Re: About diagonal matrices

[John W. Eaton]

On 20-Feb-2009, Daniel J Sebald wrote:

| If I'm a user who wants to think in terms of the extended real
| number system, then limits are of interest, i.e., Inf is the limit
| as series x_i becomes positively unbounded (but not negatively
| unbounded).  1/0 is fraught with problems though because it doesn't
| indicate how the limit is approached.  That is, how do we know that
| 1/0 is +Inf and not -Inf?  1/+0 could mean approach zero from the
| left, so 1/+0 = +Inf.  Likewise 1/-0 = -Inf.  Octave agrees with
| that train of thought, e.g.,
| 
| octave:20> 1/(+0)
| warning: division by zero
| ans = Inf
| octave:21> 1/(-0)
| warning: division by zero
| ans = -Inf
| 
| But here is where the trouble shows up:
| 
| octave:22> 1/(+0-0)
| warning: division by zero
| ans = Inf
| octave:23> 1/(-0+0)
| warning: division by zero
| ans = Inf
| 
| The problem is trying to treat 0 as both a number in the number system and 
the notion of a one-sided limit.  (Perhaps what some mathematician should have 
done long ago was to define the extended real number system as R union {+Inf, 
-Inf, +Zed, -Zed}.  Anyway...)
| 
| And how about this one, irrespective of the previous comments?
| 
| octave:42> sqrt(-0)
| ans = -0
| 
| Shouldn't that be complex 0 + 0i?  That is,
| 
| octave:44> sqrt(-2)
| ans =  0.00000 + 1.41421i
| octave:45> sqrt(-1)
| ans =  0 + 1i
| octave:46> sqrt(-0.5)
| ans =  0.00000 + 0.70711i
| ...
| sqrt(lim x-> 0 from left) = 0 + 0i.

We try to do what is right for IEEE floating point.  Maybe sqrt should
be fixed.

There are a number of other problems with the way we (and every other
language that I know of) handle complex numbers.  For example, since
we don't have pure imaginary values, I think things like i/0 or i*Inf
fail to do the right thing.

jwe