Siegfrieds recent paper
I wish everybody a happy and successful new year 2012
Let me summarize my opinion on Siegfried's interesting paper.
1. I agree with John that it will not help in diseminating interval
arithmetic
The subset definition in motion 3 is easier to comprehend.
2. We may try to handle under- and overflow more carefully by a set of
decorations
3. More algebraic properties are valid for 1788 intervals than Siegfried
assumes:
If I am not completely wrong, we have the assertions
(3.1)(3.2)(3.3)(3.9)(3.11) ad (3.15)
see below
Am 31.12.2011 08:05, schrieb Siegfried M. Rump:
For every suggestion or motion I suggest to write down a short program
showing the result with and without the suggestion. As for my proposed
interval arithmetic an example is (using INTLAB notation):
A = infsup(0,1000); % [0,1000]
B = exp(A); % [1,inf]
C = 1/B; % [T,1]
D = 1/C; % [1,inf]
E = 1/D; % [T,1]
Here T denotes "tiny" which can be coded in IEEE754 using NaN with
different mantissa bits. Note that no flag is raised. The theory,
implementation details and more examples are in the paper.
Standard interval arithmetic yields
A = infsup(0,1000); % [0,1000]
B = exp(A); % [1,inf]
C = 1/B; % [0,1]
D = 1/C; % [-inf,inf] with flag
E = 1/D; % [-inf,inf] with flag
As for 1/[0,1] the result must be [-inf,inf] because zero may be the
limit of negative numbers.
that is cset arithmetic
P1788 delivers
A = infsup(0,1000); % [0,1000]
B = exp(A); % [1,inf]
C = 1/B; % [0,1]
D = 1/C; % [1,inf] with decoration
E = 1/D; % [0,1] with decoration
On Fri, 30 Dec 2011 14:56:26 -0100, John Pryce <prycejd1@xxxxxxxxxxxxx>
wrote:
I think Siegfried Rump's recent paper is a great piece of work, whose
implications I am still absorbing. It could well lead to a practical
interval arithmetic that handles overflow and underflow better than we
currently can. But P1788 chose a different, simpler model early on.
For many reasons we cannot switch models now. A compelling reason, to
me, is that Siegfried's system is unfamiliar to the whole interval
community (not just us); it has not been worked out in detail or
implemented; experience with an implementation might uncover practical
flaws such as those that Arnold and Nate found with my own favourite,
namely cset interval arithmetic. So its time has not yet come.
The arithmetic is based on a mathematical theory given in the paper.
Properties are proved which IMHO are not valid for other definitions of
interval arithmetic. The difference to what everybody knows is the
handling of huge and tiny numbers. I think it has been worked out in
detail, and it is easy to grasp.
Best wishes for the New Year 2012,
Siegfried
4. typo in (2.14) must be 0 not in B
Jürgen
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o Prof. Dr. Juergen Wolff von Gudenberg, Lehrstuhl fuer
Informatik II
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