Towards an efficient implementation of CADNA in the BLAS : Example of DgemmCADNA routine.

Several approximations occur during a numerical simulation : physical phenomena are modelled using mathematical equations, continuous functions are replaced by discretized ones and real numbers are replaced by _nite-precision representations (oating-point numbers). The use of the IEEE-754 arithmetic generates round-o_ errors at each elementary arithmetic operation. By accumulation, these errors can affect the accuracy of computed results, possibly leading to partial or total inaccuracy. The effeect of these rounding errors can be analyzed and studied by some methods like forward/backard analysis, interval arithmetic or stochastic arithmetic (which is implemented in the CADNA validation tool). A numerical veri_cation of industrial codes, such those that are developed at EDF RD =;+;_), and a non optimized use of the memory (cache and TLB misses). We will present di_erent solutions to reduce this overhead and the results we have obtained. In order to improve the hierarchical memory usage, special data structures (Block Data Layout) are used. This allows us to improve the memory performance to reduce cache and TLB misses. A new implementation of CESTAC Method has been introduced to reduce the overhead due to the random rounding mode. Finally, we have obtained an overhead about 25 compared to GotoBLAS in a sequential mode. We will also present, briey, new extensions for CADNA : CADNA MPI and CADNA BLACS which allow to use stochastic data in programs using the communications standard routines (MPI or BLACS).