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MathGroup Archive 2006

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Re: Re: Why is the negative root?


p-valko at tamu.edu wrote:
> I am a bit surprised by the "elitism" of the responses. Paul and Andrej
> and previously Daniel Lichtbau all defend the Root objects without
> telling the whole story.

I am hard pressed to figure out what might earn one this bit of name 
calling. It's not exactly new territory (in the States one gets called 
elitist for things as simple as political major party affiliation). But 
it is a bit unusual on MathGroup.

I particularly wonder what part of the story I missed. Frankly, I can't 
even figure out what part I told: I see nothing to indicate I replied in 
MathGroup to this thread, and unless I replied from my home machine via 
webmail interface (possible, but unlikely), there is no evidence in my 
outgoing mailbox that I replied at all.


> In my opinion those objects are just
> pseudo-useful. If you plot
>     Plot[Root[-1 + b #1 + #1^3 &, 1], {b,-10,10}]
> you will see that they are defending a monster.

It is discontinuous at a root crossing. I tend to think of parametrized 
roots as an ensemble, rather than as individual root functions, and thus 
regard this as less a problem than, say, dealing with the spurious 
imaginary parts of a numericized radical formulation. All the more so in 
situations where one has roots to a quartic, or where one has no chance 
to obtain a radical form of roots.


> But I am not going to start (continue) a debate on faith. Rather I am
> trying to formulate my question on a language even hard-core Mathematica 
> defenders can accept:
> 
> "Assuming that the coefficients are real and I am interested only in
> real roots, how do I  persuade Reduce to give the formulas 69 - 72 of
> 
>  http://mathworld.wolfram.com/CubicFormula.html ?
> (I do not mind if Mathematica gives two different result depending on the sign
> of the determinant.)"
> Regards
> Peter
> [...]

Could try the method in:

http://library.wolfram.com/infocenter/Conferences/337/

See section: "Radical solutions vs. Root objects. vs. trigs for cubic 
equations". Whether this "works" will depend on what exactly you want in 
an arctrig formulation, and (I think) on some details of ComplexExpand.

An aside: if you want to think of this as some sort of battle with 
myself amongst defenders of Root[] functions, you are sort of out on 
your own. If this ever was a battle, it was won many years ago. Though 
as best I can tell it was generally recognized (both in symbolic and 
numeric computation circles), before the appearance of Root[] functions 
(and equivalents in other programs), that these were the needed 
representation for sensible handling of algebraic functions.


Daniel Lichtblau
Wolfram Research




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