Re: Solve transcendental functions

• To: mathgroup at smc.vnet.net
• Subject: [mg21138] Re: Solve transcendental functions
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 17 Dec 1999 01:21:38 -0500 (EST)
• Organization: Wolfram Research, Inc.
• References: <831v41\$g3d@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```sandmann at mip.sdu.dk wrote:
>
> If i got the function
> f[x]:=x^2-2Cos[a*x]/a
> and tries to Solve[f[x]==f'[x],x]
> I get
> The equations appear to involve transcendental functions of the
> variables in an essentially non-algebraic way.
>
> How do I solve this problem ?
> Regards Niels Sandmann
>
> Sent via Deja.com http://www.deja.com/

You can find solutions for particular values of a, provided you supply a
starting point for a Newton-type iteration.

f[x_,a_] := x^2-2Cos[a*x]/a
parametrizedSolve[f_, a_?NumberQ, init_] :=
FindRoot[Evaluate[f[x,a]==D[f[x,a],x]], {x,init},
AccuracyGoal->15, WorkingPrecision->20]

For example, if I want to handle the case where a is 1, starting near
x=2, I can do as below.

In[110]:= parametrizedSolve[f, 1, 2]
Out[110]= {x -> 2.1999981850434100730}

Starting at x=0 gives

In[111]:= parametrizedSolve[f, 1, 0]
Out[111]= {x -> -0.41895202574023425297}

Daniel Lichtblau
Wolfram Research

```

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