A9_Abhishek_1911007.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/physicsme1729/Numerical-methods-in-physics/blob/main/10_swarup_kumar_Giri_phy_P745.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HNb-EV19LZKv"
      },
      "source": [
        "# Question 1"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 455
        },
        "id": "fczaULVqLZKx",
        "outputId": "bc6ec526-31c6-4206-bd97-83996c5f91b4"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from math import *\n",
        "\n",
        "def Lennard_potetial(x,eplison=1):\n",
        "    return 4*eplison*(x**12 - x**6)\n",
        "x = np.linspace(0.5,1,1000)\n",
        "y=[Lennard_potetial(i) for i in x ]\n",
        "plt.ylim([-1.1,0])\n",
        "plt.plot(x,y)\n",
        "plt.xlabel(\"1/r\")\n",
        "plt.ylabel(\"V\")\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "oxlukiHuLZKy",
        "outputId": "3041a4e3-3695-401b-951c-81610c1230d0"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "x = 0.8908987181981318\n",
            "V(x) = -1.0000000000000002\n"
          ]
        }
      ],
      "source": [
        "import math\n",
        "def bracket(f,x1,h):\n",
        "  c = 1.618033989\n",
        "  f1 = f(x1)\n",
        "  x2 = x1 + h; f2 = f(x2)\n",
        "  # Determine downhill direction and change sign of h if needed\n",
        "  if f2 > f1:\n",
        "    h = -h\n",
        "    x2 = x1 + h; f2 = f(x2)\n",
        "# Check if minimum between x1 - h and x1 + h\n",
        "    if f2 > f1: return x2,x1 - h\n",
        "# Search loop\n",
        "  for i in range (100):\n",
        "    h = c*h\n",
        "    x3 = x2 + h; f3 = f(x3)\n",
        "    if f3 > f2: return x1,x3\n",
        "    x1 = x2; x2 = x3\n",
        "    f1 = f2; f2 = f3\n",
        "  print(\"Bracket did not find a minimum\")\n",
        "def search(f,a,b,tol=1.0e-9):\n",
        "  nIter = int(math.ceil(-2.078087*math.log(tol/abs(b-a))))\n",
        "  R = 0.618033989\n",
        "  C = 1.0 - R\n",
        "# First telescoping\n",
        "  x2 = C*a + R*b; x1 = R*a + C*b\n",
        "  f1 = f(x1); f2 = f(x2)\n",
        "# Main loop\n",
        "  for i in range(nIter):\n",
        "    if f1 > f2:\n",
        "      a = x1\n",
        "      x1 = x2; f1 = f2\n",
        "      x2 = C*a + R*b; f2 = f(x2)\n",
        "    else:\n",
        "      b = x2\n",
        "      x2 = x1; f2 = f1\n",
        "      x1 = R*a + C*b; f1 = f(x1)\n",
        "  if f1 < f2: return x1,f1\n",
        "  else: return x2,f2\n",
        "\n",
        "xStart = 0.96\n",
        "h = 0.01\n",
        "x1,x2 = bracket(Lennard_potetial,xStart,h)\n",
        "x,fMin = search(Lennard_potetial,x1,x2)\n",
        "print(\"x =\",x)\n",
        "print(\"V(x) =\",fMin)\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "q1F0WiYtLZKz"
      },
      "source": [
        "Solving it analytically\n",
        "\n",
        "\n",
        "Taking $\\epsilon$ =1 and $\\sigma$=1   \n",
        "\n",
        "\n",
        "$F(x) = 4[x^{(12)} - x^{(6)}]$\n",
        "\n",
        "\n",
        "To find minima\n",
        "$$ F'(x) = 0 \\\\\n",
        "\\implies 12x^{11} - 6x^{5} = 0 \\\\\n",
        "\\implies 2x^{6} - 1 = 0 \\\\\n",
        "\\implies x_{min}= \\left(\\frac{1}{2}\\right)^{\\frac{1}{6}} = 0.89089 $$\n",
        "\n",
        "\n",
        "$$ F(x_{min}) = -1$$\n",
        "This matches with result we obtained above\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zDFyiJHrLZKz"
