hypot with complex arguments

[giordano]

While looking at #31922 I noted that currently

julia> hypot(im)
1.0

julia> hypot(im, 1)
1.4142135623730951

which doesn't look correct given the meaning of hypot as "more precise version of \sqrt{\sum x_i^2}". I couldn't find any test about hypot with complex arguments, so I think this case was never taken into account.

[thchr]

It doesn't seem like one would ever want hypot(::Complex,::Complex) to return a complex number though: the utility of hypot is in computing lengths.
Maybe the doc-string should simply be fixed to sqrt{|x|^2+|y|^2} and sqrt{\sum |x_i|^2} for the two- and varargs method.

For what it's worth, this is what Matlab does. Numpy doesn't seem to allow complex input for hypot, which is also an option, I suppose; but technically breaking.

[giordano]

I was also thinking about disallowing complex input. The signature could be restricted from Number to Real, but yes, it's breaking

[StefanKarpinski]

Arguably, the hypotenuse of a triangle with imaginary side lengths is imaginary.

[KlausC]

An imaginary side length of a triangle is beyond my imagination. I would interpret hypot(r1+i1*im, r2+i2*im) as the length of the 4-dimensional vector (r1, i1, r2, i2), which is the equal to sqrt(abs(r1+i1*im)^2 + abs(r2+i2*im)^2).

[giordano]

@KlausC so you'd say that current behaviour is correct (and only docstrings would need to be adjusted)?

[KlausC]

right, I support @thchr. Also the 1-argument version hypot(3+4im) == hypot(3, 4) makes geometric sense, if you view a complex number as a right triangle. So no objections against complex arguments and everything for a real result and geomteric interpretation.

[StefanKarpinski]

I imagine (see what I did there?) that @stevengj or @simonbyrne might have a useful opinions on this.

[simonbyrne]

One can view hypot as the operator whose reduction gives the l2-norm, which is consistent with what @thchr and @KlausC are proposing, so it would be "more precise version of sqrt(abs2(x) + abs2(y)).

[giordano]

Indeed, before PR #25571 the vararg method of hypot was using exactly (vec)norm. Now it has been replaced by sqrt(sum(abs2(y) for y in x)), which however does overflow/underflow (#27141)

[stevengj]

Although it's possible to restrict the hypot function to real arguments, I think the extension to hypot(a,b) = norm([a,b], 2) = sqrt(abs2(a) + abs2(b)) is perfectly reasonable, and I would argue that it is the only useful extension of this function to other number types — you compute norms of complex numbers all the time, but the unconjugated sum of squares is rarely important. Since it costs us virtually nothing in code size to support the general definition, I think we should keep it. (It's what the Matlab hypot function does, for what it's worth.)

However, the documentation should be fixed to state \sqrt{|x|^2+|y|^2} instead of \sqrt{x^2+y^2}.

[giordano]

Ok, there seems to be consensus to keep current behaviour and fix the docstring. I've opened PR #31947.