Re: Siegfrieds recent paper
A happy and successful New Year 2012 to everybody!
Am 02.01.2012, 11:45 Uhr, schrieb Jürgen Wolff von Gudenberg <wolff@xxxxxxxxxxxxxxxxxxxxxxxxxxx>:
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)
(3.1), (3.2), (3.3) are the obvious ones, listed for completeness.
The interesting part is (3.9) and (3.11). Let rmin denote the smallest positive
floating-point number and define R := [rmin,rmin].
Then R*R=[0,rmin] in P1788, right? Maybe with a decoration?
In any case doesn't R*R contain zero although neither factor does?
In my arithmetic the result is R*R=[T,T], not containing zero.
As mentioned in my paper, (3.15) is valid if allowing denormalized numbers
because in this case a-b=0 is equivalent to a=b.
In my arithmetic (3.15) valid also in case all fl-pt numbers are be normalized.
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
Is the decoration "possiblyUndefined" ? It bothers me that I have to ask for that :(
In any case, the evaluation of E is a (complicated) way to evaluate f(x)=1/(1/1/exp(x)))=exp(-x).
In my proposed arithmetic E=[0,1] without decoration, so f(A) is included in A and proves that
f=exp(-x) has a fixed point in A=[0,1000]. The proof is rigorous without necessity to check a
decoration.
I don't like the idea that decorations have to be checked. My experience with it some time ago
is that I did Arnold a favour and put some decorations into INTLAB. After less than a month
I forgot about my own implementation and wrote a potentially erroneous program because
I forgot to check decorations. Consequently I removed it from INTLAB again.
I like to follow the KISS principle, i.e. all interval results should be valid as is.
4. typo in (2.14) must be 0 not in B
I guess (2.15)? Thanks.
Siegfried
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and
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