allow more generic norms on any array whose element types support norm

[stevengj]

As @mauro3 pointed out on the mailing list, we might as well use norm rather than abs when taking the absolute value of individual elements inside some of the generic norm functions, since norm is a more sensible function to define than abs on abstract Banach spaces.

This patch changes the generic Lp norms to use norm(x) instead of abs(x); there should be no performance penalty for numbers, where norm should inline to abs.

[StefanKarpinski]

Seems reasonable, although it's weird to have minabs = ... norm(...).

[stevengj]

In the context of computing a norm, it would seem weird to write minnorm = ..., too, since which "norm" is being referred to might be confusing. In context, continuing to use minabs as a variable name seemed clear enough to me to be not worth changing.

[StefanKarpinski]

Yeah, not important – LGTM, but some mathier types should maybe look too.

[mauro3]

Thanks Steven, for looking into this so quickly! Would there ever be a case to use another inside-norm apart form p=2? (I have no use-case, just wondering).

[stevengj]

@mauro3, note that the norm function is only the 2-norm for vectors and matrices. (For scalars, all norms are equivalent up to an overall scale factor.) If you have your own type, you can define whatever norm you want.

My feeling is that if you have an x::Vector{Vector{Complex}} or similar and you want some more specialized norm like the L1 norm of the L1 norms, then you have to implement it yourself. It's not practical to pass in a different elementwise norm function into vecnorm (we'd either kill performance for the common case, or require code duplication, and neither is worth it for such an unusual use-case).

[mauro3]

Yes, that sounds good.