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Re: Solution of this equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg20654] Re: Solution of this equation
  • From: John Doty <jpd at w-d.org>
  • Date: Sun, 7 Nov 1999 02:09:56 -0500
  • Organization: The Internet Access Company, Inc.
  • References: <7vrc3p$2nd@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Dave Richardson wrote:
> 
> Can anyone offer insight here?
> 
> This Mathematica expression gives 3 solutions to the equation.
> 
> Solve[Pr == (8*Tr)/(3*vr - 1) - 3/vr^2, vr]
> 
> The problem is that there are 3 Real solutions, and Mathematica is giving
> solutions with (granted a small) imaginary component.
> 
> And hitting it with  a full simplify is just not a good idea...

This is life with cubic equations. Solutions in terms of radicals
generally involve complex numbers, even when the roots are real. This
was a *major* puzzle of 16th century mathematics. See Paul Nahin's
wonderful book, "An Imaginary Tale: the Story of Sqrt[-1]" (title given
in OutputForm :-).

FullSimplify cannot generally cancel the imaginary components of the
solutions, although it can in exact calculations of cases where the
roots are all real:

In[1]:=
                  8 Tr      3
s = Solve[Pr == -------- - ---, vr]; 
                3 vr - 1     2
                           vr

In[2]:=
s /. {Tr -> -2, Pr -> 1}

Out[2]=
          5                16
{{vr -> -(-) + --------------------------- + 
          3                            1/3
               (-1188 + 324 I Sqrt[15])
 
     1                         1/3
     - (-1188 + 324 I Sqrt[15])   }, 
     9
 
           5         8 (1 + I Sqrt[3])
  {vr -> -(-) - --------------------------- - 
           3                            1/3
                (-1188 + 324 I Sqrt[15])
 
     1                                          1/3
     -- (1 - I Sqrt[3]) (-1188 + 324 I Sqrt[15])   }, 
     18
 
           5         8 (1 - I Sqrt[3])
  {vr -> -(-) - --------------------------- - 
           3                            1/3
                (-1188 + 324 I Sqrt[15])
 
     1                                          1/3
     -- (1 + I Sqrt[3]) (-1188 + 324 I Sqrt[15])   }}
     18

In[3]:=
FullSimplify[%]

Out[3]=
{{vr -> -2 + Sqrt[5]}, {vr -> -2 - Sqrt[5]}, {vr -> -1}}

In inexact numerical calculations, the cancellation of imaginary
components will generally not be exact.

Somebody seems to ask about this every few months: do we need a cubic
equation FAQ? :-)

-- 
John Doty		"You can't confuse me, that's my job."
Home: jpd at w-d.org
Work: jpd at space.mit.edu


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