[a_zaydak]

I really agree with your comments; especially number 1. Often I can't use some black box implementation of an solver (or other algorithm for that matter) without some modifications. Numerical stability is a big one but also just performance. Sometimes the mathematically correct way of doing something is not always the best in practice. Short cuts and approximations can provide huge benefits. It is difficult to make those modifications without understanding the inner workings of the original method.

[Tainnor]

One example is that GJ stops being generally efficient in arbitrary precision settings. Many people will never have to deal with this, but if you are doing cryptography, it matters.

For example, while you can use GJ to calculate the determinant of a matrix, this can easily become exponential, and for integer matrices (or generally for matrices over division rings), there is an alternative method (the Bareiss algorithm) that is actually efficient.