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 Res of K non-overlapping floating-point numbers such that Res approximates r with relative accuracy epsK , and replacing ResK by its floating-point neighbors in Res 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 cal...

• Type: Journal
• Year: 2008
• Where: SIAMSC
• Authors: Siegfried M. Rump, Takeshi Ogita, Shin'ichi Oishi
• Keywords: Algorithms, Floating-point, Rounding, SIAMSC 2008