Re: Multi-precision (was...Please give me advice)
John Pryce wrote:
On 14 Oct 2010, at 13:30, Paul Zimmermann wrote:
* would a multiple-precision implementation that only implements mid-rad be
compliant? (With a multiple-precision 'mid' and a fixed precision 'rad'.)
* would a multiple-precision implementation that only implements the triple
representation (x, inf_err, sup_err) mentioned in P1788_MAIN.pdf, paragraph
6.1, be compliant? (With multiple-precision x and fixed precision inf_err
and sup_err.)
you didn't answer those questions.
Well, I sort of asked questions back; sorry if I was too vague. No, iRRAM would not be conforming as it stands.
But on the other side, if the standard requires to implement *at least* one
inf-sup datatype then my understanding is that a package implementing only
mid-rad or (x, inf_err, sup_err) would be not conforming.
Does the set of nonempty intervals supported comprise exactly all [xlo,xhi]
where xlo,xhi belong to some given set F? (*)
(*) is not even the case for the doubles: xlo=1, xlu=0 is not allowed,
for example, although 0,1 in F. The maximum one can demand is that
the nonempty intervals present are exactly those of the form
[l,u] with l,u in F and l<=u. (**)
But for an efficient multiprecision interval arithmetic, this is too
demanding; see below.
As Arnold says, a package that implements only an (x, inf_err, sup_err) representation
could ensure this was an inf-sup idatatype. (Rather, a sequence of these, indexed by precision.)
Then it would be conforming.
But (**) is satisfied only if the error terms also have arbitrarily
long representations, which would be a severe waste of resources.
(An infsup representation with two arbitrary precision bounds
would be both more parsimonious and easier to implement, but would still
be a waste of storage and speed.)
For me, the criterion distinguishing between good and bad
representations is whether they can represent highly asymmetric
intervals such as [1,1e6] or [-1,inf] without much overestimation.
But this is difficult to cast into a nice criterion on the abstract
level.
Therefore I'd propose to require only the following:
the nonempty intervals present are among those of the form
[l,u] with l,u in F and l<=u, and all of the form
[l,u] with l,u in F_0 and l<=u are present.(**)
Here F and F_0 are different floating-point formats, and F_0
contains at least -inf, -1, 0, 1, inf.
Arnold Neumaier