Define round for complex numbers

[jiahao]

No description provided.

[ArchRobison]

round made sense to me as "round to nearest Gaussian integer". trunc sort of made sense to me as "round towards origin to a Gaussian integer". floor and ceil seemed more questionable since they cover rounding towards only two of the four corners of the complex plane. They would at least obey the identity floor(-z)=-ceil(z). But relations based on < would seem to be lost.

I'd be inclined to include only complex round for now.

[eschnett]

In addition to round, rint should also be available: round breaks ties away from zero, rint breaks ties towards even.

And there we enter a slippery slope... If we accept round with its notion of "away from zero", then trunc with its notion of "towards zero" should also make sense.

I would not be averse to having floor and ceil as well. You can use complex conjugation to cover the other two "corners". I speculate that these cases are not even necessary most of the time -- if you describe a rectangle in the complex plane, then describing two opposite corners suffices, for which you'd use floor and ceil.

[jiahao]

@simonbyrne's more recent work on exposing RoundingModes has lent more weight to the idea that only round should be exposed.

The C99 standard actually specifies separate rounding modes for the real and imaginary components, which seems a little odd at first, but I think is the only reasonable thing to do and is also what @eschnett has proposed. trunc, ceil and floor correspond to three different real rounding modes which can be freely applied to real and imaginary components separately; in principle one could define trunc(z) in terms of the (RoundToZero, RoundToZero) rounding mode, but this seems to have limited value.

[jiahao]

Following what we do now for real numbers, I've exposed 3 different round methods for complex numbers:

• The "expert" rounding mode aware round(z, RoundingMode, RoundingMode)
  which takes separate RoundingModes for real and imaginary components
• The basic round(z) which applies round to the real and imaginary
  components separately using the current global rounding mode
• round(z, d, b) for rounding the real and imaginary components
  separately to d digits each in base b.

The last method, together with float(z), is sufficient to restore the ability to round complex matrices with round.

[simonbyrne]

This seems like a reasonable approach.

[ivarne]

I think the signature should rather be

round(z, RoundingModeReal, RoundingModeImaginary)

[ArchRobison]

+1