Re: Sorting complex values

[Rik]

On 09/27/2014 11:43 AM, Daniel J Sebald wrote:
>> Should we fix this?
>
> Maybe.
>
> First, complex numbers aren't an ordered field, so sorting is fraught
> with peculiarities.  It's not really -1 being at a particular point in
> the order, it's a number (-1) whose magnitude is the same as three others
> and appearing at a perhaps arbitrary location.  Swap around the order in
> which the magnitude 1 numbers are input and it might come out with a
> different order output.

Matlab and Octave have placed an ordering on the field.  First magnitudes
are compared, and if those are equal then phase angles are compared. 
Regardless of the input order in the vector, the sort always produces the
same result.  The question is how to handle inputs which most people would
consider identical (-1-0i and -1+0i), but which Octave treats differently.

>
> The one thing that concerns me a bit is that -1 - 0i produces an angle of
> -pi.  Typically when one speaks of argument (going from the complex
> number to the argument and not vice versa) it is the principle argument,
> having a range whose lower limit is open, i.e., (-pi,pi].  Often in phase
> diagrams, the phase is unwrapped so that pi may be remapped to ..., -3pi,
> -pi, pi, 3pi, ... but that is a secondary use.  So even though
> atan(-1,-0) might produce -pi, maybe anything with -0 as the imaginary
> component should be set equal to pi for the argument.
>

That is indeed the problem that we are inheriting from the atan2 function
which is defined on the closed range [-pi, pi].  We could unwrap the phase,
we could arrange for all -0i entries to be replaced with +0i entries before
the atan2 call, we could ignore it, we could ....?

--Rik