Homework 8, due November 20, 2013.

[goodwine]

Reading: All of chapter 12 except 12.6. I have not lectured on 12.5 but you need to know it for this assignment. Even if you can often do assignments without doing the reading (I, admittedly, was that way), don't attempt it on 12.5. It will be easy to think you are doing it correctly when you are not.

Exercises: 12.7, 12.11, 12.12 and 12.xx (below).

12.xx: Write a computer program that determines an approximate numerical solution using fourth order Runge-Kutta to x' = 40 x (1 - x) where x(-1) = 1/(1 + exp(40)). You probably need to use double precision. Compare it with the exact solution (given last week). Does your program work better than ode45()?

Note: 12.7 can be very long unless you organize your program and plotting in a way to get the stuff organized from the beginning. If you do that, it's not too bad. In the past, a large number of students have totally missed the point of this problem, which isn't to make you write a bunch of different methods and run them at a bunch of different time steps (if you think that is the point of what I want you to do, you should be insulted); it's to compare and contrast the manner of convergence of the different methods and experience for yourself "If I cut the time step in half, the error goes down by XYZ..."

[epalutsis]

For 12.11, do we need to plot the x(t) solutions for both programs? Is it acceptable to just output the corresponding values at t = 3 and compare each in that manner?

[goodwine]

For 12.11, do we need to plot the x(t) solutions for both programs? Is it acceptable to just output the corresponding values at t = 3 and compare each in that manner?

You only need to compare the values at t=3 to answer the question. However, if you don't understand everything going on with the problem to fully answer it, it may be helpful to you to plot both to help you figure it out.