Re: A proposal for motion 5 "Arithmetic operations for intervals"
Ulrich Kulisch schrieb:
Arnold Neumaier schrieb:
Bo Einarsson wrote:
I submit the document "Arithmetic operations for intervals" written
by Ulrich Kulisch as a formal motion (motion 5) to be voted upon.
The document gives the definition of the arithmetic operations for
intervals in a way that is simple to read, simple to understand, and
simple to implement.
Simple???
It is short and understandable and to the point.
No. The proposal is not short, but 4 pages long, far to long
for its contents.
Simplicity should not be measured by a number of letters.
Simplicity not, but shortness, which was claimed.
There is hardly
any way to define arithmetic for floating-point intervals simpler and
shorter than by the few formulas on page 4 of the proposal.
So three pages can already be saved by deleting pages 1-3.
An implementer of interval arithmetic will not be satisfied with a
definition. He will ask for explicit formulas and a standard should
provide these.
There are different, equivalent formulas. The standard should perhaps
give some version of these, but perhaps not. The 784 standard also
ggives no explicit formulas for the results of the various rounding
modes, but leaves that to the implementors.
In any case, the early decisions should not be tied to a particular
set of explicit formulas. It should just give the semantics,
namely what the _user_ needs to know about the behavior of the
operations defined.
4. Having to write [a,b] for bounded intervals only but [a,inf), etc.
for unbounded intervals makes many table entries invalid, if one
of the computed bounds overflows under directed rounding,
resulting in an undefined [a,inf] or [-inf,b].
Moreover, it requires a separate discussion of unbounded intervals
as arguments, thus needlessly complication the exposition.
The notation [x, y) for an interval where the bound y is not an element
is quite
usual in mathemeics. The notation [x, y] may occasionally cause confusion.
I can't see any possible confusiion in the interpretation of the
interval (i.e., closed subset of the reals, as we already agreed)
[0,Inf]. It can only mean the set of all real nonnegative numbers.
However, there is considerable confusion if one insists on writing
this interval as [0,Inf). For it means that I can never refer to an
interval with lower bound zero and arbitrary upper bound u without
a clumsy statement
''[0,u] if u is finite, and [0,Inf) otherwise''.
It _must_ be possible to write an arbitrary interval with lower
bound 0 as [0,u], no matter whether u is finite or Inf, and
subsequent specialization to u=inf should not require a change
in notation.
5. Arithmetic operations like unary minus, abs, square, sqrt, etc.
are not covered. But some issues like the treatment of undefined
values of the real operation affect these as well, and hence must
be decided _before_ fixing what happens to special cases.
As the title of the proposal says it just considers arithmetic
operations for intervals.
The term is not unambiguous. For example, Wikipedia says,
''The traditional arithmetic operations are addition, subtraction,
multiplication and division, although more advanced operations (such as
manipulations of percentages, square root, exponentiation, and
logarithmic functions) are also sometimes included in this subject.''
(http://en.wikipedia.org/wiki/Arithmetic_operations)
Independently of that, my argument was that some issues like the
treatment of undefined values of the real operation affect also
elementary functions, and hence must be decided _before_ (or
simultaneously with) fixing what happens to division.
6. The set of floating-point numbers cannot be regarded as a subset
of R, since +0 and -0 are distinct floating-point numbers.
Thus already the formal setting (line 3 of the proposal) is faulty.
Interval arithmetic and floating-point arithmetic are different topics.
In interval
arithmetic -0, +0, and NaN should not be considered to be floating-point
numbers.
In interval arithmetic,
- either one does not need to consider rounding issues, but works with
exact hulls;
- or one needs to address the effects of representation by implementable
floating-point data, and then has to consider the floating-point data
actually available.
7. No symbolic notation for the set of real closed intervals is better
than a completely artificial one that puts parentheses around the
symbol for the set of real, closed and bounded intervals.
Elsewhere in mathematics, (X) is the same object as X if both have
a meaning.
If x is a number or a variable or an arithmetic expression it is correct
that (x) means the same.
But IR is none of these.
*IR may be a better notation. But its definition in the StandardNotation
may be misleading.
I*R is used for another set in modal arithmetic.
A special notation is not needed to convey all the relevant information;
hence it should not be introduced, especially since it is controversial.
Nevertheles, we are grateful for all arguments. The proposal will be
rearranged. Other duties do not allow
to do this in the three weeks timeframe. So we withdraw the motion for
the moment.
Thanks.
Proposals for motions should be error-free, short and concise.
Arnold Neumaier