       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:= SetOptions[Roots, Cubics->False];

In:= soln = Solve[Pr == (8*Tr)/(3*vr - 1) - 3/vr^2, vr]
2          2          3
Out= {{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:= vals = vr /. (soln /. {Tr->-8/3, Pr->5/4})
Out= {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:= Map[Im, vals]
Out= {0, 0, 0}

In:= N[vals]
Out= {-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