原创 Random thoughts (Part 1)

This is going to be a different kind of article for me. As I was planning it, I considered several possible topics. I couldn't decide which one to cover, so I decided to cover them all. You may find some of them incomplete—I'm giving you my thoughts as I'm having them. I welcome your thoughts and comments.

Vector/Matrix class redux

Whenever I set out to (re)create a vector/matrix math package, I always wrestle with the decision: should I include separate functions for the special case where the vectors are dimensioned 3, and the matrices, 3×3? Clearly, classes that can handle general, n-dimensional vectors and m × n matrices can also handle the case m = n = 3. On the other hand, it's certain that if we know in advance the dimensions of the objects, the code will be a little faster and more compact.

While it's important to be able to deal with general forms of the mathematical entities called vectors and matrices, there's a good argument for the specialized functions as well.

We live, after all, in a three-dimensional universe (unless your field is String Theory). Most of the computations I do—space mechanics, rocket and aircraft dynamics, robotics, dynamic simulations—involve the motions of real, physical bodies in this three-dimensional space. It's the discipline called dynamics, whose math is defined by Newton's laws of motion.

The mathematical entities called vectors and matrices weren't exactly invented specifically to deal with physics problems, but they might as well have been. The simplifications that result from their use in physical problems is so profound, it's hard to over-emphasize the point.

That being the case, I'm always tempted to define special computer functions to process 3-d vectors and matrices. Usually I succumb to the temptation. When I was developing the C++ classes called Vector and Matrix, I wrestled with the same decision. For the sake of brevity, I chose to omit the special 3-d case. For the column, I was focused more on showing you the principles rather than defining a production-quality library of functions.

However, the argument for the special case is even stronger than usual, when defining C++ classes. That's because my general-case solutions required dynamic memory allocation.

Any time we declare a new object of a certain class, whether it be a named variable or an anonymous temporary, the compiler is going to have to invoke the memory manager. That's a given. But in my general-purpose functions, I also had to dynamically allocate the storage for the object's member data, because we don't know the size of the data until the constructor is invoked. So with every declaration of one of the general-purpose objects, we get a double dose of memory allocation.

If you declare all named objects statically, you can move most of the memory allocations to the initialization phase of the programs. We might do this, for example, when writing flight software for a satellite or missile.

But if you make use of those lovely overloaded operators, you can't avoid the creation of temporary objects, and the overhead to construct and destruct them.

Finally, there's the issue of the vector cross product, which only works for 3-vectors. In our n-vector class, I had to include tests to make sure that both vectors were, in fact, dimensioned 3. That's not a problem in the special case. (Optional for extra credit: some of us gurus in the know, know that the cross product isn't really limited to 3-vectors. There's a 4-d cross product as well. If you're "one of us," drop me a line. Say the magic word, "quaternion," and I'll send you the Secret Decoder Ring Tone.)

If we know in advance that all the vectors will be dimensioned 3, and matrices, 3×3, the double dose of dynamic allocation goes away.

The situation with respect to matrices is even nicer. If you followed the creation of the Matrix class, you'll recall that we struggled with the issue of allocating matrices with arbitrary dimensions. Because C/C++ doesn't support conformant arrays, we can't just declare the array data in the form:

double x[m][n];

Instead, we had to resort to defining a single-dimensioned array and generating our own code to index into it. This isn't a problem for the 3×3 case. We simply declare the array:

double x[3][3];

and let the compiler use its own indexing, presumably optimized out the gazoo.

Recently, I had occasion to do a lot of computations in the 3-d universe. So I succumbed to the temptation and wrote a new C++ class called ThreeVec. As the name implies, the class is specialized to the case of 3-d vectors. I'm going to show it to you, but in a rather unconventional way. Instead of walking you through all the design decisions yet again one more time, I'm simply going to post the files on Embedded.com. I think you'll find that the differences between the n-vector and 3-vector cases are straightforward enough so that you won't need me leading you by the hand. Please let me know if you need more help.

Do something ...

For the same project, I also need a class for 3×3 matrices. I'll tell you about it soon, but first I have a confession to make. I have a character flaw (No! Really?).

I'm basically an optimizing sort of fellow. Before I do anything, I tend to look at the options, weigh them carefully, and choose the one that's, in some sense, optimal. I'm the guy who tries to guess which lane I should be in at a traffic light. There's no malice involved, no jamming my way in front of someone else. But given a choice, I'll choose the option that's most likely to get me where I'm going sooner with fewer hassles.

This tendency to optimize can be both a blessing and a curse. A blessing in that the code I generate is usually pretty good. A curse because it takes me longer to produce it. Because I'm looking for the best solution, I can never keep up with the software wizard who simply codes the first thought that comes to him.

In another lifetime, I built and raced midget cars. My kids also raced, and I was their chief mechanic. Naturally, optimizing the route to the front is a good habit for a driver, and I was pretty good at it. But optimization is also good when deciding how best to tune or modify the car. When it came time to make a modification, I'd find myself spending considerable time trying to optimize the design. The design optimization process goes something like this:

while(1){

if(option(a) < option(b)) return a;

if(option(b) < option(a)) return b;

}

The problem, of course, is that there's no predicate for equality. If I can't find a clear reason to choose one option over the other, I'm paralyzed into inaction, stuck in an infinite loop.

This sometimes happened when I was planning a new modification. I'd sit in the garage, holding the parts I could use, and literally weighing my options. I'd look at the parts in one hand, and see advantages in using them, but also disadvantages. I'd look at the other and see a different sets of pros and cons. I could find no clear winner.

Fortunately, my decision-making system had a priority interrupt. Not hearing any work being done, my wife would open the door behind me. She'd shout, "Do something, even if it's wrong!!!" and slam the door again. This interrupt was usually enough to jog me out of the loop.

[Continued at Random thoughts (Part 2)]