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MathGroup Archive 2006

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Re: Re: Why is the negative root?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69691] Re: [mg69656] Re: Why is the negative root?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 21 Sep 2006 07:29:11 -0400 (EDT)

I forgot to state explicitly (although this should not really be  
needed) that all I wrote root isolation applies to numerical root  
objects (without parameters). Of course the main reason why  
parametric root objects are useful is that upon substituting  
numerical values for parameters then turn into numerical root objects  
and root isolation is automatically performed. I should have also  
mentioned that root objects have one other big advantage over radical  
and other representations (particularly the one involving trig  
functions): it is very much easier to perform algebraic operations on  
them. This comes form the fact that they are defined in terms of  
minimum polynomials and there are well known algorithms for  
computing, for example, the minimal polynomial of the sum or product  
of two algebraic numbers. All such operations are performed by the  
function RootReduce, which is called up by Simplify etc.
Needless to say, such operations are much harder and in fact, usually  
impossible to carry out without using Root objects or some equivalent  
mehtod.

Andrzej Kozlowski

On 20 Sep 2006, at 20:21, Andrzej Kozlowski wrote:

>
> On 20 Sep 2006, at 15:43, Paul Abbott wrote:
>>
>>
>>> Paul and Andrej and previously Daniel Lichtbau all defend the  
>>> Root objects without
>>> telling the whole story.
>>
>> Really? What has been omitted?
>>
>>> In my opinion those objects are just pseudo-useful.
>>
>> Why do you think that?
>>
> Well, actually there is, I think, something "that has been  
> omitted". Root objects are not "tautological", "pseudo-useful"  
> objects that some imagine them to be, but each in a certain sense  
> "embodies" some pretty quite sophisticated computations. The key  
> word is "root isolation". The early versions of Mathematica  
> actually used to store the isolating informaiton as the third  
> argument to Root, but now the third argument is either 1 or 0  
> corresponding to whether exact or approximate method or root  
> isolation is used, and the relevant information is stored in some  
> other way. It is because the roots have been isolated that they can  
> be ordered and manipulated in various ways, that is impossible in  
> the case of radical expressions. So there is some truth to the  
> claim that we have not "told the whole story" but why should we? It  
> can be found in any decent book on Computer Algebra (e.g. Chee Keng  
> Yap, "Fundamental Problems in Algorithmic Algebra", Princeton  
> University Press, Chapter 6 gives all the basic necessary facts.  
> You can also look at the standard AddOn package  
> "Algebra`RootIsolation`" to see what's involved).  Articles about  
> the Mathematica implementation of Root objects have appeared more  
> than once in The Mathematica Journal. Obviously, this list is not  
> the right place for lessons on modern computer algebra. Also,  
> concerning "pseudo-usefulness": things like root objects are by no  
> means unique to Mathematica but in fact implemented in every  
> serious computer algebra system available today (including of  
> course Mathematica's main competitors in the CAS area). It's  
> curious that all these guys decided to waste so many man-hours  
> studying, researching and implementing this useless stuff, not to  
> mention writing numerous articles and books about it.
>
> Andrzej Kozlowski
>
> TOkyo, Japan


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