Which of the two norms runs is decided as follows:

============================ ========================================== arguments reduction ============================ ========================================== dim=None, ord=None flatten, Euclidean norm, any rank dim=None, ord given vector norm if 1-D, matrix norm if 2-D dim names one axis vector norm along it dim names two axes matrix norm over that plane ord is "fro"/"nuc" matrix norm regardless ============================ ==========================================

So ord means different things at different ranks, and the same order names two different quantities: ord=2 is the Euclidean norm of a vector but the spectral norm — the largest singular value — of a matrix, which is not its Frobenius norm. Pass an explicit dim, or call vector_norm or matrix_norm directly, wherever that distinction matters.

Many norm orders (spectral, nuclear) require an SVD and therefore cost $O(\min(m,n) \cdot mn)$; entry-wise norms reduce in a single pass.

Overflow is not a failure mode here: the input is rescaled by its largest magnitude before reducing, so a norm whose value is representable is computed even when its squares are not.