Rather than pseudo fields I would advocate for an "axiom" such
as
sage: Sets().Finite()
Category of finite sets
we could have
sage: Fields().Exact()
Category of exact fields
and
sage: Fields().Numerical()
Category of numerical fields
The reason is that in some situations you don't want to
differentiate between the two and being an approximate
or exact field does not matter much. Also, I am not sure
the granularity is good enough as we have two very different
kinds of "Numerical"
- truncation (floating-point): numbers are somehow exact but
operations are approximate
- guaranteed (intervals and balls): numbers are fuzzy but operations
are guaranteed
For the actual content of the ticket, to my mind "Floating-Point"
is explanatory in itself. Your proposition
> "Real Field realized by Floating-point arithmetics of precision xx"
is a bit heavy. And the "realized" is not precise enough (ie no
mathematical well-defined meaning).
Le 19/10/2020 à 16:40, Michael Jung a écrit :
Dear all,
vdelecroix schrieb am Freitag, 16. Oktober 2020 um 07:50:46 UTC+2:
I agree that these are not fields in the mathematical sense. And Sage
knows about it
sage: RR.is_exact()
False
sage: QQ.is_exact()
True
That's actually what I meant. We also have:
sage: RR in Fields()
True
We could say that RR is a non-exact realization of an exact mathematical
object which is a field, so that both outputs make somehow sense. But if we
want it as rigorously as requested, we must as well require
sage: RR in Fields()
False
If that is what we want, we perhaps need another category, e.g.
"PseudoFields" in order to maintain most algorithms, e.g. for matrices.
I'd suggest a workaround: so what about "Real Field realized by
Floating-point arithmetics of precision xx" and keep the above behavior?
And then similarly for interval arithmetics and ball arithmetics.
Best,
Michael
However, they are much more than sets as they come with approximations
of the field operations (+, x, ^-1). Maybe a reasonable terminology
would be "numerical field"? And in this regard, RealFloatingPointField
perfectly make sense. I am against RealFloats.
Le 15/10/2020 à 11:24, Samuel Lelievre a écrit :
2020-10-15 08:21:06 UTC, John Cremona:
I was expecting someone more pedantic than me to point out that this
set
is not a field in the mathematical sense. Since this is a big change
anyway
(at least to a lot of doctest outputs) should we think more carefully
about
what we want to call RR? Instead of "Real floating-point field with x
bits
of precision" we could have "Real floating-point numbers with x bits of
precision" perhaps. (With an implied "The set of" in front).
Good point!
I like "Real floating-point numbers with x bits of precision"
with short name RFN for real floating-point numbers.
Or shorter: "RealFloats" -> "Real floats with x bits of precision",
short name RF for the standard one with 53 bits of precision.
Consistency would dictate to rename and change the string representation
for all of the following:
- ComplexField -> ComplexFloats
- RealField -> RealFloats
- ComplexDoubleField -> ComplexDoubleFloats
- RealDoubleField -> RealDoubleFloats
- ComplexBallField -> ComplexFloatBalls
- ComplexBallField -> RealFloatBalls
- ComplexIntervalField -> ComplexFloatIntervals
- RealIntervalField -> RealFloatIntervals
and maybe more sort-of-fields that can be listed using:
```
sage: [g for g in globals() if 'ield' in g]
```
- ComplexLazyField -> ComplexLazyFloats?
- RealLazyField -> RealLazyFloats?
- MPComplexField -> MPComplexFloats?
What about pAdicField?
Of course we can do things one at a time, but it's good to plan ahead
and maybe have a meta-ticket to keep track of what is done and what
needs to be done.
Side remark: should ComplexIntervalFieldElement, FieldElement
and NumberFieldElement be removed from the global namespace?
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