Re: [Q] Imposing resrictions on the roots?
- To: mathgroup at smc.vnet.net
- Subject: [mg40492] Re: [mg40474] [Q] Imposing resrictions on the roots?
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 8 Apr 2003 03:02:50 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
> I have set up an equation and was able to obtain the analytical
> solutions
> for it.
> Is there any way I can get the conditions that make the root(s) have a
> certain property? (e.g. non-negative real roots)
> Thanks in advance.
If the equation is simple enough, yes, sometimes. For example, suppose
you want to find all positive real roots of
-156 + 85*x - 5*x^2 - 5*x^3 + x^4==0
Of course one way would be to find first all roots and then select the
positive ones. Another way is to use InequalitySolve or a related
function:
<< Algebra`InequalitySolve`
InequalitySolve[{-156 + 85*x - 5*x^2 - 5*x^3 + x^4 == 0,
x > 0}, x]
x==3
This is the only positive real root. Next, suppose your equation
contains a parameter, say q:
InequalitySolve[{-156 + 85*x - q*x^2 - 5*x^3 + x^4 == 0,
x > 0}, {q, x}]
x == Root[-156 + 85*#1 - q*#1^2 - 5*#1^3 + #1^4 & , 2]
This gives you an explicit formula for the positive real root of -156 +
85*x - q*x^2 - 5*x^3 + x^4 == 0, but of course the answer is expressed
in a form that you may not like since it refers to an ordering of the
roots which may not be very informative. Nevertheless, it will generate
positive values whenever you substitute real numbers for q:
Root[-156+85*#1-q*#1^2-5*#1^3+#1^4&,2]/.Table[{q->Random[]},{10}]
{2.21895,2.2686,2.2244,2.19348,2.26294,2.20147,2.2535,2.23311,2.25249,2.
21113}
You can try this approach in even in some more complicated cases.
However, you should be aware of the fact that the complexity of the
algorithms used is very much higher than in just solving polynomial
equations so normally it will be better to just use Solve and Select
(when, of course, that latter makes sense).
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Monday, April 7, 2003, at 05:54 pm, Wonsuk Doh wrote: