Math Debugging

I’ve gotten a late start on this rendering thing.  One of the pieces I’ve been struggling with is math.  I’ve improved quite a bit over the past year and I’m actively educating myself to catch up.  Recently, a friend of mine asked me for some help with a matrix problem.  Here’s what I told him to help him debug (I wish I could go back in time and tell myself this!):

• If you’ve got a series of matrix multiplications, make all of the terms identity.  Then change each to be a real value one by one.  For example: Let’s say you’re rendering an object with a lot of sub-meshes or sub-objects.  You’d set the subobject’s transforms to identity and the main object’s transform to identity.  This will make everything render on top of each other.  From here, you concentrate on getting the sub-objects transforms correct, then finally work on the main object’s transform.
• Try creating situations where the results of an equation would be identity.  For example:  When using a shadow map, you need a matrix which transforms a point from camera space to light space.  If the camera and the light have the same transform (and FOV, view distance, etc) then this matrix will be identity.  If you run it through your code and it’s not, you know you’ve got some problems to solve!
• Pay close attention to what space your source data is in.  If you feed object space points to a function that expects world space points, it’s not going to work!
• Check out Octave, it’s basically a free version of MATLAB.  You can use this to play with matrices and vectors, probably much quicker than you can do in-engine.

These are just general divide and conquer debugging tips applied to math.  But for a person new to large amounts of math in programming, it can be non-obvious to think about applying those debugging techniques to math problems.  Does anyone out there have their own favorite math debugging tips?