      },
      "source": [
        "# Question 2\n",
        "\n",
        "Again for the ease of life, we take C=1\n",
        "\n",
        "so\n",
        "$$ \\psi = (27 - 18\\sigma + 2\\sigma^2)e^{\\frac{-\\sigma}{3}}$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 435
        },
        "id": "Bp9Mp7DDLZKz",
        "outputId": "90a93763-d5a6-4bbe-8967-8c7ac0a6622a"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "def wavefunction(x):\n",
        "    return (27 -18*x + 2*x**2)*exp(-x/3)\n",
        "x = np.linspace(0,20,10000)\n",
        "y=[wavefunction(i) for i in x ]\n",
        "plt.ylim(-5,10)\n",
        "plt.plot(x,y)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "LiGPyCe8LZK0",
        "outputId": "8f165681-2fdb-4650-aeb9-eb1e9e7166c7"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "x = 3.5313730637054928\n",
            "V(x) = -3.5819473201969343\n"
          ]
        }
      ],
      "source": [
        "import math\n",
        "def bracket(f,x1,h):\n",
        "  c = 1.618033989\n",
        "  f1 = f(x1)\n",
        "  x2 = x1 + h; f2 = f(x2)\n",
        "  # Determine downhill direction and change sign of h if needed\n",
        "  if f2 > f1:\n",
        "    h = -h\n",
        "    x2 = x1 + h; f2 = f(x2)\n",
        "# Check if minimum between x1 - h and x1 + h\n",
        "    if f2 > f1: return x2,x1 - h\n",
        "# Search loop\n",
        "  for i in range (100):\n",
        "    h = c*h\n",
        "    x3 = x2 + h; f3 = f(x3)\n",
        "    if f3 > f2: return x1,x3\n",
        "    x1 = x2; x2 = x3\n",
        "    f1 = f2; f2 = f3\n",
        "  print(\"Bracket did not find a minimum\")\n",
        "def search(f,a,b,tol=1.0e-9):\n",
        "  nIter = int(math.ceil(-2.078087*math.log(tol/abs(b-a))))\n",
        "  R = 0.618033989\n",
        "  C = 1.0 - R\n",
        "# First telescoping\n",
        "  x2 = C*a + R*b; x1 = R*a + C*b\n",
        "  f1 = f(x1); f2 = f(x2)\n",
        "# Main loop\n",
        "  for i in range(nIter):\n",
        "    if f1 > f2:\n",
        "      a = x1\n",
        "      x1 = x2; f1 = f2\n",
        "      x2 = C*a + R*b; f2 = f(x2)\n",
        "    else:\n",
        "      b = x2\n",
        "      x2 = x1; f2 = f1\n",
        "      x1 = R*a + C*b; f1 = f(x1)\n",
        "  if f1 < f2: return x1,f1\n",
        "  else: return x2,f2\n",
        "\n",
        "xStart = 4\n",
        "h = 0.01\n",
        "x1,x2 = bracket(wavefunction,xStart,h)\n",
        "x,fMin = search(wavefunction,x1,x2)\n",
        "print(\"x =\",x)\n",
        "print(\"V(x) =\",fMin)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cP4mu4R3LZK0"
      },
      "source": [
        "SOLVING ANALYTICALLYT\n",
        "$$ \\psi = (27 - 18\\sigma + 2\\sigma^2)e^{\\frac{-\\sigma}{3}}$$\n",
        "\n",
        "For minima,\n",
        "$$ \\psi' = 0 \\\\\n",
        "\\implies (-18 +4\\sigma)e^{\\frac{-\\sigma}{3}} + (27 - 18\\sigma + 2\\sigma^2)e^{\\frac{-\\sigma}{3}}× \\frac{-1}{3} = 0 \\\\\n",
        "\\implies 81-30\\sigma +2\\sigma^2 = 0 \\\\\n",
        "\\implies \\sigma = 3.53137 \\text{\\ \\ }\\text{or}\\text{\\ \\ } 11.46862 \\\\ $$\n",
        "\n",
        "However after further evalution i.e. evaluating $\\psi$'', we found at 3.53137 we have a minima and at 11.46862 we have a maxima.\n",
        "$$ x_{min} = 3.53137$$\n",
        "Thus this result matches with the above\n",
        "\n",
        "\n",
        "And $$ \\psi(x_{min}) = -3.58194 $$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xfnPCyWJLZK0"
