/*[
 * Copyright 2008  Bharti Pridhnani   bharti@maia.ub.es
 *
 * Permission to use, copy, and distribute this software and its
 * documentation for any purpose with or without fee is hereby granted,
 * provided that the above copyright notice appear in all copies and
 * that both that copyright notice and this permission notice appear
 * in supporting documentation.
 *
 * This is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
 *
 * This software is provided "as is" without express or implied warranty
 * to the extent permitted by applicable law.
]*/


This program gives you the solution of 4 given points. To execute it,
you need to install the library gsl which you can get from

http://www.gnu.org/gsl

and the program mencoder which can be downloaded from

http://www.mplayerhq.hu

which makes a movie where you can see the process of the continuation
of solution to reach to your given points.

Once you have all installed, execute

./execution_program.bsh

This will compile the files and execute the program. It will create
many files which will be deleted later on.

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Structure of the program

You have 5 files:

func.h
func.c
cheb_n4.c
execution_program.bsh
README.txt

In func.c you have all the functions used for solving the problem of
chebotarev. And the main program is in cheb_n4.c. Once this is
executed, first it asks you for your 4 points. Then it orders it and
decide a configuration. If this configuration does not work it tries
the other one (remember there are two type of configurations in the
case of 4 points). It can happen that for your given points this
program does not find the solution. There are three reasons for this
to happen:

1. The harmonic measure of some critical orbit is ''small'' and we
are doing calculations near a pole. To solve it, we need to integrate
poles.
2. During the process of continuation, we are going through the change
of the two configurations. To solve this we need to program the
integration of poles.
3. The Jacobian obtained in the process of finding zeroes is singular.
This can be avoided by doing the integration as in the case of 6 points.

The integration of the poles hasn't been done for this case, but yes
in the case of 6 points. If you want to know further you can contact us.

After this, it creates a file plot.bsh which contains the instructions
to make the movie and delete the files created. The animation created
is in the file movie.avi. At this stage of the execution it will ask
you if you want to delete the images 'jpeg' and the file which
contains the capacity of the solutions obtained. At last, you will get
three more files

plot.bsh
movie.avi
sol.dat

In sol.dat you have the solution for your 4 points.
Remark: maybe you need to give permisions to the file plot.bsh, in this
case write in execution_program.bsh the sentence

chmod 777 plot.bsh

just before ./plot.bsh

On the screen, we print each solution obtained in the process of
continuation of the solutions.
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Examples:

You can try these two examples:

1. a1=-2+3i, a2=-3-i, a3=4+4i, a4=5+i
2. a1=-1+6i, a2=-2-4i, a3=2+5i, a4=1-3i

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This program simply ilustrates the method used to solve the problem of
Chebotarev. 

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Bharti Pridhnani