Information for Instructors


Because iterative methods are so important in solving so many of today’s real world problems, it is important that a student’s first exposure to iterative methods be as positive and simple, as well as informative and educational, as possible. With this in mind, the main purpose of the tutorials and Java applet is to allow for easy visualization and experimentation. In more detail, here are the primary ideas of the module:

Features of the applet:

One note regarding the fact that the applet deals only with 2 x 2 sytems. Obviously we would never use an iterative method to solve a 2 x 2 sytems. Indeed, we would not use an iterative method to solve any small (64 x 64, 512 x 512, etc.) system. In my own image processing research I typically deal with (sparse) systems of 65536 x 65536 (2562 x 2562) or 262144 x 262144 (5122 x 5122). Also, we don't generally use any of the three methods (Jacobi, Gauss-Seidel, SOR) discussed in the tutorial or included in the applet, although some of their basic ideas are found in any iterative method and sometimes the ideas unique to each idea are actually found in preconditioners. Still, these methods are very straightforward, which makes them relatively easy to understand, which is why they are often a student's first taste of iterative methods for solving linear systems.

Furthermore, when dealing with the 2 x 2 case (as opposed to 3 x 3, or especially larger systems), it is much easier to visualize the results of these methods, which in my experience both as a learner and a teacher is very valuable. With this said, the tutorial and the experiementation and visualization that are possible via the applet are good as introductions to iterative methods for solving Ax = b, but of course for a numerical analysis or numerical linear algebra course, further treatment of these topics would be both appropriate and necessary. In particular, in further treatment of these topics, it is important to point out that some of the simple and elegant results seen in the 2 x 2 case are no longer true when dealing with larger systems.

Often when first introducing iterative methods for solving linear systems, only the Jacobi and Gauss Seidel Methods are considered. The SOR Method is consequently presented at the end of Part 2 of the Tutorial (page 9), as an optional topic.

Here are a few suggestions for you on how to use this module and the applet. You can:

Since creating this tutorial and applet, I have taught linear algebra twice. Both times I had to go to a conference about a month into the semester. I had the students read the first part of the tutorial and use the applet to do the Exercises (page 6) completely on their own in my absence. Then later in the semester, after we had covered eigenvalues and eigenvectors, we discussed in class the ideas covered in the second half of the tutorial, and the students did Exercises from the second set (page 10). With this approach, students get two tastes of iterative methods and numerical linear algebra: At first, they simply see what the methods do, and later they get to understand how they work.

The homework exercises comprise problems corresponding to Part 1 of the tutorial (Exercises 1 – 6, page 6) and to Part 2 of the tutorial (Exercises 7 – 18, page 10). I designed each homework problem carefully and thoughtfully to both pique students' curiosity and to challenge them, taking advantage of the applet, which does most of the tedious computation, and which allows for clear visualization. There are several possibilities for student exercises. You can:

See the Exercises Goal Chart for more details on the particular ideas each homework problem was designed to explore.

For the applet and all other files:
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