Error estimates for the summation of real numbers with application to floating-point summation

Publikationstyp
Journal Article
Date Issued
2017-05-03
Sprache
English
Author(s)
Lange, Marko  
Rump, Siegfried M.  orcid-logo
Institut
Zuverlässiges Rechnen E-19  
TORE-URI
http://hdl.handle.net/11420/3835
Journal
BIT  
Volume
57
Issue
3
Start Page
927
End Page
941
Citation
BIT Numerical Mathematics 3 (57): 927-941 (2017-09-01)
Publisher DOI
10.1007/s10543-017-0658-9
Scopus ID
2-s2.0-85018980233
Publisher
Springer Science + Business Media B.V
Standard Wilkinson-type error estimates of floating-point algorithms involve a factor γk: = ku/ (1 - ku) for u denoting the relative rounding error unit of a floating-point number system. Recently, it was shown that, for many standard algorithms such as matrix multiplication, LU- or Cholesky decomposition, γkcan be replaced by ku, and the restriction on k can be removed. However, the arguments make heavy use of specific properties of both the underlying set of floating-point numbers and the corresponding arithmetic. In this paper, we derive error estimates for the summation of real numbers where each sum is afflicted with some perturbation. Recent results on floating-point summation follow as a corollary, in particular error estimates for rounding to nearest and for directed rounding. Our new estimates are sharp and unveil the necessary properties of floating-point schemes to allow for a priori estimates of summation with a factor omitting higher order terms.
Subjects
Error analysis
Floating-point
Real numbers
Summation
Wilkinson-type error estimates
DDC Class
510: Mathematik