Demos/Ch1_SymbolicTaylorSeries.ipynb

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        "<a href=\"https://colab.research.google.com/github/stephenbeckr/numerical-analysis-class/blob/master/Demos/Ch1_SymbolicTaylorSeries.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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        "id": "LNDLwuliDDql",
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      "source": [
        "Manipulate Taylor series using the **SymPy** package in Python, following the tutorial at [scipy-lectures.org/packages/sympy.html](https://scipy-lectures.org/packages/sympy.html).  SymPy is used in [SageMath](https://www.sagemath.org/).\n",
        "\n",
        "You can do this in Maple, which is a bit old-school, or Mathematica (which we have a license for at CU).  Wolfram Alpha is another choice (by the company that makes Mathematica) though the free version does have some limitations.\n",
        "\n",
        "The nice thing about Python is that it is free and community supported, so if you take the time to use it, it won't go away. If you learn Mathematica now, you may not be able to use it in your job, since it's an expensive license."
      ]
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      "source": [
        "# Load the package\n",
        "import sympy as sym\n",
        "from sympy import init_printing\n",
        "init_printing()  # This will make output look very nice!\n",
        "\n",
        "import math  # has things like math.pi (which is the numerical value of pi)"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
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        "outputId": "28095ce3-048e-4587-e09c-0e82835625f7"
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      "source": [
        "# First, all symbolic variables must be declared. This is in contrast to Mathematica\n",
        "x = sym.Symbol('x')\n",
        "\n",
        "# Now, let's ask for a Taylor series expansion\n",
        "sym.series( sym.cos(x), x )"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
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            "text/latex": "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{6}\\right)$$",
            "text/plain": [
              "     2    4        \n",
              "    x    x     ⎛ 6⎞\n",
              "1 - ── + ── + O⎝x ⎠\n",
              "    2    24        "
            ]
          },
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            "tags": []
          },
          "execution_count": 3
        }
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      "metadata": {
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      "source": [
        "# We can see the Docstring for the function to see all the options\n",
        "?sym.series"
      ],
      "execution_count": 4,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Jw6zt3zMExbp",
        "colab_type": "text"
      },
      "source": [
        "Before going further, let's play with cos(pi/4)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_IxXjVhHE_H6",
        "colab_type": "code",
        "colab": {
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          "height": 126
        },
