# Error estimation of floating-point summation and dot product

@article{Rump2012ErrorEO,
title={Error estimation of floating-point summation and dot product},
author={Siegfried M. Rump},
journal={BIT Numerical Mathematics},
year={2012},
volume={52},
pages={201-220}
}
• S. Rump
• Published 1 March 2012
• Mathematics
• BIT Numerical Mathematics
We improve the well-known Wilkinson-type estimates for the error of standard floating-point recursive summation and dot product by up to a factor 2. The bounds are valid when computed in rounding to nearest, no higher order terms are necessary, and they are best possible. For summation there is no restriction on the number of summands. The proofs are short by using a new tool for the estimation of errors in floating-point computations which cures drawbacks of the “unit in the last place (ulp…
39 Citations

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## References

SHOWING 1-10 OF 12 REFERENCES
Accurate Floating-Point Summation Part I: Faithful Rounding
• Mathematics, Computer Science
SIAM J. Sci. Comput.
• 2008
This paper presents an algorithm for calculating a faithful rounding of a vector of floating-point numbers, which adapts to the condition number of the sum, and proves certain constants used in the algorithm to be optimal.
Ultimately Fast Accurate Summation
• S. Rump
• Computer Science, Mathematics
SIAM J. Sci. Comput.
• 2009
Two new algorithms to compute a faithful rounding of the sum of floating-point numbers and the other for a result “as if” computed in $K$-fold precision, which are the fastest known in terms of flops.
Accuracy and stability of numerical algorithms
This book gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic by combining algorithmic derivations, perturbation theory, and rounding error analysis.
Handbook of Floating-Point Arithmetic
The Handbook of Floating-point Arithmetic is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.
Average-case stability of Gaussian elimination
• Mathematics
• 1990
Gaussian elimination with partial pivoting is unstable in the worst case: the “growth factor” can be as large as $2^{n - 1}$, where n is the matrix dimension, resulting in a loss of $n - 1$ bits of
On Floating Point Errors in Cholesky
• Mathematics
• 1989
Let H be a symmetric positive deenite matrix. Consider solving the linear system Hx= busing Cholesky, forward and back substitution in the standard way, yielding a computed solution ^ x. The usual
Handling floating-point exceptions in numeric programs
It is argued that the cheapest short-term solution would be to give full support to most of the required (as opposed to recommended) special features of the IEC/IEEE Standard for Binary Floating-Point Arithmetic.
Fast Inclusion of Interval Matrix Multiplication
• Mathematics, Computer Science
Reliab. Comput.
• 2005
Numerical results are presented to illustrate that the new algorithms to calculate an inclusion of the product of interval matrices using rounding mode controlled computation are much faster than the conventional algorithms and that the guaranteed accuracies obtained are comparable to those of the conventional algorithm.
The vector floating-point unit in a synergistic processor element of a CELL processor
• S. M. Müller, +9 authors S. Dhong
• Computer Science
17th IEEE Symposium on Computer Arithmetic (ARITH'05)
• 2005
The floating-point unit in the synergistic processor element of the 1st generation multi-core CELL processor is described, optimizing the performance critical single precision FMA operations, which are executed with a 6-cycle latency at an 11FO4 cycle time.