Physics 780.20: Assignment #3

• The updated version of derivative_test.cpp is available here.
• Note that extrap_diff calls central_diff with two different h's and then extrap_diff2 (which you are to write) calls extrap_diff twice (and NOT central_diff twice). Your error plot should show a slope of h⁶.
• When writing an adaptive code (first bonus problem), you can assume that the round-off error part of the error is given by the machine precision divided by h (as described in Chapter 8). In practice this may overestimate the actual error.
• If you know that the relative error scales in the "approximation error region" as b*h² for the central difference formula. How can you determine b from evaluating the derivative at two different h values (without knowing the exact answer)?
• The results are most interesting is you pick a value for the harmonic oscillator parameter b that is not optimal. Or, if possible, compare results with two different values of b.
• Assign variables to the minimum, maximum, and increment values for the radial value r. For example, rmin=0., rmax=10., delta_r=0.05 are reasonable choices.
• Plot the results for different dimension bases on the same gnuplot plot. This is made much easier by using a plot file that uses a different datafile for each basis size, rather than trying to make one datafile with all of the results. (You can rename the output files from your code each time you change the basis size.)
• When commenting on the nature of the convergence, you might note what part of the wave function is reproduced first (e.g., when is the tail reproduced?). If you chose a different value for b, how might the convergence change? (What is most important to get a fit with dimension size 1?)
• For the second bonus problem, think about how you measure the goodness of fit when you fit a straight line to some data points. Can you do something analogous when you have a function of r like a wave function?