Before you compile the program, make sure you have installed all the packages required for the version you want to built (as listed above). Assuming that the package was extracted into the ../capd
directory, go to the subdirectory ../capd/make and call
make target=X11
Running instructions
- Windows
change current directory into the bin directory.
Under the command line ( Start→Run… : cmd ) write proof.exe "input data file name",
for example proof.exe paper_example.txt,
- Linux
change current directory into the bin directory.
Under the terminal write ./proof "input data file name",
for example ./proof paper_example.txt,
where
"input data file name" is the input data file name.
Input file structure
Refer the paper for the exact meaning of the input parameters. All the numbers should be entered explicitly, using the exponential notation, for
instance
1e-02 is allowed.
- m - symmetric Galerkin projection dimension, the projection will contain modes a_k such that
|k| <= m.
- nu - viscosity constant of the Burgers equation, should be positive, refer (1) in the article for the
exact meaning. One may extend a single value to an interval by setting the next value nonzero.
- nu_interval_right_bound - a nonnegative value. The interval
nu+[0,nu_interval_right_bound] defines a set of the viscosity constant values for which the algorithm will
attempt to perform the proof. If one likes to obtain a result for a single value of the viscosity constant, this parameter should equal to
0.
- a_0 - value of the zero indexed mode (it is constant).
- s - order of polynomial decay of coefficients that is required from the constructed bounds and trapping
regions, have to be an integer satisfying s ≥ 4.
- order - order of the Taylor method used by C0 Lohner algorithm.
- step - step size of the Taylor method used by C0 Lohner algorithm,
a lot of attention should be given to the proper choice of the step size, especially at high
nu values step size should be decreased, because the system is becoming stiff
and blow ups may result.
- forcing perturbation diameter - diameter (diam) of the uniform forcing perturbation,
namely the interval [-diam/2, diam/2] is added to each forcing mode appearing in the list that follows.
- forcing - list of the floating point numbers indicating forcing function that is used.
Values corresponds to the modes in ascending order starting at a_1, each mode is defined by two consecutive values,
first is the value of the real part of the mode, second is the value of the complex part of the mode. Number of elements in this list should
be less or equal 2m.
Note that only reasonable values should be used, refer
Section 7 in the article for the example sets of parameters for which the algorithm
completes successfully. For certain values a blow up during the integration process may occur, if so, a proper message is being outputted. More specific
technical parameters can be changed by editing the
config.h file and recompiling the program.
The output
The program saves the output data from the proofs into four output files. When the proof is complete four files appear in the
bin directory. The files are as follows
- proof_part1_the_trapping_region.txt this file contains a trapping region enclosing the fixed point,
along with corresponding data (matrices, eigenvalues).
- proof_part2_exact_fixed_point_location.txt this file contains calculated exact rigorous fixed point location
(self-consistent bounds containing the fixed point).
- proof_part3_the_l.txt this file contains the upper bound of the logarithmic norm on the trapping region.
- proof_part4_basin_of_attraction.txt This file contains data regarding the basin of attraction of the fixed point.
- a collection of files named n_0.txt, where n is a number. Each of
those files stores the record from the rigorous integration of a set that has occurred when verifying if the all solutions in an absorbing set
belongs to the basin of attraction of a fixed point.
License
This program is an open research software dedicated for performing computer assisted proof
Copyright (C) 2010-2013 Jacek Cyranka
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program 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. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see
.
Please consult the webpage www.cyranka.net,
or contact me on jcyranka (at) gmail.com for further details.