Motivation

The Mandelbrot set is a set of complex numbers. When a certain recurrence relation is applied to the numbers

outside the set, quickly approach infinity. The values for numbers inside the set remain finite. Near

the edge of the set, the speed at which the terms approach infinity varies wildly.

The recurrence relation is

$z_{n+1}=z_n^2+C$

where $z_0=C$ and $C$ is a complex number. As it turns out, if the size ${?}^1$ of $z_i$ exceeds $2$ for some value of $i,$ then

it can be shown that the terms will eventually become infinite. The value of i when $z_i$ exceeds $2$ varies wildly for

values near the set members.

Plots of the Mandelbrot set can produce beautiful images. For this project, you will take an existing program that

plots the Mandelbrot set and parallelize it$.$

Existing Code

The program you are given is invoked as:

mandelPlot tlr tli real$-$side$-$length imag$-$side$-$length max$-$iterations

horizontal$-$pixels vertical$-$pixels image$-$file

It identifies whether the points within a rectangle of the complex plane are in the mandelbrot set by apply$-$

ing the recurrence relation above to each point. The coordinate of the top left point in the rectangle is:

tlr $+$ tiiI. The length of each side in the real and complex plane is given by real$-$side$-$length and

imag$-$aide$-$length respectively. There is a one$-$to$-$one correspondance between the values tested for inclu$-$

sion in the Mandelbrot set and pixels in the graph that is created by the program. Hence the number of

points explored is horizontal$-$pixels$*$vertical$-$pixels and the distance between two points along any axis

is side$-$length$/($pixels$-1).$

The recurrence relation given earlier is applied to each point in turn. At most max$-$iterations terms are

calculated. If after this number of terms the value is still below two, the point is taken to be in the set.

The program also plots the points through creation of a PPM file named image$-$file. The number of iterations

performed at a particular point determines its color in the final plot. The PPM file format specifies the number

of pixels in each row and column as part of a header written to the file. Following the header are three bytes for

each pixel that represent the red, green, and blue color values for the point. The value of each component must

be $255$ or lower. The pixel positions are implicit; the color values are listed in row major order. The first color

value corresponds to the upper left corner of the image. The remaining colors are applied row wise until all the

points have been specified.

You can use the command

gi$=$p filename

to view the PPM files you create on a departmental Linux system.

${?}^1$ The size of $z_i$ is the length of the hypotenuse of a triangle where the sides are the complex and imaginar