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Re: Re: Why is the negative root?

• To: mathgroup at smc.vnet.net
• Subject: [mg69688] Re: [mg69656] Re: Why is the negative root?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 21 Sep 2006 07:29:04 -0400 (EDT)

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