        "outputId": "a52a0feb-b5d8-4cb9-cb4e-1277866cbc72"
      },
      "source": [
        "print( math.cos( math.pi/4 ) )\n",
        "# or, ask to print out 20 decimal places\n",
        "print( '%.20f' % math.cos( math.pi/4 ) )  # this is old-school: see https://docs.python.org/3/tutorial/inputoutput.html#old-string-formatting\n",
        "print( math.cos( sym.pi/4) )\n",
        "print( sym.cos( math.pi/4) )\n",
        "print( sym.cos( sym.pi/4) )\n",
        "sym.cos( sym.pi/4)  # same as above, but use fancy formatting"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.7071067811865476\n",
            "0.70710678118654757274\n",
            "0.7071067811865476\n",
            "0.707106781186548\n",
            "sqrt(2)/2\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\frac{\\sqrt{2}}{2}$$",
            "text/plain": [
              "√2\n",
              "──\n",
              "2 "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gC1aH5zrGNYm",
        "colab_type": "text"
      },
      "source": [
        "Now let's explore the series. In particular, let's try to evaluate cos( .75 ), using only addition and multiplication, and a few \"memorized\" values of sine/cosine (let's assume that we memorize the fact that $\\cos( \\pi/4 ) = \\sqrt{2}/2$ and that $\\sin( \\pi/4 ) = \\sqrt{2}/2$.\n",
        "\n",
        "Since .75 is not that far from $\\pi/4 \\approx 0.785$, let's do a Taylor expansion of $\\cos$ around $\\pi/4$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XFery7hKFwYb",
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          "height": 47
        },
        "outputId": "8bb68cd7-eb53-47b0-9137-94906b41e9de"
      },
      "source": [
        "Taylor = sym.series( sym.cos(x), x, x0=sym.pi/4, n=5)\n",
        "Taylor"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": 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QjpaHO+ZrojEWqqHlLN4xQBZ6moL8P9YiL/8rkEpHsKcqCQ3a1/IY4BZEWy8hlZv7gNOBTRzO3xmZdxlE9dqTJtnGXpAGt7VID3sY6n+7/z+JLAxXOJcihm5ftCEV4XzEX7sXbUiKZKGBM02ZB+RwrbpSR61BvnpzRXWZjDpqU3XZ96ijjqF8WlYdJ+NqWiMz/Khe3ek09m6GVBJdtqGqOln4/zpKMuVwCflMyq4Lg5BNvecUbUiKpK2BjZDl5cOG5Kje3Kmj1iBfvbmiukxGHbWpuux71FHHUD4tq46TsQnh83ZVr26kEXuPQLb1CWtsqBtp+38CMrRaqSgHARchm38rvRmNDGUYXqwZtUC1Fo/qrRhUm9GoLquB6jge1XL+DKe1grQf1Ws8qtdi2aloAxRFURRFURRFURRFURRFURRFURRFURRFURRFURRFURRF6VP8H+yQMrwNDWS2AAAAAElFTkSuQmCC\n",
            "text/latex": "$$\\frac{\\sqrt{2}}{2} - \\frac{\\sqrt{2}}{2} \\left(x - \\frac{\\pi}{4}\\right) - \\frac{\\sqrt{2}}{4} \\left(x - \\frac{\\pi}{4}\\right)^{2} + \\frac{\\sqrt{2}}{12} \\left(x - \\frac{\\pi}{4}\\right)^{3} + \\frac{\\sqrt{2}}{48} \\left(x - \\frac{\\pi}{4}\\right)^{4} + \\mathcal{O}\\left(\\left(x - \\frac{\\pi}{4}\\right)^{5}; x\\rightarrow \\frac{\\pi}{4}\\right)$$",
            "text/plain": [
              "                            2             3             4                     \n",
              "        ⎛    π⎞      ⎛    π⎞       ⎛    π⎞       ⎛    π⎞                      \n",
              "     √2⋅⎜x - ─⎟   √2⋅⎜x - ─⎟    √2⋅⎜x - ─⎟    √2⋅⎜x - ─⎟     ⎛       5       ⎞\n",
              "√2      ⎝    4⎠      ⎝    4⎠       ⎝    4⎠       ⎝    4⎠     ⎜⎛    π⎞       π⎟\n",
              "── - ────────── - ─────────── + ─────────── + ─────────── + O⎜⎜x - ─⎟ ; x → ─⎟\n",
