Re: the "set paradigm" is harmful
Intlab provides an intermediate midrad representation
in a fast matrix multiply option, but at the expense of an
overestimation factor of 1.5 for the results widths.
To get rid of this factor would make the midrad representation
slower than the standard version!
This is not the whole truth, the factor 1.5 is the WORST case.
It ONLY happens for WIDE input intervals.
For example, let two interval matrices A,B be given such that
each component interval has a relative accuracy of 1e-3.
Denote the narrowest inclusion of A*B by ABnarrow, the inclusion
computed by midpoint-radius arithmetic by ABmidrad.
Then the diameter of all components of ABnarrow are AT MOST
overestimated relatively by 5e-7.
In other words, we can expect the narrow result intervals to
have relative diameter 2e-3, say, for example
[ 3.138451 , 3.144735 ]
for pi. The "overestimated" result would then be
[ 3.138450 , 3.144736 ]
The gain is that mid-rad interval matrix multiplication
is MUCH MUCH faster than inf-sup arithmetic.
I understand that in Arnold's applications wide intervals often
occur. So he cannot accept any overestimation.
In my applications wide intervals occur, but not that frequent.
However, large matrices occur. Multiplication of interval matrices
of large dimension in realistic time needs mid-rad arithmetic:
n=500; A=midrad(randn(n),1e-3); B=midrad(randn(n),1e-3);
intvalinit('SharpIVmult'); tic, A*B; toc,
===> Slow but sharp interval matrix multiplication in use
Elapsed time is 77.518951 seconds.
intvalinit('FastIVmult'); tic, A*B; toc
===> Fast interval matrix multiplication in use (maximum overestimation
maximally factor 1.5 in radius)
Elapsed time is 1.418354 seconds.
The reason is first that matrix mid-rad IV-multiplication can use BLAS
routines. Second it needs only 2 times to switch the rounding mode
compared to (at best) n^2 times for inf-sup multiplication.
Best wishes
Siegfried
--
=====================================================
Prof. Dr. Siegfried M. Rump
Institute for Reliable Computing
Hamburg University of Technology
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21071 Hamburg
Germany
phone +49 40 42878 3027
fax +49 40 42878 2489
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and
Visiting Professor at Waseda University
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