Re: motion elementary functions
> From: John Pryce <j.d.pryce@xxxxxxxxxxxx>
> Subject: Re: motion elementary functions
> Date: Mon, 19 Oct 2009 21:13:59 +0100
> To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
>
> Dan, P1788
>
> On 19 Oct 2009, at 16:36, Dan Zuras Intervals wrote:
> > Both are reasonable functions to demand we have lying
> > around for use by 1788.
> >
> > But both are Real functions of Real or integer powers.
> >
> > In 1788 we are in the business of defining interval
> > functions of interval arguments.
>
> I am wondering if there is a misunderstanding going on. Of course,
> very possibly I am the one who is misunderstanding...
Perhaps we misunderstand each other motivation but I'm
fairly sure you understand the technical implications
far better than I do.
Let's explore those & see what we can discover:
>
> It is fairly explicit in Motion 6 that an interval elementary
> function ee is, by definition, an interval extension of a point
> elementary function e. (Things like hull(xx,yy) and intersection
It IS fairly explicit, isn't it.
3.3.11. INTERVAL ELEMENTARY FUNCTION. An interval version
of a point elementary function, that is provided by an
implementation. The set of these is the implementation's
interval elementary function library (interval library for
short). These terms may be qualified by a format, e.g.
"binary64 interval library".
3.3.12. INTERVAL EXTENSION of a point function. See Section
3.5.5.
then later
3.5.5. Unless otherwise specified, an interval mapping is a
... interval extension (of a point elementary function)...
where the parenthetical text is mine.
It is also fairly explicit that extensions of real point
functions are not the only interval functions to be
defined. All of hull, interval intersection, & interval
union are all cited as counter examples.
For the purposes of this argument, let us admit the
possibility that there might be more.
> (xx,yy) are not of this kind, and are called interval mappings, not
> functions.) So for each e, there are basically just two things for
> P1788 to decide:
Well, I would agree that this is very nearly all that is
needed for any given point elementary function 'e' ('f' is
used in Motion 6).
Still, prior to that decision one must decide just WHICH
functions 'e' should be specified, what are their definitions,
are they suitable candidates for point elementary function
extension in this manner, & THEN their domain & accuracy as
well as issues of exceptional behavior are to be defined.
> (1) a precise definition of e's domain, and value at each point of
> the domain;
> (2) how tight we require the interval extension to be, as measured
> e.g. by Vienna's "tight", "accurate" or "valid".
> If we decide "tight" then ee is unique: it has to be the "natural
> interval extension".
Motion 6 is also careful to avoid issues of accuracy by
distunguishing between interval extensions & sharp interval
extensions.
>
> I think Motion 7 states it is not about (2), which is to be decided
> later (by a subgroup, I personally hope).
Agreed. This will be a difficult issue for both us as
writers of 1788 & the 1788 implementers & users as our
customers.
>
> Hence Motion 7 is just about (1), that is,
> about agreeing a list of precisely specified point
> functions of real variables.
While this is a resonable conclusion, it is not the one
I would draw. Or if it is, then I guess I would argue
that xx^yy be outside the domain of functions suitable
for point function extension to the interval domain.
>
> Note that, (as Vienna says) pown(a,p) or in Vienna's notation a
> intpow p, is not regarded as one function of 2 variables (real,int),
> but as a family of functions pown(. , p) of one real variable.
>
> John
This is a perfectly reasonable family of functions to
extend to intervals. But please remember that it IS an
infinite family of functions being defined.
This family of (now) interval functions, indexed by 'p'
would be defined to take its interval inputs from the
whole of IR & return interval results to the whole of
IR.
Note that these functions are perfectly continuous in
their domains (with the exception of those functions
indexed by negative p called with intervals containing
zero as an element). While they are all well defined
for intervals with negative elements, those indexed by
an odd 'p' return intervals with negative elements &
those indexed by an even 'p' return positive intervals.
And all issues of continuity in the 'p' argument have
been defined out of consideration by its use as an
index rather than an element of R.
Again, it is a perfectly reasonable family of functions
to have lying around but is leaves much to consider
before defining an entirely interval xx^yy.
>
> >
> > It was that issue I was trying to address & that about
> > which my argument was made.
> >
> > Demand any useful set of power functions you feel is
> > necessary to aid in the development of an interval
> > library.
> >
> > But I would still like to hope that we can get past
> > the mistakes of 754 & come to some agreement on a
> > single x^y in the domain & range of the intervals.
> >
> > It is possible, I believe.
> >
> > In a way that:
> > is mathematically correct,
> > is as general as possible, &
> > was NOT possible for floating-point.
> >
> >
> > Dan
You have made no similarly supporting argument for the
point extension of a real x^y to an interval xx^yy
which is restricted to x > 0 or xx > 0.
Let me make one: It is ALSO a perfectly reasonable
function to have lying around. Everyone knows how to
compute it. And, if the french complete their work,
it will soon be possible to compute it both efficiently
& accurately ('tightly', in our jargon).
There are issues of what is to be done with such a
function when called outside its domain. But given our
current discussion of decorated intervals, I'm sure we
can come to some agreement involving an empty interval
suitably decorated with some 'out of domain' indicator.
Both this real-to-real function & the family of real-to-
integer functions are well known & well defined.
But if one attempts to combine them into a real-to-real
function that covers the entire domain including x < 0,
such an attempt is doomed to encounter contradictions.
In the domain of the REALS.
HOWEVER, among the intervals the answer is different.
Or, to be perfectly candid, it CAN be different.
We have it within our power (no pun intended :-) to
define an 'interval' x 'interval' --> 'interval'
function that:
(1) agrees with the real-to-real function
where it is defined,
(2) agrees with each & every real-to-integer
function where it is defined,
(3) is the interval extension of a power
function defined on the rationals,
(4) is the interval extension of the real
projection of the complex x^y,
(5) and agrees with all of these functions
everywhere.
In the parlance of Motion 6 this would NOT be an
interval extension of a point function but a function
defined for intervals in a way that is unique to the
interval domain much like hull has no counterpart
among the reals.
It would be, as someone pointed out, something of an
all-things-to-all-people. It could, in principle,
replace all of the other functions as it is likely
that, in practice, it would call one of the other
functions within their domain anyway.
As I realized at the end of my argument last week,
while it would be at the top of the food chain, so
to speak, it would not be the answer to everyone's
problems.
In particular, defining an interval xx^yy in this way
means that certain contractive mapping algorithms
would never converge to an arbitrarily tight result
in the negative domain due to the split into two
distinct manifolds.
Thus, it might be necessary to provide helper functions
along the lines of negativeManifoldPow &
positiveManifoldPow to aid those algorithms.
So, FAR from attaining the goal of having
One Ring to rule them all,
One Ring to find them,
One Ring to bring them all and
in the darkness bind them
I have created 3.
It is not what I was seeking but it was what came out
of the argument. I could hardly deny the result.
Whether this is the sort of thing you 'expect to vote
for' or not is up to you.
I don't know what will be needed as well as most of
you.
Its up to you now.
Yours,
Dan