Re: Draft: P1788.1 Standard for Interval Arithmetic (Simplified)

[Walter Mascarenhas <walter.mascarenhas@xxxxxxxxx>]

Dear Colleagues,

  I am implementing the content of the P1788_1_MAIN.pdf file
posted by Ned and I have a couple of questions:

1) What should be returned by the interval version of the function

sign?  For instance, what should be returned by sign( [-1,1] )?

2) Regarding decorations, suppose f: R -> R is a continuous function

such that, due to rounding,  f([0,1]) (without decoration) returns

 [-oo,+oo] (= entire).  In my reading, in this case the standard requires
that the decorated version returns [-oo,+oo]_dac.

Consider now g(x) = 2 * arctan(x) / pi.  Again, I believe that
the standard requires that g( [-oo,+oo]_dac ) should return [-1,1]_dac,
but this implies that g( f([0,1]) ) returns [-1,1] dac,  and I find
this odd because g(f(.)) is continuous and (numerically) bounded
in [0,1], and due to the unboundedness of the numerical f(I)

we loose the information that I is bounded.

 Note that the same will happen for every continuous functions

f and g and bounded interval I such that the (numerical) range of
g is bounded and f(I) is unbounded due to rounding. In other words,
in this case we can say for sure that g( f(I) ) is com but
the standard allows only the weaker conclusion that g(f(I)) is dac,

because the information about the boundedness of I is lost

along the way due to the unboundedness of the numerical f(I).

Of course, we could check that g(f(I))  is bounded and promote
dac to a "dac+" which would also ensure that the interval
g(f(I)) is bounded, but by looking only at the decorated g(f(I))
we would not be able to tell whether I is bounded.

However, if there is a  need to check the boundedness of g(f(I))
then I do not see the point of having the (inaccurate) information
regarding the boundedness of g(f(I)) encoded in the decoration
to begin with, specially when this inaccurate information
confounds the information regarding the boundedness of I.

In other words, why should we encode information

regarding the boundedness of f(I) in the decoration

given that we can check this easily given f(I)?

Wouldn't things be simpler if the decorations

propagated only information regarding the

relation of x and the domain of f or the set

in which f is continuous?

For instance, by removing the requirement that the
computed f(x) is bounded from the definition of com

(or having a decoration com* without this

requirement) the evaluation of g( f(I) ) in the example

above would yield naturally a com (or com*) interval,

and the users interested in information regarding

the boundedness of g( f(I) ) could obtain it by

inspecting g( f(I) ), as they will need to do
anyway under the current definition of com.

Am I missing something?

                     walter.

On Mon, Jul 27, 2015 at 10:28 AM, Ned Nedialkov <nedialk@xxxxxxxxxxx> wrote:

Dear Colleagues, 

I have attached a draft of the basic standard. IEEE did not like “Basic Standard” 

so it is "(Simplified)”. 

I would like to invite everybody for comments and suggestions. 

Best regards, 

Ned Nedialkov

PS Juergen you could post it in the private area of the official web server