Error Analysis of Various Forms of Floating Point Dot Products




Castaldo, Anthony M.
Whaley, R. Clint
Chronopoulos, Anthony T.

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UTSA Department of Computer Science


This paper discusses both the theoretical and statistical errors obtained by various dot product algorithms. A host of linear algebra methods derive their error behavior directly from dot product. In particular, most high performance dense systems derive their performance and error behavior overwhelmingly from matrix multiply, and matrix multiply’s error behavior is almost wholly attributable to the underlying dot product that it is built from (sparse problems usually have a similar relationship with matrix-vector multiply, which can also be built from the dot product). With the expansion of standard workstations to 64-bit memories and multicore processors, much larger calculations are possible on even simple desktop machines than ever before. Parallel machines built from these hugely expanded nodes can solve problems of almost unlimited size. Therefore, assumptions about limited problem size that used to bound the linear rise in worst-case error due to canonical dot products can no longer be assumed to be true today, and will certainly not be true in the near future. Therefore, this paper discusses several implementations of dot product, their theoretical and achieved error bounds, and their suitability for use as performance-critical building block linear algebra kernels.



dot product, inner product, error analysis, BLAS, ATLAS



Computer Science