      },
      "source": [
        "# Question 20"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ATXie6EMLZK0",
        "outputId": "3921412f-219f-40bb-c3d3-c56e4ec4c300"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[-0.125  0.8  ]\n",
            "[0.089375 3.439375]\n",
            "[20.47936289 32.20967539]\n",
            "[1684.75048641 1715.61689872]\n",
            "[3051637.1194273  3050902.82662716]\n",
            "[9.30353311e+12 9.30353158e+12]\n",
            "[8.65556715e+25 8.65556715e+25]\n",
            "[7.49188426e+51 7.49188426e+51]\n",
            "[5.61283298e+103 5.61283298e+103]\n",
            "[3.1503894e+207 3.1503894e+207]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "[nan nan]\n",
            "Solution: x = [nan nan]\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "<ipython-input-6-060a267a2f67>:3: RuntimeWarning: overflow encountered in double_scalars\n",
            "  f0 = x[0]**2 - 2*x[0] - 2*x[1]**2 + x[1]\n",
            "<ipython-input-6-060a267a2f67>:3: RuntimeWarning: invalid value encountered in double_scalars\n",
            "  f0 = x[0]**2 - 2*x[0] - 2*x[1]**2 + x[1]\n",
            "<ipython-input-6-060a267a2f67>:4: RuntimeWarning: overflow encountered in double_scalars\n",
            "  f1 = x[0]**2  + x[0] - 2*x[1]**2 -1.5*x[1] - 0.05\n",
            "<ipython-input-6-060a267a2f67>:4: RuntimeWarning: invalid value encountered in double_scalars\n",
            "  f1 = x[0]**2  + x[0] - 2*x[1]**2 -1.5*x[1] - 0.05\n"
          ]
        }
      ],
      "source": [
        "import numpy as np\n",
        "def F(x):\n",
        "    f0 = x[0]**2 - 2*x[0] - 2*x[1]**2 + x[1]\n",
        "    f1 = x[0]**2  + x[0] - 2*x[1]**2 -1.5*x[1] - 0.05\n",
        "    return np.array([f0, f1])\n",
        "\n",
        "x = np.array([0., 0.25])\n",
        "\n",
        "def fixed_point_iteration(x, kmax=100, tolerance=1e-8):\n",
        "    for i in range(kmax):\n",
        "        x_new = x-F(x)\n",
        "        print(x_new)\n",
        "        if np.all(np.abs(x_new - x) < tolerance):\n",
        "            return x_new\n",
        "        x = x_new\n",
        "    return x\n",
        "\n",
        "sol_x = fixed_point_iteration(x)\n",
        "print(f\"Solution: x = {sol_x}\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 450
        },
        "id": "mHfS3OjuLZK1",
        "outputId": "7319705b-05c5-4291-edfc-6528de44048e"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "def f0(x0,x1):\n",
        "    return x0**2 - 2*x0 - 2*x1**2 + x1\n",
        "def f1(x0,x1):\n",
        "    return x0**2  + x0 - 2*x1**2 + -1.5*x1 - 0.05\n",
        "x0 = np.arange(-1,1.,0.01)\n",
        "x1 = np.arange(-1.,1.,0.01)\n",
        "X0, X1 = np.meshgrid(x0,x1)\n",
        "c1=plt.contour(x0, x1, f0(X0, X1),[0],colors=['b'])\n",
        "c2=plt.contour(x0, x1, f1(X0, X1),[0],colors=['r'])\n",
        "plt.xlabel('$x_0$')\n",
        "plt.ylabel('$x_1$')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "e3acLZ_KLZK1"
      },
      "source": [
        "Not able to get to any of the solutions. Because the fixed point iteration is not a good method for determing the roots of multi dimensional function."
      ]
    }
  ],
  "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": "3.10.0"
    },
    "orig_nbformat": 4,
    "colab": {
      "provenance": [],
      "include_colab_link": true
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}