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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Further analysis of Kahan's algorithm for the accurate computation of $ 2\times 2$ determinants


Authors: Claude-Pierre Jeannerod, Nicolas Louvet and Jean-Michel Muller
Journal: Math. Comp. 82 (2013), 2245-2264
MSC (2010): Primary 65G50
Published electronically: March 4, 2013
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Abstract: We provide a detailed analysis of Kahan's algorithm for the accurate computation of the determinant of a $ 2 \times 2$ matrix. This algorithm requires the availability of a fused multiply-add instruction. Assuming radix-$ \beta $, precision-$ p$ floating-point arithmetic with $ \beta $ even, $ p \geq 2$, and barring overflow or underflow we show that the absolute error of Kahan's algorithm is bounded by $ (\beta +1)/2$ ulps of the exact result and that the relative error is bounded by $ 2u$ with $ u=\frac {1}{2}\beta ^{1-p}$ the unit roundoff. Furthermore, we provide input values showing that i) when $ \beta /2$ is odd--which holds for $ 2$ and $ 10$, the two radices that matter in practice--the absolute error bound is optimal; ii) the relative error bound is asymptotically optimal, that is, for such input the ratio (relative error)/$ 2u$ has the form $ 1-O(\beta ^{-p})$. We also give relative error bounds parametrized by the relative order of magnitude of the two products in the determinant, and we investigate whether the error bounds can be improved when adding constraints: When the products in the determinant have opposite signs, which covers the computation of a sum of squares, or when Kahan's algorithm is used for computing the discriminant of a quadratic equation.


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Additional Information

Claude-Pierre Jeannerod
Affiliation: INRIA, Laboratoire LIP (CNRS, ENS de Lyon, INRIA, UCBL), Université de Lyon — 46, allée d’Italie, 69364 Lyon cedex 07, France
Email: claude-pierre.jeannerod@ens-lyon.fr

Nicolas Louvet
Affiliation: UCBL, Laboratoire LIP (CNRS, ENS de Lyon, INRIA, UCBL), Université de Lyon — 46, allée d’Italie, 69364 Lyon cedex 07, France
Email: nicolas.louvet@ens-lyon.fr

Jean-Michel Muller
Affiliation: CNRS, Laboratoire LIP (CNRS, ENS de Lyon, INRIA, UCBL), Université de Lyon — 46, allée d’Italie, 69364 Lyon cedex 07, France
Email: jean-michel.muller@ens-lyon.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02679-8
PII: S 0025-5718(2013)02679-8
Received by editor(s): December 7, 2011
Received by editor(s) in revised form: January 17, 2012
Published electronically: March 4, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.