C

Visualization of zeros of polynomals

This folder contains the C program for calculating zeros in parallel.

Compilation

Compile it by running make. You make have to edit the Makefile if the compiler does not find the libtiff and gsl librariers.

Usage

The C program is used as follows:

./zeroes <input-file> <output-file>

The input file has the following format:

xmin -2.0
xmax 2.0
ymin -2.0
ymax 2.0
xres 1024
coeffs -1.0 1.0
degrees 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

The xmin, xmax, ymin, ymax values determine the region in the complex plane that should be displayed. The horizontal resolution of the output image is xres, and the vertical resolution is compute accordingly. The coefficients to be used are given by coeffs and the degrees by degrees. The above example will generate all polynomials with coefficients -1 and 1 of degrees up to 17.

The output file is a 32-bit grayscale TIFF image. The gray levels are log-scale frequency counts of how many times each pixel was hit by a zero. Thus whiter regions have a greater density of zeroes. In order to make a nice picture from the output we need to fiddle with the image colors. ImageMagick is a wonderful tool for doing precisely that.

For example, using the above parameters (also found in littlewood.txt) we run the program as follows:

./zeros littlewood.txt picture.tiff

Then we process the resulting picture.tiff using ImageMagick:

convert picture.tiff -evaluate log 400000 -normalize -fill orange -tint 100 picture.png

You can also experiment with -contrast-stretch, and for really good quality you will need some competence with ImageMagick.

Generating a movie

If you are planning to generate movies, you can generate a really big picture and then cut out movie frames from it using ImageMagick. Then you can combine the frames into a movie with ffmpeg. The movie.py program helps you generate a movie, as follows.

First we create a very large image and process it to give it some colors:

./zeros movie-example.txt big-picture.tiff
convert big-picture.tiff -evaluate log 400000 -normalize -fill cyan -tint 100 big-picture.png

Then you look at movie.py and edit it to define whatever flyovers and zooms you are interested in, and also make sure that you use the correct image (sorry, it's not a very user-friendly program). Then, to generate a movie for one of the flyovers, say zoom, you run:

./movie.py zoom