              "2        2             4             12            48        ⎝⎝    4⎠       4⎠"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "n96sKGoaGofK",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        },
        "outputId": "26f1274e-7d35-4b94-a065-f862df0d54a2"
      },
      "source": [
        "Taylor.subs(x,.75) # Not what we want"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\mathcal{O}\\left(1\\right)$$",
            "text/plain": [
              "O(1)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YoFBLV1uIhIV",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 70
        },
        "outputId": "b78268a7-3a8b-4bf0-97da-63fa55ea2612"
      },
      "source": [
        "Taylor.evalf( subs={x: .75})"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": 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\n",
            "text/latex": "$$1.26246714845634 - 0.353553390593274 \\left(x - 0.785398163397448\\right)^{2} + 0.117851130197758 \\left(x - 0.785398163397448\\right)^{3} + 0.0294627825494395 \\left(x - 0.785398163397448\\right)^{4} - 0.707106781186548 x + \\mathcal{O}\\left(\\left(x - \\frac{\\pi}{4}\\right)^{5}; x\\rightarrow \\frac{\\pi}{4}\\right)$$",
            "text/plain": [
              "                                                                              \n",
              "                                                            2                 \n",
              "1.26246714845634 - 0.353553390593274⋅(x - 0.785398163397448)  + 0.117851130197\n",
              "                                                                              \n",
              "\n",
              "                                                                              \n",
              "                           3                                             4    \n",
              "758⋅(x - 0.785398163397448)  + 0.0294627825494395⋅(x - 0.785398163397448)  - 0\n",
              "                                                                              \n",
              "\n",
              "                      ⎛       5       ⎞\n",
              "                      ⎜⎛    π⎞       π⎟\n",
              ".707106781186548⋅x + O⎜⎜x - ─⎟ ; x → ─⎟\n",
              "                      ⎝⎝    4⎠       4⎠"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2Mz9Y2OHJP8Q",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 41
        },
        "outputId": "248c1ce2-97d3-4237-d0f7-87468cdd6951"
      },
      "source": [
        "# Numerically evaluating this was a problem, because of the O( x^5 ) term. Let's remove it.\n",
        "# See https://docs.sympy.org/latest/tutorial/calculus.html\n",
        "Taylor.removeO()"
      ],
