{"cells": [{"cell_type": "markdown", "metadata": {"tags": ["module-mm1", "module-mm2", "module-nm"]}, "source": ["(taylor_series)=\n", "# Taylor series\n", "[Mathematics Methods 1](module-mm1) [Mathematics Methods 2](module-mm2) [Numerical Methods](module-nm) \n", "```{index} Taylor series\n", "```\n", "A Taylor series is a method by which we can approximate an arbitrary function by a sum of terms made up of that function and its derivatives at a single point.\n", "\n", "This concept underlies many things (especially in computational science) so it's worth us spending some time reviewing it.\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["## Motivation - a constant approximation\n", "\n", "Consider a function of a single independent variable: $f(x)$\n", "\n", "Suppose this function is really really expensive to evaluate. Suppose I have already evaluated it at a single $x$ value, call it $x_0$, i.e. I know the value of $f(x_0)$.\n", "\n", "Given only this information, what is the best guess (estimation/approximation) I can make for $f(x)$ for a choice $x\\ne x_0$?\n", "\n", "Well I can't really do anything better than \n", "\n", "$$ f(x) \\approx f(x_0)$$\n", "\n", "This may seem a hopeless approximation, but it can be used in some situations. Pretend $f$ is a weather forecast and $x$ is time. This just says that for a forecast of the weather tomorrow use what we see today - this is a real technique called [*persistent forecast*](https://en.wikipedia.org/wiki/Weather_forecasting#Persistence) (\"today equals tomorrow\").\n", "\n", "\"... This makes persistence a 'hard to beat' method for forecasting longer time periods\": \n", "\n", "Let's consider an example:"]}, {"cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": ["%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt"]}, {"cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": ["def f(x):\n", " return np.exp(x)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["and plot our current situation, assuming that $x_0=0$. We call this the expansion point - we are constructing an approximation \"around this point\"."]}, {"cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [{"data": {"text/plain": [""]}, "execution_count": 7, "metadata": {}, "output_type": "execute_result"}, {"data": {"image/png": 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\n", 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