Many algorithms in numerical analysis are affine equivariant: they are immune to changes of affine coordinates. This is because those algorithms are defined using affine invariant constructions. There is, however, a crucial ingredient missing: most algorithms are in fact defined regardless of the underlying dimension. As a result, they are also invariant with respect to non-invertible affine transformation from spaces of different dimensions. We formulate this property precisely: these algorithms fall short of being natural transformations between affine functors. We give a precise definition of what we call a weak natural transformation between functors, and illustrate the point using examples coming from numerical analysis, in particular B-Series.
CITATION STYLE
Verdier, O. (2018). Full affine equivariance and weak natural transformations in numerical analysis—The case of B-series. In Springer Proceedings in Mathematics and Statistics (Vol. 267, pp. 349–361). Springer New York LLC. https://doi.org/10.1007/978-3-030-01397-4_11
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