System of equations incl. complex and absolute values

[KanedaSyndrom]

Homework Statement

I'm currently implementing an algorithm in Matlab, however, I've hit a bump, I'm trying to solve the following system of equations:

Known variables = a, b, c, d[complex]
Unknown variabls = fs, fd, fv, alpha

My problem is what to do with the |alpha| since I can't get Matlab to accept that term analytically, only when solving numerically, which isn't an option in this instance.

Homework Equations

a = fs + fd*|alpha|^2 + fv
b = fs + fd + fv
c = fs + fd*alpha + fv/3
d = fv/3

The Attempt at a Solution

I've gone as far as solving to the point where I had alpha isolated and dependent on |alpha| alone.

So, I've no problem using the solve function in Matlab (to my knowledge), but I'm not able to find an analytical solution for this system due to the term |alpha|, since solve takes a string and won't recognize the 'abs(alpha)' as a string, and solving numerically isn't an option since there's more than a million iterations, so I have to find the analytical solution first before implementing.

I'm really rusty in solving for complex numbers by hand as well. I've searched the net, but what do you do when one of the factors in the system is an unknown as well?

 

[kru_]

Is alpha real, or possibly complex?

 

[KanedaSyndrom]

Alpha is assumed complex, but really, all values would follow from the d value, and thus some values will become complex.