       finding roots of user defn. functions

• To: MATHGROUP at yoda.physics.unc.edu
• Subject: finding roots of user defn. functions
• From: "James M. Rotenberry" <H5RR1001%SMUVM1 at vm.cis.smu.edu>
• Date: Mon, 09 Mar 92 09:41:34 CST
• Organization: Southern Methodist University

```      I am using Mathematica 1.2 on a Mac II under 6.0.7.

In my work I often need to find the root of a function,
and I thought that Mma would be what I required.  Below I have
the results of a test program.  I am seeking the value of alpha
where the real part of the root of the quadratic becomes zero.
(A Hopf bifurcation occurs at this value, hence the name of the
user defined function.  I am aware that there are other ways to
determine this value for this simple test case.)

I have searched through the Mathematica 1.2 and 2.0 manuals,
Maeder's book, Blachman's book and have tried various
combinations of Release[], Hold[], and HoldAll[].  How can I get
this program to treat my function like one of its built in functions
such as BesselJ[]?  The function Plot[] works fine, probably since
a table of values is created, but FindRoot does not work.

This seems like a very natural thing for a symbolic manipulation
to do, so I apologize if this is a frequently asked question.
Thanks.

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Hopf[alpha_,B_,eps_,beta_]:=Re[ -y-3 /.
FindRoot[0==y^2+
(beta+alpha B/(1+eps alpha B/(1+beta))^2
-alpha/(1-alpha))y +alpha beta B/
(1+eps alpha B/(1+beta))^2,{y,-.1+.1I}] ]

eqn1=Hopf[.2,16,.1,2]

-0.818569

plt1=Plot[Hopf[x,16,.1,2],{x,.2,.3}]

-Graphics-

eqn2=FindRoot[Hopf[x,16,.1,2], {x,{.25,.27}}]

FindRoot::fnotnum:
Function number 1 is not numerical at
32. x
{-0.1 + 0.1 I}=0.02 I - ------------------ +
2
(1. + 0.533333 x)
1. x
(0.1 - 0.1 I) (2. - --------- +
1. - 1. x
16. x
------------------).
2
(1. + 0.533333 x)

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

James M. Rotenberry
Dept. of Math.
Southern Methodist University
rotenber at alumni.caltech.edu

```

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