Euler's Method.ipynb

euler-s-method.ipynb

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      "name": "Euler's Method.ipynb",
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        "<a href=\"https://colab.research.google.com/gist/KhanradCoder/64e79ae98f48768ada0b78417746624f/euler-s-method.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "import numpy as np  \n",
        "import matplotlib.pyplot as plt"
      ],
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      "outputs": []
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      "cell_type": "markdown",
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      "source": [
        "In the y_actual function where it says return, you can input the function you want to estimate. In this case, I'm using x^2 for simplicity, however you can also use an equation in terms of y and x. The y_derivative function is simply the derivative of the y_actual function, which in this case is 2x "
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      "source": [
        "def y_actual(x=0,y=0):\n",
        "  return(x**2)\n",
        "\n",
        "def y_derivative(x=0, y=0):\n",
        "  return (2*x)"
      ],
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      "outputs": []
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      "cell_type": "markdown",
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      "source": [
        "Next, I wrote the function for Euler's method.there are many parameters I could have taken in, but I ultimately decided on the x value we want to estimate, the number of steps we want to take, and an initial y value."
      ]
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      "source": [
        "def eulers_method(x, n):\n",
        "  y_hat = 0\n",
        "  h = x/n\n",
        "  for i in range(n):\n",
        "    y_hat += h*y_derivative(h*i,0,0)\n",
        "  return(y_hat)"
      ],
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      "outputs": []
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      "cell_type": "markdown",
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      "source": [
        "Finally, I printed the results of Euler's approximation, the actual solution and the percent error. You can see that as the steps increase, the error decreases."
      ]
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      "source": [
        "x = 5\n",
        "steps = 1000000\n",
        "\n",
        "print(\"Euler's method approximation: \"+str(eulers_method(x, steps)))\n",
        "print(\"Actual solution: \"+str(y_actual(x)))\n",
        "print(\"Error: \"+str(np.abs((y_actual(x)-eulers_method(x, steps))/eulers_method(x, steps))*100)+\"%\")"
      ],
      "execution_count": 53,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Euler's method approximation: 24.999975000000006\n",
            "Actual solution: 25\n",
            "Error: 0.00010000009997499789%\n"
          ],
          "name": "stdout"
        }
      ]
    }
  ]
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