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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: [mg69679] Re: [mg69642] Re: Why is the negative root?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Wed, 20 Sep 2006 02:44:51 -0400 (EDT)

I did not explain my last sentence because I did not think  any  
exlanation should be needed, but any who needs an explanation should  
take a look at the following animation (which has nothing to do with  
root objects)  and then compare it with the supposed "monster example" .

Do[Plot[Evaluate[x^3 - b*x - 1], {x, -3, 1}, PlotRange -> {{-3,
     1}, {-10, 10}}], {b, -10, 10}]

The first part of my comment should also now be clear.

Andrzej Kozlowski



On 19 Sep 2006, at 21:03, Andrzej Kozlowski wrote:

> I also do not wish to enter into useless disputes, however, in my  
> opinion the essence of this dispute was not about "faith" or even  
> Mathematica but about mathematics, which by its very nature is an  
> "elitist" subject. The example you give below illustrates this  
> perfectly.
>
> Andrzej Kozlowski
>
>
> On 19 Sep 2006, at 18:44, 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. 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.
>>
>> 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
>>
>>
>> Paul Abbott wrote:
>>>> Paul Abbott wrote:
>>>>> In this case, the single root can be represented by this  
>>>>> radical. But
>>>>> modify your example slightly:
>>>>>   Reduce[{z^3 - z^2 - b z + 3 == 0, b > 0, z > 0}, z] //  
>>>>> FullSimplify
>>>>> How would you prefer the result to be expressed now?
>>>>
>>>> The answer is:
>>>> b > (-1 - 647/(50867 + 5904*Sqrt[82])^(1/3) + (50867 +
>>>> 5904*Sqrt[82])^(1/3))/12 &&
>>>>  ((Sqrt[1 + 3*b] + (2 + 6*b)*Cos[(Pi/2 - ArcTan[(-79 +
>>>> 9*b)/(3*Sqrt[3]*Sqrt[-231 + 54*b + b^2 + 4*b^3])])/3])/
>>>>    (3*Sqrt[1 + 3*b]) ||
>>>>   (Sqrt[1 + 3*b] - (1 + 3*b)*Cos[(Pi/2 - ArcTan[(-79 +
>>>> 9*b)/(3*Sqrt[3]*Sqrt[-231 + 54*b + b^2 + 4*b^3])])/3] +
>>>>     Sqrt[3]*(1 + 3*b)*Sin[(Pi/2 - ArcTan[(-79 +
>>>> 9*b)/(3*Sqrt[3]*Sqrt[-231 + 54*b + b^2 + 4*b^3])])/3])/
>>>>    (3*Sqrt[1 + 3*b]))
>>>>
>>>> The answer is in every engineering handbook.
>>>
>>> Then every engineering handbook is deficient! Radical  
>>> formulations are
>>> prone to numeric problems. Root objects do not have this liability.
>>>
>>> Why do you object to Root objects? Is this an "engineering" fetish?
>>>
>>>> They call it the "Cardano formula".
>>>
>>> I too learnt how to compute the roots of cubics and quartics in high
>>> school, and I know about the Cardano formula. However, the above
>>> expression is _not_ the (standard) Cardano formula as it involves  
>>> trig
>>> and inverse trig functions. See
>>>
>>>   http://mathworld.wolfram.com/CubicFormula.html
>>>
>>> Actually, the above expressions are, effectively, Chebyshev  
>>> radicals:
>>>
>>>   http://en.wikipedia.org/wiki/Cubic_equation#Chebyshev_radicals
>>>
>>> In general, the Cardono formula is _not_ practically useful. Any
>>> computation that you need to do involving roots of polynomials is  
>>> better
>>> done using Root objects (or using Chebyshev radicals).
>>>
>>> Also, consider solving a quintic instead of a quartic ...
>>>
>>> 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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