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Re: Solvability sensitive to small changes in numerical input

• To: mathgroup at smc.vnet.net
• Subject: [mg47568] Re: Solvability sensitive to small changes in numerical input
• From: "Peter Pein" <petsie at arcor.de>
• Date: Fri, 16 Apr 2004 05:20:40 -0400 (EDT)
• References: <c5j7sp$r29$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

"Gareth J. Russell" <gjr2008 at columbia.edu> schrieb im Newsbeitrag
news:c5j7sp$r29$1 at smc.vnet.net...
> Hi,
>
> I have a function that includes the Solve command
>
> f[a1_, a2_, b_] := Module[{c},
>
>     c = 2*b/(b - 1);
>
>     Solve[b/(1 + (a1*n)^c) + b/(1 + (a2*n)^c) == 2, n]
>
> ]    ]
>
> but it seems very sensitive to the numerical input. For example,
>
> f[2., 0.05, 6.19]
>
> and
>
> f[2., 0.05, 6.21]
>
> Produce the error "The equations appear to involve the variables to be
> solved for in an essentially non-algebraic way."
>
> Whereas
>
> f[2., 0.05, 6.20]
>
> produces the solutions
>
> {{n -> -23.8725 - 13.2503i]}, {n -> -23.8725 + 13.2503i}, {
>   n -> 0.183281 - 0.707873i}, {n -> 0.183281 + 0.707873i]}, {n -> 27.
> 3032}}
>
> Can anyone enlighten me as to what is going on here? Other peoples' work
> suggests that, given a1 = 2., it should be possible to get numerical
> solutions for a large range of values of a2 and b.
>
> If it makes a differences we make the assumptions a1 > 0, a2 > 0, a2 <
> a1 and b >= 1.
>
> Oh, and v5.0.1.0 on Mac OS X
>
> Thanks!
>
> Gareth Russell
>
> Columbia University
>
Hi Gareth,
if you're interested in the real solution, try

f[a1_, a2_, b_] := With[{c = (2*b)/(b - 1)},
   8^(1/c)*(1/(Sqrt[(a1^c + a2^c)^2*(c - 4)^2 + 32*(a1*a2)^c*(c - 2)] +
       (a1^c + a2^c)*(c - 4)))^(1/c)]

-- 
Peter Pein, Berlin
to write to me, start the subject with [