A program for finding and validating all zeros to a system of two analytic functions in a rectangular domain.
To get the program running you will need an installed copy of C-XSC. This is a C++ library for handling interval arithmetic's. Information about the library and instructions for installing can be found on their website. The version used for this program is 2.5.4.
Unfortunately the complex interval arithmetic in C-XSC is not thread safe and to be able to use it in a multi-threaded environment a small change to the code is needed. We will here describe the steps to do so.
In the directory were you installed C-XSC, usually ~/cxsc, you will
need to change in the file path-to-cxsc/include/cinterval.inl. On
line 644 change from
return mult_operator(a,b);
to
return cinterval(Re(a)*Re(b)-Im(a)*Im(b), Re(a)*Im(b)+Im(a)*Re(b));
and on line 657 change from
return div_operator(a,b);
to
return (a * cinterval(Re(b), -Im(b))) / (sqr(Re(b)) + sqr(Im(b)));
This all you need to change and you will not need to recompile the library.
To be able to compile the program you will first need to add the path
to your C-XSC installation in the makefile. Set the variable
CXSCDIR to point to the path of your installation, usually this is
CXSCDIR=/home/USERNAME/cxsc
To compile the program call make all while in the root of the
project. This will create the two programs integrate and
generateZeros in the directory build. You can also compile
optimized versions of the program, this gives longer compiling times
but improves the performance of the program. To do this call make build/integrateOpt or make build/generateZerosOpt which will create
the two programs integrateOpt and generateZerosOpt in the build
directory.
We will here give a minimal example to check if the program compiled correctly works.
When you download the code the preset function is f(z_1, z_2) = (z_1, z_2), i.e the identity in the two dimensional complex space. This of course have only one zero, at the origin. We will here use the program to prove this.
After having ran make all go into the build directory and run the
program with
./integrate -- -1 1 -1 1 -1 1 -1 1
This tells the program compute the logarithmic integral for the domain z_1 = [-1, 1] x i[-1, 1], z_2 = [-1, 1] x i[-1, 1]. The answer is the number of zeros, counting multiplicity, in the domain. Since we know that the number of zeros in the domain is exactly one it should output an interval containing the only integer 1. The output is
([ 0.603480, 1.524247],[ -0.409304, 0.409304])
indicating that the integral has a value with the real part inside the first interval and the imaginary part inside the second interval. We see that the value 1 is indeed inside this interval.
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