I know the factorization method of Leopold Infeld and his student Thomas Hull (I-H). The quest for exact solutions since the time of Evariste Galois has become an industry. I am pursuing the proposal that Theoretical Physics algorithms, can be systematized by the following two principles:
Fit Principle (FP)
Leibniz Principle (LP)
The first could also be called Gauss Principle, the first example was the Minimum Mean Square Error Fit algorithm. Leibniz advised to use the simplest algorithm to represent data. Put together they are useful, and a guide to understand the four Forces of Nature.
I-H has an intermediary step with the first variation operator additively modified by a gauge function. This function encodes the forces.