Definition of intervals as subsets of R - the bad news
Dear all,
now I found time to work on the change of INTLAB according to the
anticipated new definition of intervals as subsets of R. However,
a number of problems occur.
Theoretically, we have the simple formula
[ainf,asup] + [binf,bsup] = [ainf+binf,asup+bsup] .
In practice, I distinguish various cases such as
IR op R
R op IR
IR op IR .
Now [ainf,asup]+inf may occur, and I have to check whether
ainf=-inf. You can imagine that many case distinctions are
necessary. Moreover, I often optimize the code when one
operand is scalar, not to speak of complex intervals, which
are treated in INTLAB as well. Really awkward becomes the
multiplication of two matrices where some of the n^3 products
may be 0*inf or inf-inf may occur.
The case distinctions are all done for the sake of speed.
If only scalar intervals were involved, this may not be
beneficial. However, INTLAB is the toolbox for verified
computations in Matlab, and as the name says, this is a
Matrix Laboratory.
In a nutshell, defining interval to consist only of finite
numbers is a nice concept, but various exceptional operations
such as 0*inf or inf-inf have to be identified and redefined.
For example, this slows down addition by about a factor 2.
I am afraid, a penalty might also apply to a C- or Fortran
implementation.
At least for my INTLAB this is hardly acceptable. The difference
between the current state and the anticipated change is only
the handling of infinity. In many applications infinity hardly
occurs, so all computations are slowed down for not much benefit.
I want to stress again that these are my implications for INTLAB,
forcing me not to do the intended change. From a mathematical
point of view I am much in favor of intervals being sets of
finite numbers; just 0*A for any interval A is such a nice
property.
However, for the time being interval arithmetic depends on the
IEEE 754 floating-point standard. Hence special care is necessary
for exceptional operations such as 0*inf or inf-inf.
Best wishes
Siegfried M. Rump
--
=====================================================
Prof. Dr. Siegfried M. Rump
Institute for Reliable Computing
Hamburg University of Technology
Schwarzenbergstr. 95
21071 Hamburg
Germany
phone +49 40 42878 3027
fax +49 40 42878 2489
http://www.ti3.tu-harburg.de
and
Visiting Professor at Waseda University
Faculty of Science and Engineering
Shinjuku Lambdax Bldg. 902
2-4-12 Okubo, Shinjuku-ku
Tokyo 169-0072
Japan
phone/fax in Japan +81 3 5286 3414