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