better names for norm functions?

[StefanKarpinski]

I've always found the convention of calling norm with a string or symbol argument indicating different norms both ugly and inconvenient and on the implementation side, checking various symbols is awful. It seems to me that vector and matrix norms deserve their own function names, so I'd like to propose better names, such as:

norm2(A)
norm1(A)
normp(p,A)
norminf(A)
normmax(A)
normfrob(A)
normscha(p,A)
normspec(A)
normnucl(A)

We can either get rid the Matlab-style norm function or leave it for compatibility define it by calling the various primitive norm functions listed here.

[johnmyleswhite]

I actually really like the Matlab style norm function, but agree that its implementation is kind of awful. Why not just have norm(A, p) as the Lp norm be the default norm and only use extended names for the others?

[stevengj]

It is pretty convenient to just be able to type norm(expression) to get a reasonable default norm of the expression regardless of type.

It's also not clear to me that matrix and vector norms should necessarily have different names, since the two are closely related. Two possibilities:

• If normfoo(v) gives a vector norm, normfoo(M) could give the corresponding induced matrix norm. Problem: the induced norm may be hard to compute efficiently in general. (For the L2 norm you use the SVD, and for the L1 or Linf norm I guess you can get the induced norm by solving an LP, but for general Lp norms I'm not sure how you do it.)
• Alternatively, if normfoo(v) computes a vector norm, normfoo(A) could simply compute the equivalent of normfoo(reshape(A,numel(A))), which is a valid matrix norm and can always be done in linear time. Other matrix norms could then have different names. Note that the Frobenius norm in this case is just an alias for norm2.

[toivoh]

Actually, the induced 1-norm (inf-norm) of a matrix is even easier to compute: sum the absolute values along the columns (rows), then maximize over the sums. (This can also be used to overestimate the induced 2-norm, since norm2(A)^2 <= norm1(A)*norminf(A))

I have no idea about how to compute induced p-norms in general, though. I would still prefer for norm to compute induced matrix norms by default; I think that is what you most often want.

[ViralBShah]

I think the proposed norm names are worse than what matlab does! I think that using symbols for stuff like Frobenius norm is a bit ugly, but a series of new names is uglier.

[StefanKarpinski]

I guess I'm alone on this one. I for one define frob(A) = norm(A,'frob') in all my Matlab codes. I'll leave this for a bit and see if anyone else chimes in, and if not, I'll just close it.

[timholy]

I don't have strong preferences, but I confess to being sympathetic to this proposal. So you're not alone.

[toivoh]

I'm with you too, Stefan. Those norms are not different versions of the same function, they are just different functions.

In matlab, the setup with a single norm function might possibly be useful for writing functions that use a norm specified by the user. In Julia, I think that such a thing would be done much better by accepting a norm function as an argument to the function.

[StefanKarpinski]

It seems like a good compromise here might be to give names to the different ways we're going to provide for computing norms – i.e. the different if/else cases in the norm function so that one can call them directly and then implement norm but checking the symbol parameter and calling the implementation functions.

[stevengj]

I don't think implementing both styles is an improvement; better to pick one style and stick with it than have a lot of redundant functions.

I still think that norm(x, p) (defaulting to p=2) is a reasonable style at least for Lp norms on vectors and the corresponding induced norms (at least for p=1,2,inf) on matrices.

[StefanKarpinski]

The only redundant function then would be norm; the main change would be having individual entrypoints for specific norm functions instead of cramming them all into the body of norm. Even just as an internal implementation detail, it seems like better style. This would just be a matter of exposing those entrypoints in Base.

[StefanKarpinski]

Following Matlab, we would then have the following norm functions:

norm2(a)
norm1(a)
norminf(a)
normfrob(a)
normp(p,v)

[johnmyleswhite]

And would lose norm? That seems like a mistake to me.

[StefanKarpinski]

No, that could stay.

[ViralBShah]

I am not excited about having all these norm functions. The only ugly case is the Frobenius norm. In that case, I am happier having norm(a, :fro) as compared to 5 new normx functions.

[JeffBezanson]

The most discoverable name for that case is probably frobeniusnorm().

[johnmyleswhite]

Why not just frobenius()?

[ViralBShah]

I still think we just leave it as norm(a, :fro). Also, once we have keyword arguments, this will become a whole lot prettier.

[JeffBezanson]

I increasingly think things should be called What They Are Called. To get the frobenius norm you call frobeniusnorm(). The syntax norm(a, :fro) is not obvious to a matlab user either. I'm not sure what the keyword would be. norm(a, flavor=:fro)? If I'm being crazy here I'll just go back to my corner though.

[StefanKarpinski]

Obviously, I agree that norm(a, :fro) is not a good spelling. forbeniusnorm is a bit long-winded, but better.

[pao]

Because this needs more opinions, I'm sympathetic to the argument for non-p norms, but prefer that the norm(a, p) be used for p-norms including the infinity-norm, defaulting to 2. Those really can all be expressed by parameterizing a single function, even if they're not implemented that way.

[johnmyleswhite]

I agree with @pao that norm(a, p) is what norm should stand for. I think longer names for other functions is better, although I'm not sure why we can't have frobenius instead of frobeniusnorm.

[JeffBezanson]

I'm ok with frobenius. It's equally tab-completable.

[ViralBShah]

I do not like frobenius as that does not even suggest that it is a norm. I suggest the following names, which are also tab-completable and discoverable in the REPL:

norm(x, p) # for 1,2,3... norms and Inf norm
normfro(x)

[nolta]

I think the problem w/ frobenius is that it could refer to either the norm or the number (http://en.wikipedia.org/wiki/Coin_problem).

I mildly prefer not splitting up norm, since even Mathematica follows Matlab's convention (e.g., Norm[x, "Frobenius"]). But frobeniusnorm is ok w/ me too.

[StefanKarpinski]

I like Viral's suggestion here: norm(x,p) for p = 1, 2, ..., Inf and normfro for the Frobenius norm. Other radically different norm functions can get their own function name and maybe be defined in a Norms package/module.

[ViralBShah]

Ok, I will update the implementation accordingly. A Norms package for everything else is perfect.