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Accurate floating-point summation part I: Faithful rounding
Publikationstyp
Journal Article
Date Issued
2008
Sprache
English
Author(s)
Institut
TORE-URI
Volume
31
Issue
1
Start Page
189
End Page
224
Citation
SIAM Journal on Scientific Computing 1 (31): 189-224 (2008)
Publisher DOI
Scopus ID
Publisher
SIAM
Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s, i.e., the result is one of the immediate floating-point neighbors of s. If the sum a is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e., it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number. All statements are also true in the presence of underflow. The algorithm does not depend on the exponent range. Our algorithm is fast in terms of measured computing time because it allows good instructionlevel parallelism, it neither requires special operations such as access to mantissa or exponent, it contains no branch in the inner loop, nor does it 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 algorithm are proved to be optimal. © 2008 Society for Industrial and Applied Mathematics.
Subjects
Distillation
Error analysis
Error-free transformation
Extended and mixed precision basic linear algebra subprograms
Faithful founding
High accuracy
Maximally accurate summation
XBLAS
DDC Class
004: Informatik
510: Mathematik