Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest

In this Part II of this paper we first refine the analysis of error-free vector transformations presented in Part I. Based on that we present an algorithm for calculating the rounded-to-nearest result of s := ∑ pi for a given vector of floatingpoint numbers pi, as well as algorithms for directed rounding. A special algorithm for computing the sign of s is given, also working for huge dimensions. Assume a floating-point working precision with relative rounding error unit eps. We define and investigate a K-fold faithful rounding of a real number r. Basically the result is stored in a vector Resν of K non-overlapping floating-point numbers such that ∑ Resν approximates r with relative accuracy epsK , and replacing ResK by its floating-point neighbors in ∑ Resν forms a lower and upper bound for r. For a given vector of floating-point numbers with exact sum s, we present an algorithm for calculating a K-fold faithful rounding of s using solely the working precision. Furthermore, an algorithm for calculating a faithfully rounded result of the sum of a vector of huge dimension is presented. Our algorithms are fast in terms of measured computing time because they allow good instruction-level parallelism, they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require some extra precision: The only operations used are standard floating-point addition, subtraction and multiplication in one working precision, for example double precision. Certain constants used in the algorithms are proved to be optimal.