Re: the "set paradigm" is harmful

[Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>]

Siegfried M. Rump schrieb:

> 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.

This only shows an an inefficient implementation of the exact
matrix-matrix multiply. One can use BLAS and gets by with two
rounding mode changes if one uses X*Y = sum_j X(:,j)*Y(j,:) and first
computes all upper bounds, and then all lower bounds.

I'd be interested in how this compares in timing with the above.


Arnold Neumaier