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    "# Numerical Methods 1\n",
    "### [Gerard Gorman](http://www.imperial.ac.uk/people/g.gorman), [Matthew Piggott](http://www.imperial.ac.uk/people/m.d.piggott), [Nicolas Barral](http://www.imperial.ac.uk/people/n.barral)"
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   "source": [
    "# Revision exercises"
   ]
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   "source": [
    "## Question 1\n",
    "\n",
    "Below is a extract of shot data from a seismic survey.\n",
    "\n",
    "|  Time (ms)  |  Shot  |\n",
    "|-----|-----|\n",
    "| 0 | -0.021373 |\n",
    "| 4 | -0.024578 |\n",
    "| 8 | -0.023914 |\n",
    "| 12 | -0.018227 |\n",
    "| 16 | -0.00781 |\n",
    "| 20 | 0.005602 |\n",
    "| 24 | 0.019264 |\n",
    "| 28 | 0.030235 |\n",
    "| 32 | 0.036059 |\n",
    "| 36 | 0.035334 |\n",
    "\n",
    " 1. Calculate the Lagrange polynomial for these points. Plot both the Lagrange polynomial and the raw data points.\n",
    " 2. The full shot is available in the file [shot.txt](data/shot.txt) (in the data folder) - where the sample interval is 4ms as above. Note that the file only contains one column as you can calculate the time column yourself. Use cubic-polynomial splines to re-interpolate the data for a sample interval of 7.07ms. Plot both the original shot data and the interpolated time series."
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   "source": [
    "## Question 2\n",
    "\n",
    "1. Calculate the upper triangular form for the matrix:\n",
    "\\begin{align*}\n",
    "A = \\begin{bmatrix}\n",
    "−5 &  3 &  4\\\\\n",
    "10 & −8 & −9\\\\\n",
    "15 &  1 &  2\n",
    "\\end{bmatrix}\n",
    "\\end{align*}\n",
    "\n",
    "2. Consider the matrix:\n",
    "\\begin{align*}\n",
    "C = \\begin{bmatrix}\n",
    "−5 &  3 &  4\\\\\n",
    "10 & −8 & −9\\\\\n",
    "15 &  1 &  2\n",
    "\\end{bmatrix}\n",
    "\\end{align*}\n",
    "Does matrix $C$ have an inverse? If not, then why not? "
   ]
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   "metadata": {},
   "source": [
    "## Question 3\n",
    "\n",
    "Consider the function\n",
    "$$f(x) = \\dfrac{1}{(x − 0.3)^2 + 0.01} - \\dfrac{1}{(x − 0.8)^2 + 0.04}$$.\n",
    "\n",
    "\n",
    " 1. Write a function that computes the second derivative of $f(x)$ using central differencing. The interface to your function should look like *central_diff2(f, x, h)* where $f$ is the function to be differentiated, $x$ is the position at which the derivative should be estimated, and $h (= \\Delta x)$ is the step size.\n",
    " 2. Use this function to compute the derivative at $x = 0.5$ for decreasing values of $\\Delta x$. Start with $\\Delta x=1.2$ and keep halving until the relative difference between solutions falls below $1.0^{-6}$ Plot the convergence of the method, i.e. plot $\\Delta x$ against the absolute difference between the analytical value and the finite difference approximation."
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   "source": [
    "## Question 4\n",
    "\n",
    "Consider the integral:\n",
    "$$\\int_0^{2\\pi} x^2\\cos(x) dx.$$\n",
    "\n",
    "Show how the absolute error between the exact solution and the numerical solution varies with the size of the integration step for the four numerical integration methods: trapezoid rule; Simpson's rule; composite Simpson's rule; and Weddle's rule. Show your result by plotting error against integration step, $dx$."
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