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From: Leland Brown (lelandbrown_at_[hidden])
Date: 2006-06-09 01:28:24

Andy Little wrote (in an earlier post):

> OK, what I was going onto say was that this is only the beginning of the
> complexities involved in strong type checking on physical quantities .

Yeah, and wait till you hear this one! It gets worse - though I also have a
solution that works for me. Read on, if you dare :-)

Warning - this post is primarily for the more mathematically inclined - though
I think it brings up an important issue that anyone wanting to implement a
matrix library on dimensioned quantities needs to think about.

John Phillips <phillips <at>> writes:
> Maybe I'm not getting it yet, but I don't think this gets you out of
> the woods.
> The problem I see is that matrices are used for many things other
> than just transformations. For example, a user might want to diagonalize
> a matrix, or find the eigenvalues, or any number of other things. If so,
> having units on the matrix becomes problematic, I think.

With certain restrictions, this can be done. But it brings up other problems.

It's a basic tenet of matrix algebra that for a square matrix A (as long as it
has an inverse), both of these are true:

   A * inv(A) = I
   inv(A) * A = I

where I is the identity matrix of the same size as A. Now suppose A is a
matrix of elements having different dimensions. For instance, if A is a 2x2
covariance matrix of position and velocity values, it might have units like

      m^2 m^2/s
      m^2/s m^2/s^2

Its inverse has well-defined dimensions, with the units you might expect, like:

      1/m^2 s/m^2
      s/m^2 s^2/m^2

But if you multiply A * inv(A) or inv(A) * A, you'll get two different
products, neither of which is the identity matrix!!! The results are:

   A * inv(A) =
      1 0 s (I call this "I1" below)
      0/s 1

   inv(A) * A =
      1 0/s (I call this "I2" below)
      0 s 1

   I =
      1 0
      0 1

Of course, we all learned in high school physics that "0" and "0 seconds" are
*not* the same thing, and Andy's library faithfully embodies that truth. Thus,
we have three *different* matrix values above. To make matters worse - for the
same reason, neither A * I = A nor I * A = A. In fact, neither expression
(A * I) nor (I * A) will compile, because they both involve addition of
quantities with different dimensions. This addition would be (correctly, I
think) not allowed by the library, even though in this particular case, one of
the addends has a numerical value of zero. To multiply A by the identity, we
would need (I1 * A) or (A * I2). What a mess!

BTW, I believe John's example of Guassian elimination may fail not only because
of things like pivoting (as he points out), but even without that, because of
this issue - addition or subtraction of values where one is numerically exactly
zero but has the wrong dimensions.

So what's a programmer to do? In my case, I was already using something akin
to "t3_quantity" for these "mixed" matrices, and in that context I came up with
the following solution:

I introduced a special value that a t3_quantity can hold, called an
adimensional zero. Normally, a value of my t3_quantity has specific dimensions
(or is specifically dimensionless), even if its numerical value happens to be
zero. An adimensional zero is a special case. What this means is that its
dimensions are undefined, ambiguous. Semantically, this means it can be added
to a value of any dimension (zero + x = x, of course, with the dimensions of
the result the same), and when multiplied by any value the result is another
adimensional zero (zero * x = zero).

I define a constant called "zero," and then define the identity matrix as

   I =
       1 zero
      zero 1

Now (I1 == I) and (I2 == I) both evaluate to true, (A * I) and (I * A) both
compile and give A as the result, etc. Problem solved! It works out quite
well in all my linear algebra. But sadly, at the price of increasing the run-
time overhead already required by my "t3_quantity."

Anybody got a better idea?

-- Leland

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