There is a flowchart in the mldivide, \ documentation page that describes at a high level what it does to solve a system.
But if you're publishing your results, do you need to explain what it does or would showing that it gives a correct result (if x = A\b then A*x should be close to b for some definition of "close to") be sufficient?
By the way, I wouldn't consider a system of size 3^5-by-3^5 to be "really large".
3^5
Or did you mean 3e5?
3e5
