Re: About diagonal matrices

[Daniel J Sebald]

Daniel J Sebald wrote:
> John W. Eaton wrote:
>
> > 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.
>
>
> octave:2> i/0
> warning: division by zero
> ans = NaN - NaNi
> octave:3> i*Inf
> ans = NaN + Infi
>
> That is a bit messy, isn't it?  As a slight fix for consistency, I'd 
> think these two should match, i.e., i/0 = NaN + Infi.

Should "Infi" be allowable syntax?  Otherwise the only way of creating 0 + Infi 
is

octave:16> complex(0,Inf)
ans =    0 + Infi

Here's a bug:

octave:18> complex(0,Inf) - Inf
ans = -Inf + Infi
octave:23> complex(0,-Inf) + Inf
ans = Inf - Infi

That should be -Inf + NaNi, I'd think.  Same sort of thing here:

octave:28> complex(0,-Inf) / 0
warning: division by zero
ans = NaN - NaNi

should be NaN - Infi.


Perhaps there should be some test functions that come along with the function 
"complex", once we figure out (i.e., guess or arbitrarily assign) what the 
answer should actually be.

Dan