It is actually really simple. I just suck at using Matlab.
Basically I have several 8x8 matrices, the contents of which contain lots of constants and 2 unknown variables, w and n.
I need to multiply these matrices together, which will give me an overall transfer matrix, [U]. This will still be an 8x8 matrix. It needs to be done symbolically since I have 2 variables in there.
From here I have some boundary conditions set that will simplify the 8x8 matrix to a 4x4. Basically I am just pulling elements {{U31,U41,U71,U81},{U32,U42,U72,U82},{U35,U45,U75,U85},{U36,U46,U76,86}} from the 8x8 matrix and creating a new 4x4 matrix, [U1].
Then I set the entire matrix = 0 and find the determinant, which gives me a rather large polynomial with unknowns w and n.
At this point I need 3 things done with this polynomial:
1. Assume w=n, so instead of a large polynomial with 2 unknowns, it is just a large polynomial with 1. I then need to find all of the roots of this polynomial.
2. I need to solve for all roots of w if given a range of n to input. For instance, say my range of n is (-10000, 10000) with an interval of 100. For each ith increment of n, I need the program to find the roots (of w) of the equation. If my numbers are correct I have a 16th order polynomial right now so I should have 8 roots of w for each n.
3. Finally, I need it to plot all of this and make it look pretty
No I work at Halliburton now.