Question

2. (5 points) Do the same thing as in the previous problem, but now include the fourth-order Runge-Kutta method (that you have written yourself) as well. In this problem, treat the fourth-order Runge-Kutta solution with $n = 10^5$ as the true solution. Also, in addition to plotting the $\log_{10}$ of the errors (i.e., $|y_n - \hat{y}(t_n)|$) versus the $\log_{10}(n)$, plot the $\log_{10}$ of the errors versus the $\log_{10}$ of the number of function evaluations (i.e., 1 the number of times that you evaluated $f(t, y)$) for each method. Has this changed the slopes compared with the plot where the horizontal axis was the number of steps ($\log_{10}(n)$)? 3. (3 points) Do the same thing as in the previous problem, but now include Euler's method. From the log-log plot, estimate the error in Euler's method with $n = 1000$ steps.

          2. (5 points) Do the same thing as in the previous problem, but now
include the fourth-order Runge-Kutta method (that you have written
yourself) as well. In this problem, treat the fourth-order Runge-Kutta
solution with $n = 10^5$ as the true solution. Also, in addition to plotting
the $\log_{10}$ of the errors (i.e., $|y_n - \hat{y}(t_n)|$) versus the $\log_{10}(n)$, plot the $\log_{10}$
of the errors versus the $\log_{10}$ of the number of function evaluations (i.e.,
1
the number of times that you evaluated $f(t, y)$) for each method. Has
this changed the slopes compared with the plot where the horizontal
axis was the number of steps ($\log_{10}(n)$)?
3. (3 points) Do the same thing as in the previous problem, but now
include Euler's method. From the log-log plot, estimate the error in
Euler's method with $n = 1000$ steps.

2. (5 points) Do the same thing as in the previous problem, but now
include the fourth-order Runge-Kutta method (that you have written
yourself) as well. In this problem, treat the fourth-order Runge-Kutta
solution with n = 10^5 as the true solution. Also, in addition to plotting
the log10 of the errors (i.e., |yn - ŷ(tn)|) versus the log10(n), plot the log10
of the errors versus the log10 of the number of function evaluations (i.e.,
1
the number of times that you evaluated f(t, y)) for each method. Has
this changed the slopes compared with the plot where the horizontal
axis was the number of steps (log10(n))?
3. (3 points) Do the same thing as in the previous problem, but now
include Euler's method. From the log-log plot, estimate the error in
Euler's method with n = 1000 steps.

Added by Stephanie P.

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