30 June 2008

(Pseudo)random numbers matter

The newly released LAPACK Working Note #206 gives yet another reason why generating good pseudorandom numbers matters:

In May 2007, a large high performance computer manufacturer ran a twenty-hour long High Performance Linpack benchmark. The run fails with the following output:

|| A x - b ||_oo / ( eps * ||A||_1 * N ) = 9.22e+94 ...... FAILED

What happened was that the benchmark's matrix generator uses a lame linear congruential pseudorandom number generator, which causes generated matrices to have repeated columns for certain unfortunate choices of matrix dimension. This of course makes one wonder why the generator doesn't just make a matrix which is known to be invertible, say, by generating a sufficiently nonzero diagonal matrix and hitting it on both sides with orthogonal transforms until the zeros are filled in. Regardless, the bug meant 20 hours of very expensive, intensely power-consuming supercomputer time were wasted on computing the wrong answer to a problem which at such sizes very few people need to solve. So, random numbers do matter ;-)

26 June 2008

Contemplation leads to messes

I like to make espresso using one of these gadgets. When you press out all the water, it leaves a "puck" of compressed grounds, which you can pop out straight into the garbage or compost just by pushing on the handle. I was looking at the puck this morning and saw some lovely patterns made by different layers of grounds. One thin layer in the puck appeared more compressed than the other layers, but this more compact layer wasn't a straightforward horizontal cross section -- it had a ridged topography like one sees in the sedimentary rock hills around here. It made me wonder about the bulk properties of solid particles, about which I had seen an interesting presentation a few months before, on simulating such materials.

As I was holding the puck in my hand and examining this layer from all angles, the puck suddenly broke into its constituent coffee grounds and made a huge mess all over the floor. I realized then that I had fallen into the nerd's usual trap of neglecting practical matters for the sake of contemplating lovely abstractions ;-)