Error Analysis of Various Forms of Floating Point Dot Products

This thesis 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 the dot product. In particular, many 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 it uses. As the standard commercial workstation expands to 64-bit memories and multicore processors, problems are increasingly parallelized, and the size of problems increases to meet the growing capacity of the machines. The canonical worst-case analysis makes assumptions about limited problem size which may not be true in the near future. This thesis discusses several implementations of dot product, their theoretical and achieved error bounds, and their suitability for use as performance-critical building block of linear algebra kernels. It also statistically analyzes the source and magnitude of various types of error in dot products, and introduces a measure of error, the Error Bracketing Bit, that is more intuitively useful to programmers and engineers than the standard relative error.