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

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( √ 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.

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