Re: Solution of this equation
- To: mathgroup at smc.vnet.net
- Subject: [mg20728] Re: Solution of this equation
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Wed, 10 Nov 1999 00:17:43 -0500
- Organization: Wolfram Research, 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...
>
> Any help?
>
> Thanks,
>
> Dave!
>
> --
> Dave Richardson
> University of Maryland
> Department of Mechanical Engineering
> Center for Environmental Energy Engineering
> (301) 405-8726
> dhr at glue.umd.edu
It may be useful to express the result in terms of Root[...] objects
rather than radicals, as the latter of necessity must be expressed in
terms of Sqrt[-1] for generic cases. To get Root solutions one uses
SetOptions.
In[21]:= SetOptions[Roots, Cubics->False];
In[22]:= soln = Solve[Pr == (8*Tr)/(3*vr - 1) - 3/vr^2, vr]
2 2 3
Out[22]= {{vr -> Root[-3 + 9 #1 - Pr #1 - 8 Tr #1 + 3 Pr #1 & , 1]},
2 2 3
> {vr -> Root[-3 + 9 #1 - Pr #1 - 8 Tr #1 + 3 Pr #1 & , 2]},
2 2 3
> {vr -> Root[-3 + 9 #1 - Pr #1 - 8 Tr #1 + 3 Pr #1 & , 3]}}
An advantage is that now you get solutions that are known to be
explicitly real when appropriate values for Pr and Tr are used.
In[28]:= vals = vr /. (soln /. {Tr->-8/3, Pr->5/4})
Out[28]= {Root[-36 + 108 #1 + 241 #1 + 45 #1 & , 1],
2 3
> Root[-36 + 108 #1 + 241 #1 + 45 #1 & , 2],
2 3
> Root[-36 + 108 #1 + 241 #1 + 45 #1 & , 3]}
In[31]:= Map[Im, vals]
Out[31]= {0, 0, 0}
In[32]:= N[vals]
Out[32]= {-4.82362, -0.752372, 0.220437}
It is also typically faster and more accurate to compute numerical
values for solutions expressed as algebraic numbers rather than in terms
of radicals.
Daniel Lichtblau
Wolfram Research