On the Accuracy of the Solution of Linear Problems on the CELL Processor

International audienceSeveral super computers have been designed as massively parallelcomputers using the CELL processor as their main component. Such isfor example the IBM Roadrunner which broke the world computing speedrecord in June 2008. However, even if machines of this kind areabsolutely necessary to solve numerical problems that could not be solved otherwise, the question of the accuracy of thesolution may become critical when obtained with amonstrous amount of computation. Concerning the question of accuracy, thearithmetic of the eight on chip parallel processors of CELL havetwo drawbacks: i) rounding is towards zero and not tonearest, ii) division is very inaccurate. The paperdeals with the effect of these two particularities on the resultof scientific computations. First, it is shown that the classicalcomputation of the inner product of two $n$-dimensional vectorshas an accuracy which is$O(\sqrt{n})$ for rounding to nearest and $O(n)$ for allother rounding modes. Thus the fast rounding to zero mode of the CELL arithmetic is certainly not the best concerning theaccuracy of results when solving linear problems. Second, it is shown thatin algorithms using divisions, it is necessary to be careful inprogramming as standard low level functions do not includedivision but only (multiplicative) inverse with a low precision.The consequence is that solving large linear systems on supercomputers using the CELL with unsuitable methods may be prone tosignificant errors and therefore the results must be carefully controlled. Numerical examples are given