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtgAAAAeCAYAAAAW/WmFAAAABHNCSVQICAgIfAhkiAAAC8lJREFUeJztnX+wFVUdwD/gw0AytF/QD+vlgMRYzgvSslHmYaRj4kjF9MNsvEaRM6FjVhpOjrcfE1b+wH6o2Y95Vs5ovIKGymIYYQJTy5ICI0PoVSRIpKIpWa/oj++5c/ct++Ps7jl7z7vv+5nZeXD27O73nv3cs+fsnj0XFEVRFEVRFEVRFEVRFEVRFCVExpXY5mCHj68oLpgM/NPh/tRlpVO4rJNBXVbcoHWsUiejvh6cA1xY90EVxTHqsdItqMtKiKiXSp0E6dv4gvnPAn7sIxBFqRH1WOkW1GUlRNRLpU6C9K1oA/sVwF99BKIoNaIeK92CuqyEiHqp1EmQvhVpYE8F9vgKRHHKMmQ80lc6HUiAqMfh82Hgd8CTZrkHuUOhjERdDp9lwK8Qj/8OrAFe09GI/KNeKnUSrG9FGthnAj/xFYjijDcCS5AGinIo6nH47AIuB2YDrwfuAlYDJ3QyqABRl8OnH7gReBNwGjAMrAOe38GYfKNeKnXSFb7dRPEhJUq9TAF2APOADegd7CTU49HJY8CHOh1EYKjLo4/nAv8Fzu50IB5RL5U6CdY326AmIJXC/zzFcSuwF5nWR0lmDjLs4wMZeW4BBoH1tUQ0+vDtMajLNti43OIw4N1Iw+QXPoMaZajLYVDEZYAjkevu494i6izqZRgU9XK0Uodv3pmHXOTivAA5gauAh4EDwH5gE7AYuwb8iUjhXOok0nCZiojw5ZT15yFfiKwvxSpgN9LYiPNB4NeIcGB3B9tFTCDjY9cij/YPADuBlcDJOcevmzSPQV0ugm+XAV6LzKE7DDxB9hjspHgWmf9vRMa/HgS+m7EPF+e/TtTl6uR5DG5cjvI94AGk4+gzpk6hXlanbi+jx9N68FC8l8/VwNEJ6RciJ/gR4DZgOfAt5IJ4ELmbmjdZ91qTf1KZwEYRS5AymZew7hikDJ4i+wtzkll/RSx9JvICzcxI2gbyG9guYvq8Wb8P+AbiyiDwb6QiPC8nhjpJ8xjU5SL4dLnF4cB05E7McsSvtJfDkuLZbNKeAraR38B2cf7rRF2uTpbH4M7lFtch5+TYGmLqFOplder2Mno8rQcPxVn5jAd6EtK/lpL/NGQsWbzVPg34izngOzICPw5phN2SkadbuBNpJMTvXIxDXnrZAXyR/EpzG/BnRpZ5w2w3HFkOImU7DDzHU0zTkJ7dHuDFsXXzzHY7Mz5LFg2zfX+JbYt6DOpyEXy6nMY64JsF4pkHzDAx9ZPfwK56/n2hLvsjzWNw7/L1yN3EV9cYk0/US3/U6WX8eL7qwQblr+fQGd9alC6faKajkTcx3xvb8FjkZCZxFzLtUHz8yx7gZvPv/ozA348Ic0fK+rXxgA3jgAGz7uqM/bvmLNqPZfKW10W2ex5yYtYgDdIoF5t1FwBPW8RwOzLn41siaauRR+p9keV+k7cPuZscx0VMr0Qcug8ZExdlPdLDflHeB3JMGY9hbLlc1mPw73Ia40nuKKbFsx7Yjv3P51Y9/z5Ql/PxUSeDW5dvAN5j9veHjH24jskX6mU+o8HLpONpPTiSSuUTbWAfDzwKLIhtuIByv5DzH/N3OCPPfCToe1PWfxz5IJ9hZG/uGuB8pCf7iRKxlWU/8sYqwIPApyJL60u/CbgK+G1kuwXII+9Vsf3NQr7wNwA/t4zhbvM3+oV5AtgaW55GZl7YSnIjw0VM25HG+0nAC2Pr5iIv9KzL2YdrXHsM3edyWY/Bv8uY/ZwK9CIdx+VIpXVbwj7S4nGJzfn3gbqcj+s6Gdy6/FWkMXQu8mLjNLMkjYt1HZMv1Mt8QvfS5nhxxlo92DqGs/KZgYgxIZJ2c0reLHqALUjD7oyUPJNNIFty9jVg9tMw/7/C/P8OOjPgfqE5/mdj6ZeZ9MUJ26xEXtiaGEnrQe4yP0R7PFmT/Ec+U0yeX+bEuYHsMdiuYroEqdT2IhXYcuRFnn8hdxTiQ0dsaVD+kZIrj6F7XS7jMdTj8gDyWPNZxKt1pJd9Ujxx+skfIpKGzfn3ibqcj6s6Gdy7nHbXsllDTD5RL/MJ2cu848WpWg82qDZEpG7fWjgvn4eAN5t/TwauzdlxEteYA2X1Lo4zedbm7OsY5E3NPwFLzTY/RXoVnaBpYlgUS7/dpM+JpU9EhkqsjKV/GumNR2faaO07r9I8QLVfLnId00Lkbnn0ArIduWtTlgbVvpAuPIbudblJMY8hPJfT4onTT/kGts359426nE0TN3UyhOlylZh8ol5m0yRsL23rT6heDzaodj2Hen0DT+VzLfJCBsA5yK/kFOFic6BtZP9a1cm0e5Z5LKfdcLsbOMIylqHIdjaLzQV4tck7I5b+R+QRQXyc6Nkmf7Sx+QakN/6FWN4mdl+Yv1HtMY3LmC4z212HjIk6AvkFvp+Z7eL7S2KIYudpwGKfVT2G7na5qMcQnstJ8STRT7kGtu35jzKE+zpHXc7GRZ0MYbpcNaYWQ6iXeQwxtry0rT+L1oNDuL+eQ72+gYPySXor80fIm5kfQS5My/IijrAUGR/0e6Sn8VhG3gPmb96td5Ap6FosBp6xjGcHMlTBlkcs8vQhjwwejqQdiUwp9iDyWDvK25Fxyq2eTQ/wbeRLdmWB2KJMol1+ZXAVUz8yTd8qRs5L+hvgbWZ/H0Ue5WTNJrICOCqW1od8iW5FvrBRNlvEVsVj6H6Xi3oM4bkcj8clRc5/FB91jrqcTdU6GcJ02UVMLdTLfMaalzb1Z5l60Mf1HOr1DTyVTw/y0txM4MbckNtcgrTit2A37valJv+mnHznImN8d5v8N2Vn98pRtHvFUU416d+JpR+GTO9yZ8I+bJYVCTGMR8oj6+3ZLFzG1HosclHKsX5g1ndiWp+yHkP3u1zUYwjP5aR40uin2B3souffN+pyOi7q5Oh+QnK5aky+US/TCd1Lm/rTZT3YoNr1HOr1zUn5JN3BHkYe738S+RU0Gy5H3m7djLytus9im90c+uMocd6KPD7YivQONiKPQ1Yg43Hqps/8jc+sMNv8jffE5iK//BN9A/VZ0ufznY1M2bMJ+Xz3JOSZiUw7ZNvri+MyptYjrrSp+FrpSdME+qaMxzA2XC7qMYTnclI8Lihz/n2jLqfjok6GMF2uGpNv1Mt0Qvcyr/4c6/Wg1/J5H9LreblF3iuRVvz92I9TbDFotp2esO4U5NHOTuAlJm2Ryb+64HFc0eqxLImlf92knx5L/xLyYsJUy/03yR9TdYHJs9Ryn3FcxvROs24P8LLYujMRhw4gohalQfUebxGPYey4XNRjCM/lIvH0Y3cHu8r59426nIzvOhnCcrlITHWgXiYTupdZx/NRDzaofj2H+nxzUj5Jd7BBJvV+ANiVE8T5tN9u3YgM9o4zRPog9u8jwwfOYOQ4pT5kvM1+pHew26QPIh/qHORRS5FejAvSeqWtQj4eGYuzC+k1LkR6lY86jOF0pLx/WGJb1zENIlOozUcG+K9CGtuzkPkjxyHzjv7DwbHKYOsxjC2Xi3gM4blsE89Cs4DMOwzyotSA+fc+4GOR/FXPv2/U5WRCqJPBr8sho14mE7KXWcfTerCm8unNy0C7B5W1bMjY/nDkA9wXSZuONNIeB05I2Ga+2W/aZPM+2YwUbPyt5HchF+xnaF+0T0TivBR7mmT3SKcgd4TL9sh9xDQB6a3fCzyJPMbZi1R4SXdCbWngpsfba5mvydhxuYjHEJ7LNvG0jp+2DBXMn3f+66DXMl8TddlVnQydd7lMTHXSa5mviXoZgpdZx2vt03U92MDN9Rz8+9aJ8vHGMiSo+E8yj3Y+h3yuVznc50Vmn6eU3N5HTEobddmeKi6rx/7pRpd9eaMu14d6aU+al+pcNl1VPhORX21b0+lAHLON8i8iJjEJmRZosMI+XMekjERdtqOqy+qxf7rRZR/eqMv1ol7akeWlOpdN15XPXOAq5Nd5lGRmIY8nejsbhpKDupyPujw6UJfzUZfrR73MR71UFEVRFEVRFEVRFEVRFEVRFEVRFEVRFEVRFP/8H5gl1fRoCTWoAAAAAElFTkSuQmCC\n",
