Proceedings of the 2009 conference on Symbolic numeric computation
Computing correctly rounded integer powers in floating-point arithmetic
ACM Transactions on Mathematical Software (TOMS)
Proceedings of the 24th ACM International Conference on Supercomputing
Algorithm 908: Online Exact Summation of Floating-Point Streams
ACM Transactions on Mathematical Software (TOMS)
Accurate evaluation of a polynomial and its derivative in Bernstein form
Computers & Mathematics with Applications
Accuracy versus time: a case study with summation algorithms
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
PPAM'09 Proceedings of the 8th international conference on Parallel processing and applied mathematics: Part II
Algorithm engineering: bridging the gap between algorithm theory and practice
Algorithm engineering: bridging the gap between algorithm theory and practice
Accurate Matrix Factorization: Inverse LU and Inverse QR Factorizations
SIAM Journal on Matrix Analysis and Applications
PerPI: a tool to measure instruction level parallelism
PARA'10 Proceedings of the 10th international conference on Applied Parallel and Scientific Computing - Volume Part I
A robust algorithm for geometric predicate by error-free determinant transformation
Information and Computation
Accurate evaluation algorithm for bivariate polynomial in Bernstein-Bézier form
Applied Numerical Mathematics
Verified Bounds for Least Squares Problems and Underdetermined Linear Systems
SIAM Journal on Matrix Analysis and Applications
Accurate solution of dense linear systems, part I: Algorithms in rounding to nearest
Journal of Computational and Applied Mathematics
Accurate evaluation of the k-th derivative of a polynomial and its application
Journal of Computational and Applied Mathematics
Self-Alignment Schemes for the Implementation of Addition-Related Floating-Point Operators
ACM Transactions on Reconfigurable Technology and Systems (TRETS)
Minimizing synchronizations in sparse iterative solvers for distributed supercomputers
Computers & Mathematics with Applications
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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 $s$ 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 instruction-level 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.