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Re: using FindRoot to find multiple answers in a domain?

• To: mathgroup at smc.vnet.net
• Subject: [mg69261] Re: using FindRoot to find multiple answers in a domain?
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Mon, 4 Sep 2006 02:41:25 -0400 (EDT)
• Organization: The University of Western Australia
• References: <ed0rn9$sm6$1@smc.vnet.net> <ed3rou$rv5$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <ed3rou$rv5$1 at smc.vnet.net>, "akil" <akomur at wanadoo.nl> 
wrote:

> IntervalRoots package and RootSearch do not work for the kind of formulas I 
> have.

Are you sure?

> For example in the domain [0,Pi/2] we want to solve (I only used FindRoot 
> because this was the fastest):
> 
> I put the package I test in on 
> http://home.wanadoo.nl/akomur/testRootSearch.nb so that you can see the 
> formulas I use.
> 
> Copying them In here ruins the formulas.

Copy as Plain text. Your first formula is

  nlv = Max[-81.24275115593154  Cot[beta] - 48.489352280270914,   
        Min[-46.844746272761526 Cot[beta] - 71.4213555357176, 
            -46.480384751324436 Cot[beta] - 71.66426321667566]]

Note that PiecewiseExpand can be applied to such formulae. If you 
simplify the result

  Simplify[PiecewiseExpand[nlv]]

you deduce that nlv is just

  -81.24275115593154  Cot[beta] - 48.489352280270914 if Cot[beta] <= 2/3

  -46.844746272761526 Cot[beta] - 71.4213555357176 if Cot[beta] >= 2/3

You can check that these values are consistent when Cot[beta] == 2/3. 
Moreover, the catch-all condition

  -46.480384751324436 Cot[beta] - 71.66426321667566

is redundant for real beta since it only applies when Cot[beta] == 2/3. 
A brief examination of nuvb shows that it involves ratios of (powers of) 
expressions similar to nlv. I expect that simplification of these 
expressions prior to constructing nuvb would lead to a final form that 
is not that hard to handle.

I expect that if you step back and tell us more about the original 
problem, then a more elegant approach would present itself.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
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