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MathGroup Archive 2007

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Re: Interpreting the solutions... better this time

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75280] Re: Interpreting the solutions... better this time
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 23 Apr 2007 05:43:48 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f0ejof$pcq$1@smc.vnet.net>

Apostolos E. A. S. Evangelopoulos wrote:
> Right, here goes the whole thing again, since I didn't present it well last time...
> 
> So, what I'm trying to solve is
> 
> (8*Pi*R^3*S)/(3*h^2) + (2*h*Pi*(S + 3*\[Gamma]))/3 + 
>  ((R^2*(-h + 2*R)^2*(-h/(2*R^2) - (2*R)/h^2))/
>     (Sqrt[3]*Sqrt[R^2*(-h^2/(4*R^2) + (2*R)/h)]) - 
>    (4*(-h + 2*R)*Sqrt[R^2*(-h^2/(4*R^2) + (2*R)/h)])/
>     Sqrt[3] - (8*R*(-h + 2*R)*(-h/(2*R^2) - (2*R)/h^2)*
>      Sqrt[R^2*(-h^2/(4*R^2) + (2*R)/h)])/(3*Sqrt[3]) + 
>    (16*(R^2*(-h^2/(4*R^2) + (2*R)/h))^(3/2))/
>     (9*Sqrt[3]*R) + (16*(-h/(2*R^2) - (2*R)/h^2)*
>      (R^2*(-h^2/(4*R^2) + (2*R)/h))^(3/2))/
>     (9*Sqrt[3]))*\[CapitalKappa] == 0
> 
> with respect to h.
> 
> What Mathematica returns is
[snip: ... root objects deleted...]

> I do not understand the # symbols and I do not know what I'm supposed to do with this, generally. How can I further evaluate these 'Root[blah]' forms?

What is returned is a list of *Root* objects [1]. According to the 
Online Help [2, 3]:

"There are some equations, however, for which it is mathematically 
impossible to find explicit formulas for the solutions. Mathematica uses 
Root objects to represent the solutions in this case."

In[1]:=
Solve[2 - 4*x + x^5 == 0, x]

Out[1]=
{{x -> Root[2 - 4*#1 + #1^5 & , 1]},
   {x -> Root[2 - 4*#1 + #1^5 & , 2]},
   {x -> Root[2 - 4*#1 + #1^5 & , 3]},
   {x -> Root[2 - 4*#1 + #1^5 & , 4]},
   {x -> Root[2 - 4*#1 + #1^5 & , 5]}}

"Even though you cannot get explicit formulas, you can still find the 
solutions numerically."

In[2]:=
N[%]

Out[2]=
{{x -> -1.51851}, {x -> 0.508499}, {x -> 1.2436},

   {x -> -0.116792 - 1.43845 I},

   {x -> -0.116792 + 1.43845 I}}

Regards
Jean-Marc

[1] "Built-in Functions / Numerical Computation / Number Representation 
/ Root", http://documents.wolfram.com/mathematica/functions/Root

[2] _The Mathematica Book_, "1.5.7 Solving Equations", 
http://documents.wolfram.com/mathematica/book/section-1.5.7

[3] _The Mathematica Book_, "3.4.2 Equations in One Variable", 
http://documents.wolfram.com/mathematica/book/section-3.4.2


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