            "text/latex": "$$\\frac{\\sqrt{2}}{48} \\left(x - \\frac{\\pi}{4}\\right)^{4} + \\frac{\\sqrt{2}}{12} \\left(x - \\frac{\\pi}{4}\\right)^{3} - \\frac{\\sqrt{2}}{4} \\left(x - \\frac{\\pi}{4}\\right)^{2} - \\frac{\\sqrt{2}}{2} \\left(x - \\frac{\\pi}{4}\\right) + \\frac{\\sqrt{2}}{2}$$",
            "text/plain": [
              "          4             3             2                  \n",
              "   ⎛    π⎞       ⎛    π⎞       ⎛    π⎞       ⎛    π⎞     \n",
              "√2⋅⎜x - ─⎟    √2⋅⎜x - ─⎟    √2⋅⎜x - ─⎟    √2⋅⎜x - ─⎟     \n",
              "   ⎝    4⎠       ⎝    4⎠       ⎝    4⎠       ⎝    4⎠   √2\n",
              "─────────── + ─────────── - ─────────── - ────────── + ──\n",
              "     48            12            4            2        2 "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3IAVBOLaKrn1",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 68
        },
        "outputId": "f5337a99-593d-4553-947b-74f3e5365493"
      },
      "source": [
        "guess = Taylor.removeO().subs(x,.75).evalf()\n",
        "print(\"Our guess is: \\t%.15f\" % guess)\n",
        "print(\"True value is: \\t%.15f\" % math.cos( .75) )\n",
        "print(\"Discrepency is: \\t%.3e\" % abs( guess - math.cos(.75) ))"
      ],
      "execution_count": 31,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Our guess is: \t0.731688868548266\n",
            "True value is: \t0.731688868873821\n",
            "Discrepency is: \t3.256e-10\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Z3sKMUtuLldm",
        "colab_type": "text"
      },
      "source": [
        "Our *absolute* error is about $3\\times 10^{-10}$. Is this what we'd expect? Let's use math to be able to **guarantee** a small error (without having known the true answer first)\n",
        "\n",
        "Let's look at the remainder term in the Taylor series"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CzffLsu2K2e6",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 47
        },
        "outputId": "de699bfb-62fb-4f71-d8af-7ccae5c15173"
      },
      "source": [
        "x0=sym.Symbol('x0')\n",
        "sym.series( sym.cos(x), x, x0=x0, n=5)"
      ],
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\cos{\\left (x_{0} \\right )} - \\left(x - x_{0}\\right) \\sin{\\left (x_{0} \\right )} - \\frac{1}{2} \\left(x - x_{0}\\right)^{2} \\cos{\\left (x_{0} \\right )} + \\frac{1}{6} \\left(x - x_{0}\\right)^{3} \\sin{\\left (x_{0} \\right )} + \\frac{1}{24} \\left(x - x_{0}\\right)^{4} \\cos{\\left (x_{0} \\right )} + \\mathcal{O}\\left(\\left(x - x_{0}\\right)^{5}; x\\rightarrow x_{0}\\right)$$",
            "text/plain": [
              "                                     2                   3                   4\n",
              "                             (x - x₀) ⋅cos(x₀)   (x - x₀) ⋅sin(x₀)   (x - x₀) \n",
