Re: Why is the negative root?

*To*: mathgroup at smc.vnet.net*Subject*: [mg69537] Re: Why is the negative root?*From*: p-valko at tamu.edu*Date*: Fri, 15 Sep 2006 06:44:49 -0400 (EDT)*References*: <200609130803.EAA18412@smc.vnet.net><eebd6d$l97$1@smc.vnet.net>

"Why do you need to use ToRadicals here?" I need the positive root(s) of the equation z^3 - z^2 - b*z-1==0 where b>0. (This is a cubic equation of state problem.) I know the answer: there is only one positive root and it is given by 1/3 - (2^(1/3)*(-1 - 3*b))/(3*(29 + 9*b + 3*Sqrt[3]*Sqrt[31 + 18*b - b^2 - 4*b^3])^(1/3)) + (29 + 9*b + 3*Sqrt[3]*Sqrt[31 + 18*b - b^2 - 4*b^3])^(1/3)/(3*2^(1/3)) but I still do not have any idea how to persuade Mathematica to give me this (or any reasonable) result. (Regarding Root and ToRadicals, now I understand that Root is numbering the roots differently for low and high b values: Plot[Root[-1 - b*#1 - #1^2 + #1^3 & , 1] , { b, 0, 10} ] but when I am applying ToRadicals, it freezes the symbolic form of the root with respect to the low b. Thanks for helping me to understand that.) Peter Andrzej Kozlowski wrote: > On 13 Sep 2006, at 17:03, p-valko at tamu.edu wrote: > > > (Reduce[{z^3-z^2- b z-1==0, b>0, z>0}, z] //ToRadicals)/.b->3.//Chop > > > > gives a negativ z: > > > > z == - 0.414214 > > > > What am I doing wrong? > > > > Regards > > Peter > > > > Why do you need to use ToRadicals here? There is no point in it and, > in fact you, are only asking for trouble. I am not sure why > Mathematica even tries to convert Root objects with parameters to > radicals, since it can't be done reliably due to branching problems > and is of no practical use whatever, but at least the Help tells you > that: > > â?¼ If Root objects in expr contain parameters, ToRadicals[expr] may > yield a result that is not equal to expr for all values of the > parameters. > > And, of course, if you do not use ToRadicals, there is no problem: > > Reduce[{z^3 - z^2 - b*z - 1 == 0, b > 0, z > 0}, z] /. b -> 3 > > z == 1 + Sqrt[2] > > > Andrzej Kozlowski > Tokyo, Japan

**References**:**Why is the negative root?***From:*p-valko@tamu.edu