On the Accuracy of the Solution of Linear Problems on the CELL Processor
Résumé
Several super computers have been designed as massively parallel
computers using the CELL processor as their main component. Such is
for example the IBM Roadrunner which broke the world computing speed
record in June 2008. However, even if machines of this kind are
absolutely necessary to solve numerical problems that could not
be solved otherwise, the question of the accuracy of the
solution may become critical when obtained with a
monstrous amount of computation. Concerning the question of accuracy, the
arithmetic of the eight on chip parallel processors of CELL have
two drawbacks: i) rounding is towards zero and not to
nearest, ii) division is very inaccurate. The paper
deals with the effect of these two particularities on the result
of scientific computations. First, it is shown that the classical
computation of the inner product of two $n$-dimensional vectors
has an accuracy which is
$O(\sqrt{n})$ for rounding to nearest and $O(n)$ for all
other rounding modes. Thus the fast rounding to zero mode of the
CELL arithmetic is certainly not the best concerning the
accuracy of results when solving linear problems. Second, it is shown that
in algorithms using divisions, it is necessary to be careful in
programming as standard low level functions do not include
division but only (multiplicative) inverse with a low precision.
The consequence is that solving large linear systems on super
computers using the CELL with unsuitable methods may be prone to
significant errors and therefore the results
must be carefully controlled. Numerical examples are given.