[martinhath]

The fact that they're all cubic isn't really the notable part of the runtime of computing the different decompositions, because the constants involved are really different. In practice, a common reason for computing many of these decompositions is to solve a linear system `Ax=b`, because with the decomposition in hand it is really easy to solve the whole system (using e.g. backsubstitution). For instance, with C++s Eigen, look at the 100x100 column of [1], and we can see that there's orders of magnitude difference between the fast and slow approaches. THey're all still cubic, sure, but we're talking 168x difference here.

(of course, it's not so clear cut, since robustness varies, not all methods are applicable, and the benchmark is for solving the system, not computing the decomposition, but overall, knowledge of which decomposition is fast and which is not is absolutely crucial to practitioners)

[1]: https://eigen.tuxfamily.org/dox/group__DenseDecompositionBen...