cleared-demos/quadrature_and_diff/Finite Differences vs Noise.ipynb

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      "source": [
        "# Finite Differences vs Noise\n",
        "Copyright (C) 2020 Andreas Kloeckner\n",
        "\n",
        "<details>\n",
        "<summary>MIT License</summary>\n",
        "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
        "of this software and associated documentation files (the \"Software\"), to deal\n",
        "in the Software without restriction, including without limitation the rights\n",
        "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
        "copies of the Software, and to permit persons to whom the Software is\n",
        "furnished to do so, subject to the following conditions:\n",
        "\n",
        "The above copyright notice and this permission notice shall be included in\n",
        "all copies or substantial portions of the Software.\n",
        "\n",
        "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n",
        "IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n",
        "FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n",
        "AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n",
        "LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n",
        "OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n",
        "THE SOFTWARE.\n",
        "</details>"
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      "source": [
        "import numpy as np\n",
        "import numpy.linalg as la\n",
        "import matplotlib.pyplot as pt"
      ]
    },
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      "source": [
        "def f(x):\n",
        "    return np.sin(2*x)\n",
        "def df(x):\n",
        "    return 2*np.cos(2*x)"
      ]
    },
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      "source": [
        "Here's a pretty simple function and its derivative:"
      ]
    },
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        "plot_x = np.linspace(-1, 1, 200)\n",
        "\n",
        "pt.plot(plot_x, f(plot_x), label=\"f\")\n",
        "pt.plot(plot_x, df(plot_x), label=\"df/dx\")\n",
        "pt.grid()\n",
        "pt.legend()"
      ]
    },
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      "source": [
        "Now what happens to our numerical differentiation if\n",
        "**our function values have a slight amount of error**?"
      ]
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        "\n",
        "# set up grid\n",
        "n = 10\n",
        "x = np.linspace(-1, 1, n)\n",
        "h = x[1] - x[0]\n",
        "x_df_result = x[1:-1] # chop off first, last point\n",
        "\n",
        "# evaluate f, perturb data, finite differences of f\n",
        "f_x = f(x)\n",
        "\n",
        "f_x += 0.025*np.random.randn(n)\n",
        "\n",
        "df_num_x = (f_x[2:] - f_x[:-2])/(2*h)\n",
        "\n",
        "# plot\n",
        "pt.plot(x, f_x, \"o-\", label=\"f\")\n",
        "pt.plot(plot_x, df(plot_x), label=\"df/dx\")\n",
        "pt.plot(x_df_result, df_num_x, label=\"df/dx num\")\n",
        "pt.grid()\n",
        "pt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* Now what happens if you set `n = 100` instead of `n = 10`?"
      ]
    }
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