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Fast and accurate computation of the Euclidean norm of a vector
Citation Link: https://doi.org/10.15480/882.8023
Publikationstyp
Journal Article
Date Issued
2023-09
Sprache
English
TORE-DOI
Volume
40
Issue
3
Start Page
1391
End Page
1419
Citation
Japan Journal of Industrial and Applied Mathematics 40 (3): 1391-1419 (2023-09)
Publisher DOI
Scopus ID
Publisher
Springer
The numerical computation of the Euclidean norm of a vector is perfectly well conditioned with favorite a priori error estimates. Recently there is interest in computing a faithfully rounded approximation which means that there is no other floating-point number between the computed and the true real result. Hence the result is either the rounded to nearest result or its neighbor. Previous publications guarantee a faithfully rounded result for large dimension, but not the rounded to nearest result. In this note we present several new and fast algorithms producing a faithfully rounded result, as well as the first algorithm to compute the rounded to nearest result. Executable MATLAB codes are included. As a by product, a fast loop-free error-free vector transformation is given. That transforms a vector such that the sum remains unchanged but the condition number of the sum multiplies with the rounding error unit.
Subjects
Error-free transformation
Euclidean norm
Faithful rounding
Rounding error
DDC Class
510: Mathematics
Publication version
publishedVersion
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s13160-023-00593-8.pdf
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Format
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