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 >> >