The exact dot product as basic tool for long interval arithmetic
Computing with guarantees is based on two arithmetical features. One is fixed (double) precision interval arithmetic. The other one is dynamic precision interval arithmetic, here also called long interval arithmetic. The basic tool to achieve high speed dynamic precision arithmetic for real and interval data is an exact multiply and accumulate operation and with it an exact dot product. Pipelining allows to compute it at the same high speed as vector operations on conventional vector processors. Long interval arithmetic fully benefits from such high speed. Exactitude brings very high accuracy, and thereby stability into computation. This document, which has been incorporated into the draft standard for interval arithmetic being developed by IEEE P1788, specifies the implementation of an exact multiply and accumulate operation.
exact dot product computer arithmetic arithmetic standard floating-point arithmetic scientific computing