              "cos(x₀) - (x - x₀)⋅sin(x₀) - ───────────────── + ───────────────── + ─────────\n",
              "                                     2                   6                   2\n",
              "\n",
              "                               \n",
              "⋅cos(x₀)    ⎛        5        ⎞\n",
              "──────── + O⎝(x - x₀) ; x → x₀⎠\n",
              "4                              "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Z9pmRsOdK-sM",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "outputId": "3e9bc62f-9ebb-4b0d-dbe2-ad023093007e"
      },
      "source": [
        "xi = sym.Symbol('xi')\n",
        "Remainder = -(x-x0)**5*sym.sin(xi)/120\n",
        "Remainder"
      ],
      "execution_count": 27,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$- \\frac{1}{120} \\left(x - x_{0}\\right)^{5} \\sin{\\left (\\xi \\right )}$$",
            "text/plain": [
              "         5        \n",
              "-(x - x₀) ⋅sin(ξ) \n",
              "──────────────────\n",
              "       120        "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 27
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CqhBjSNzM7Jz",
        "colab_type": "text"
      },
      "source": [
        "By Taylor's remainder theorem, we know $\\xi \\in [x,x_0]$ (or $\\xi \\in [x_0,x]$ if $x_0 < x$) such that the above remainder is the error in the Taylor series. We don't know exactly what $\\xi$ is, but that's OK, because for any $\\xi$, we can just bound $|\\sin(\\xi)| \\le 1$.  So using this,..."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kbO87uahMoZC",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "outputId": "6cc7770e-f90b-47ba-ebf3-71d9ef059a6c"
      },
      "source": [
        "RemainderBound = -(x-x0)**5/120\n",
        "RemainderBound"
      ],
      "execution_count": 29,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$- \\frac{1}{120} \\left(x - x_{0}\\right)^{5}$$",
            "text/plain": [
              "         5 \n",
              "-(x - x₀)  \n",
              "───────────\n",
              "    120    "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 29
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "csDn1GNMNdqW",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        },
        "outputId": "072428e2-1a20-432a-886e-a0463b6dc180"
      },
      "source": [
        "# See https://docs.sympy.org/latest/tutorial/basic_operations.html\n",
        "RemainderBound.subs( [ (x,.75), (x0,math.pi/4) ])"
      ],
      "execution_count": 30,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$4.6315078243002 \\cdot 10^{-10}$$",
            "text/plain": [
              "4.63150782430020e-10"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 30
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-EbkbRFTN9mP",
        "colab_type": "text"
      },
      "source": [
        "To summarize, our Taylor series approximation had error $3.25 \\times 10^{-10}$.  We were able to use math to give an *a priori* bound that the error was no more than $4.63 \\times 10^{-10}$ (though in that bound, we disregarded potentially cancellation errors when summing the series)"
      ]
    }
  ]
}