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Re: Limit and Root Objects

• To: mathgroup at smc.vnet.net
• Subject: [mg72305] Re: Limit and Root Objects
• From: Andrzej Kozlowski <andrzej at akikoz.net>
• Date: Tue, 26 Dec 2006 04:38:05 -0500 (EST)
• References: <em606c$2o1$1@smc.vnet.net> <4586C045.2020805@metrohm.ch>

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.

What I am not really convinced of is that Limit really couldn't deal  
with this problem, at least partially, in the case of real roots of   
cubics and quartics. I have been told that it cannot be done because  
Limit relies on Series - and that I of course completely agree that  
Series can't possibly deal with this.  Even worse  problems of this  
type will inevitably  happen in the case of non-real roots, which can  
switch order in intractable ways. But it seem to me that in the case  
of real roots one could use a simple numeric-symbolic method to get  
the right answer- which is indeed what I did by hand before posting  
this problem. The key point is, that problem can only occur at a  
double root, because two conjugate complex roots must first collide  
on the real line before a real root is formed. The double roots can  
be found by using the derivative and Resultant, as I did in this case:


Reduce[Resultant[poly[b,x], D[poly[b,x]], x] == 0, b, Reals]

where b is a parameter. Now, if a Limit is taken at a point that is  
not a double root continuity can be assumed and we are home. If the  
Limit point is a double root, then (in the case of a cubic) we that  
the only possibilities for the limit will be the value of the first  
or the third real root at this point (where there will be three real  
roots). We can use NLimit with significance arithmetic to settle  
this. Significance arithmetic is important since the two branches can  
be arbitrarily close (we may have a near tripe root). Actually, we  
would have to first check algebraically that we do not have a triple  
root at this point - if we do, there is no problem though.

I can see that doing all this would considerably increase the  
complexity of Limit whenever any kind of Root objects were present in  
an expression - which may be strong argument against it. On the other  
hand, I think these kind of situations are quite interesting and it  
would be a good idea to be able to get them right without having to  
resort to manual computations, as I did in this case.
Another argument for leaving things as they are, the same kind of  
phenomenon for complex roots (and perhaps polynomials of high  
degree)   seems quite un-manageable so in that sense the problem  
would still remain.  But I think real roots are interesting and  
important, and even if this could be done only for cubic and  
quartics, it would be worth while.
If all this is not practical, I think a warning message should always  
be issued by Limit whenever such Root objects are encountered  (I  
think Series already does that).


Andrzej Kozlowski




On 19 Dec 2006, at 01:22, dh wrote:

> *This message was transferred with a trial version of CommuniGate 
> (tm) Pro*
> Hi Andrzej,
> It is definitly a bug. The reason for the bug may be that not only  
> the function is not continuous at b == -(3/2^(2/3)), but also the  
> numbering changes. There are three different real roots for b<-(3/2^ 
> (2/3)), therefore, the first is the smallest. For b=-(3/2^(2/3))  
> the two lowest roots merge and for b>-(3/2^(2/3)) the two    
> "former" lowest become complex. And now the first root is the  
> "former" highest. MMA seems to keep the number of the root in the  
> limit process. To make the bug even worse, the wrong first root at  
> b=-(3/2^(2/3)) is a double root and therefore, reduced to a  
> quadratic root object.
> Daniel
>
>
> Andrzej Kozlowski wrote:
>> It is easy to check that the function
>> f[b_] := Root[#1^3 + b*#1 - 1 & , 1]
>> is discontinuous at b, where
>> Reduce[Resultant[x^3 + b*x - 1, D[x^3 + b*x - 1, x], x] == 0, b,  
>> Reals]
>> b == -(3/2^(2/3))
>> indeed this was not so long ago discussed in connection with  a   
>> little argument about "usefulness' of Root objects. In view of  
>> this,  isn't the following a bug?
>> u = Limit[f[b], b -> -(3/2^(2/3)), Direction -> 1]
>> Root[2*#1^3 + 1 & , 1]
>> v = Limit[Root[#1^3 + b*#1 - 1 & , 1], b -> -(3/2^(2/3)),  
>> Direction -  > -1]
>> Root[2*#1^3 + 1 & , 1]
>> u == v
>> True
>> It looks like Limit is making life too easy for itself by  
>> assuming  continuity.
>> Using NLimit shows that things are not as simple:
>> w = NLimit[f[b], b -> -(3/2^(2/3)), Direction -> -1]
>> 1.5874010343874532
>> z = NLimit[Root[#1^3 + b*#1 - 1 & , 1], b -> -(3/2^(2/3)),  
>> Direction -  > 1]
>> -0.7937180869283765
>> Andrzej Kozlowski
>