Equation for determining amount of matrix solutions for a given Collatz-Matrix equation?

[Unity+]

This is a problem faced within another topic and I made this topic separate because I wanted to focus on this specific problem, which would deviate from the other topic. However, if moderators feel otherwise then it will be fine if the topic is moved to the other topic.

 

So the problem(which I still haven't solved) is dealing with finding an equation to predict the amount of matrix solutions for a given Collatz-Matrix equation, which is defined by as the following:

 

[math]C(x)_{k\times d}\begin{Bmatrix} a_{f} &b_{f} \\ u_{f}&v_{f} \end{Bmatrix},s(k_{p},d_{p})[/math]

 

Now, how these work is there could be multiple or just 1 matrix solution for a given Collatz-Matrix equation. For each matrix solution, the initial element lies in the coordinate given in s(). Then, the next element's placement is determine by which function(within the brackets) is used. If a_f is used, the element is placed above(if it can be done) the element at hand. If it is b_f, it goes to the left. If it is u_f, then it goes below, and if it is v_f it goes to the right. The next element's value will be based off the function used. Also, in this case every element must end up being a whole number.

 

Since there are multiple ways to do this, there can be multiple matrix solutions. Now, here is some more information.

Though I am assuming the weak version of the solution will also apply to the strong version as well.

 

The weak version assumes there to be a ratio [math]\frac{d_{e}}{d_{i}}[/math] to exist within each function, where the Collatz-Matrix equation is set up as the following:

 

[math]C(x)_{k\times d}\begin{Bmatrix} \frac{x}{r}&\frac{x-d_{e}}{d_{i}} \\ d_{i}x+d_{e}&rx \end{Bmatrix},s(k_{p},d_{p})[/math]

 

The variable r is equal to [math]\frac{d_{i}\times\frac{\frac{\mathrm{d} }{\mathrm{d} x}\left ( u_{f}b_{f} \right )}{d_{e}}}{(1+d_{i}x)}[/math]

 

If you need more information to solve this, please let me know. I will also continue to work on this problem.

Edited May 3, 2014 by Unity+