200787711, 0.000006200787711, 0.000012401613872\n",
            "17,0.693145751953125,0.693153381347656,0.693149566650391,-0.000002386090445, 0.000002386090445, 0.000004772186584\n",
            "18,0.693145751953125,0.693149566650391,0.693147659301758,-0.000000478741813, 0.000000478741813, 0.000000957483854\n",
            "19,0.693145751953125,0.693147659301758,0.693146705627441, 0.000000474932504, 0.000000474932504, 0.000000949864782\n",
            "20,0.693146705627441,0.693147659301758,0.693147182464600,-0.000000001904654, 0.000000001904654, 0.000000003809308\n",
            "21,0.693146705627441,0.693147182464600,0.693146944046021, 0.000000236513925, 0.000000236513925, 0.000000473027794\n",
            "22,0.693146944046021,0.693147182464600,0.693147063255310, 0.000000117304635, 0.000000117304635, 0.000000234609257\n",
            "23,0.693147063255310,0.693147182464600,0.693147122859955, 0.000000057699990, 0.000000057699990, 0.000000115399978\n",
            "24,0.693147122859955,0.693147182464600,0.693147152662277, 0.000000027897668, 0.000000027897668, 0.000000055795335\n",
            "25,0.693147152662277,0.693147182464600,0.693147167563438, 0.000000012996507, 0.000000012996507, 0.000000025993014\n",
            "26,0.693147167563438,0.693147182464600,0.693147175014019, 0.000000005545926, 0.000000005545926, 0.000000011091853\n",
            "27,0.693147175014019,0.693147182464600,0.693147178739309, 0.000000001820636, 0.000000001820636, 0.000000003641272\n"
          ]
        }
      ]
    }
  ]
}