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MathGroup Archive 2007

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Re: Limit and Root Objects

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72620] Re: Limit and Root Objects
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Fri, 12 Jan 2007 06:05:49 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <em606c$2o1$1@smc.vnet.net> <4586C045.2020805@metrohm.ch> <em8jfr$pfv$1@smc.vnet.net>

In article <em8jfr$pfv$1 at smc.vnet.net>,
 Andrzej Kozlowski <andrzej at akikoz.net> wrote:

> What you describe, including the fact that the numbering or roots  
> changes is inevitable and none of it is not a bug. There cannot exist  
> an ordering of complex roots that does not suffer from this problem.  
> What happens is this.
> Real root objects are ordered in the natural way. A cubic can have  
> either three real roots or one real root and two conjugate complex  
> ones. Let's assume we have the latter situation. Then the real root  
> will be counted as being earlier then the complex ones. Now suppose  
> you start changing the coefficients continuously. The roots will  
> start "moving in the complex plane", with the real root remaining on  
> the real line the two complex roots always remaining conjugate  
> (symmetric with respect to the real axis). Eventually they may  
> collide and form a double real root. If this double real root is now  
> smaller then the the "original real root" (actually than the root to  
> which the original real root moved due the the changing of the  
> parameter), there will be a jump in the ordering; the former root  
> number 1 becoming number 3.
> This is completely unavoidable, not any kind of bug, and I am not  
> complaining about it. It takes only elementary  topology of  
> configuration spaces to prove that this must always be so.

But is there a continuous root numbering if the roots are not ordered?

What I mean is that if you compute the roots of a polynomial, which is a 
function of a parameter, then if you assign a number to each root, can 
you follow that root continuously as the parameter changes? Two examples 
are presented below.

Here is some code to animate numbered roots using the standard root 
ordering, displaying the root numbering:

 rootplot[r_] := Table[ListPlot[
   Transpose[{Re[x /. r[a]], Im[x /. r[a]]}], 
   PlotStyle -> AbsolutePointSize[10],     
   PlotRange -> {{-3, 3}, {-3, 3}}, 
   AspectRatio -> Automatic,     
   PlotLabel -> StringJoin["a=", ToString[PaddedForm[Chop[a], {2, 1}]]],
   Epilog -> {GrayLevel[1], 
    MapIndexed[Text[#2[[1]], {Re[#1], Im[#1]}] & , x /. r[a]]}], 
      {a, -6, 10, 0.5}]

First, we have a polynomial with real coefficients:

  r1[a_] = Solve[x^5 - a x - 1 == 0, x]

Animating the trajectories of the roots using

  rootplot[r1]

we observe that, as you mention above, when the complex conjugate roots 
2 and 3 coalesce, they become real roots 1 and 2 and root 1 becomes root 
3. But, ignoring root ordering, why isn't it possible for these roots to 
maintain their identity (I realise that at coelescence, there is an 
arbitrariness)?

Second, we have a polynomial with a complex coefficient:

  r2[a_] = Solve[x^5 + (1+I) x^4 - a x - 1 == 0, x]

Animating the trajectories of the roots using

  rootplot[r2]

we observe that, even though the trajectories of the roots are 
continuous, the numbering switches:

  2 -> 3 -> 4
  5 -> 4 -> 3
  3 -> 4 -> 5
  4 -> 3 -> 2

and only root 1 remains invariant. Again, ignoring root ordering, why 
isn't it possible for all these roots to maintain their identity and so 
be continuous functions of the parameter? And wouldn't such continuity 
be nicer than enforcing root ordering?

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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